#72209956

order	29160
isoclinism	isoclinic class of #72209956
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

Fitting

H(5,𝔽₃) ⋊ SL(2,𝔽₅)

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₃	#72209637
commutator, hypocenter	G	C₁
Fitting	H(5,𝔽₃)	SL(2,𝔽₅)
cosocle	#92404	A₅
LCS	G
UCS	C₁ ◁ C₃

151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

(2,12,31,7)(3,21,19,6)(4,32,48,9)(5,41,40,8)(13,70,28,45)(14,44,73,36)(15,52,25,58)(16,93,128,57)(17,69,27,102)(18,68,138,105)(20,54,30,59)(22,112,64,50)(23,55,84,51)(24,121,101,53)(26,108,107,104)(29,109,60,145)(33,150,178,117)(34,143,153,158)(35,79,56,78)(37,46,47,163)(38,148,62,171)(39,63,49,67)(42,127,157,98)(43,75,166,126)(61,170,66,65)(71,133,173,92)(72,129,110,100)(74,154,114,176)(76,195,116,205)(77,193,196,140)(80,185,118,168)(81,161,146,156)(82,130,120,85)(83,96,137,216)(86,136,131,191)(87,115,227,208)(88,186,243,198)(89,177,183,155)(90,167,182,184)(91,149,132,111)(94,224,235,210)(95,159,242,180)(97,192,124,141)(99,165,113,181)(103,194,125,218)(106,187,134,209)(119,139,190,202)(122,151,215,175)(123,230,204,200)(135,142,220,197)(144,189,172,164)(147,162,169,188)(152,179,213,228)(160,229,217,206)(174,241,238,203)(199,221,225,236)(201,212,240,237)(207,222,231,233)(211,232,226,219)(214,234,239,223);(2,13,14)(3,22,23)(4,33,34)(5,42,43)(15,85,86)(16,94,95)(17,76,82)(18,77,83)(19,64,84)(20,78,60)(24,122,123)(25,130,131)(26,115,119)(27,116,120)(28,73,31)(29,30,79)(35,148,162)(36,154,163)(37,155,70)(38,109,164)(39,156,165)(40,157,166)(44,176,46)(45,47,177)(48,178,153)(49,161,181)(50,61,168)(51,184,170)(55,167,65)(56,171,188)(62,145,189)(66,185,112)(68,192,186)(69,136,194)(74,199,200)(75,202,203)(80,210,211)(81,213,205)(87,91,220)(88,193,92)(89,151,221)(90,219,159)(96,217,158)(97,150,206)(98,174,106)(99,103,179)(100,214,207)(101,215,204)(102,191,218)(104,187,197)(105,141,198)(107,208,190)(108,209,142)(113,125,228)(114,225,230)(117,229,124)(118,224,226)(126,139,241)(127,238,134)(128,235,242)(129,239,231)(132,135,227)(133,243,140)(137,138,196)(143,216,160)(144,240,233)(146,152,195)(147,234,237)(169,223,212)(172,201,222)(175,236,183)(180,182,232);(1,2,3)(4,15,24)(5,16,25)(6,17,26)(7,18,27)(8,60,22)(9,64,23)(10,19,30)(11,20,31)(12,68,69)(13,73,41)(14,79,32)(21,102,108)(28,63,84)(29,67,78)(33,151,152)(34,159,160)(35,87,124)(36,88,125)(37,169,143)(38,172,146)(39,91,128)(40,92,129)(42,174,175)(43,179,180)(44,96,130)(45,97,131)(46,81,182)(47,75,183)(48,100,132)(49,101,133)(50,86,134)(51,103,135)(52,186,115)(53,187,119)(54,104,138)(55,85,139)(56,106,140)(57,190,116)(58,191,120)(59,107,105)(61,89,113)(62,150,114)(65,90,117)(66,158,118)(70,193,195)(71,197,136)(72,76,194)(74,127,163)(77,198,173)(80,126,148)(82,196,149)(83,209,110)(93,192,137)(94,225,188)(95,226,184)(98,144,170)(99,147,171)(109,220,216)(111,208,142)(112,205,227)(121,218,141)(122,155,199)(123,164,211)(145,202,243)(153,203,233)(154,212,242)(156,210,238)(157,223,229)(161,213,234)(162,201,239)(165,222,228)(166,200,241)(167,221,231)(168,219,235)(176,204,236)(177,214,232)(178,206,224)(181,217,230)(185,215,240)(189,207,237);(1,4,5)(2,15,16)(3,24,25)(6,50,51)(7,55,56)(8,61,62)(9,65,66)(10,39,48)(11,40,49)(12,70,44)(13,74,75)(14,80,81)(17,86,103)(18,85,106)(19,91,100)(20,92,101)(21,109,45)(22,113,114)(23,117,118)(26,134,135)(27,139,140)(28,143,144)(29,146,147)(30,128,132)(31,129,133)(32,148,46)(33,153,57)(34,161,53)(35,145,54)(36,112,59)(37,170,63)(38,171,67)(41,163,47)(42,93,165)(43,121,157)(52,167,177)(58,168,176)(60,89,150)(64,90,158)(68,193,96)(69,195,130)(71,154,188)(72,166,178)(73,127,183)(76,200,206)(77,199,207)(78,172,99)(79,126,182)(82,211,215)(83,210,217)(84,169,98)(87,202,104)(88,205,107)(94,197,212)(95,208,201)(97,102,220)(105,125,227)(108,216,131)(110,156,181)(111,162,184)(115,231,232)(116,152,233)(119,160,234)(120,235,236)(122,237,198)(123,240,196)(124,243,138)(136,242,225)(137,228,175)(141,229,180)(142,239,226)(149,164,185)(151,203,190)(155,189,173)(159,213,187)(174,192,222)(179,218,223)(186,221,214)(191,219,204)(194,241,224)(209,238,230);(1,6,7)(2,17,18)(3,26,27)(4,35,36)(5,44,45)(8,52,57)(9,53,58)(10,54,21)(11,12,59)(13,76,77)(14,82,83)(15,87,88)(16,96,97)(19,104,102)(20,68,107)(22,115,116)(23,119,120)(24,124,125)(25,130,131)(28,136,141)(29,137,142)(30,138,108)(31,69,105)(32,149,110)(33,148,154)(34,162,163)(37,167,156)(38,166,168)(39,70,55)(40,50,109)(41,72,173)(42,176,47)(43,46,177)(48,51,112)(49,145,56)(60,186,190)(61,164,157)(62,188,181)(63,71,121)(64,187,191)(65,165,155)(66,178,184)(67,93,111)(73,194,198)(74,159,201)(75,174,204)(78,192,208)(79,196,209)(80,212,151)(81,214,179)(84,197,218)(85,91,193)(86,220,92)(89,211,223)(90,222,199)(94,217,150)(95,158,206)(98,215,203)(99,213,207)(100,103,205)(101,202,106)(113,123,229)(114,225,230)(117,228,122)(118,224,226)(126,242,152)(127,160,239)(128,195,139)(129,134,216)(132,135,227)(133,243,140)(143,231,238)(144,240,233)(146,241,235)(147,234,237)(153,170,185)(161,189,171)(169,221,210)(172,200,219)(175,236,183)(180,182,232);(1,8,9)(2,14,13)(3,28,29)(4,37,38)(5,46,47)(6,52,53)(7,57,58)(10,32,67)(11,63,41)(12,71,72)(15,89,90)(16,98,99)(17,82,76)(18,83,77)(19,84,64)(20,60,78)(21,110,111)(22,73,30)(23,31,79)(24,126,127)(25,114,118)(26,136,137)(27,141,142)(33,155,109)(34,70,164)(35,167,166)(36,156,168)(39,61,163)(40,148,65)(42,44,177)(43,176,45)(48,170,66)(49,62,171)(50,154,165)(51,185,178)(54,149,93)(55,157,162)(56,181,189)(59,121,173)(68,186,192)(69,196,119)(74,92,169)(75,100,150)(80,172,91)(81,158,101)(85,151,219)(86,221,159)(87,211,222)(88,223,199)(94,174,103)(95,106,179)(96,215,213)(97,203,207)(102,218,191)(104,197,187)(105,209,120)(107,190,208)(108,116,198)(112,153,184)(113,146,129)(115,194,138)(117,128,143)(122,139,238)(123,241,134)(124,242,160)(125,152,239)(130,225,224)(131,230,226)(132,182,147)(133,144,183)(135,232,234)(140,233,236)(145,188,161)(175,243,240)(180,237,227)(193,212,200)(195,231,228)(201,220,210)(202,214,206)(204,205,217)(216,229,235);(1,10,11)(2,19,20)(3,30,31)(4,39,40)(5,48,49)(6,54,12)(7,21,59)(8,32,63)(9,67,41)(13,64,78)(14,84,60)(15,91,92)(16,100,101)(17,104,68)(18,102,107)(22,79,28)(23,29,73)(24,128,129)(25,132,133)(26,138,69)(27,108,105)(33,156,157)(34,165,166)(35,70,50)(36,55,109)(37,61,148)(38,163,65)(42,178,161)(43,153,181)(44,51,145)(45,112,56)(46,170,62)(47,66,171)(52,149,71)(53,93,72)(57,110,121)(58,111,173)(74,90,172)(75,158,99)(76,187,192)(77,191,208)(80,169,89)(81,98,150)(82,197,186)(83,218,190)(85,220,88)(86,87,193)(94,214,215)(95,207,204)(96,103,202)(97,205,106)(113,126,143)(114,182,144)(115,196,136)(116,209,141)(117,146,127)(118,147,183)(119,137,194)(120,142,198)(122,235,239)(123,242,231)(124,195,134)(125,139,216)(130,135,243)(131,227,140)(151,210,223)(152,238,229)(154,167,164)(155,168,162)(159,222,200)(160,228,241)(174,206,213)(175,224,234)(176,184,189)(177,185,188)(179,203,217)(180,233,230)(199,219,201)(211,212,221)(225,232,240)(226,237,236)

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((2,12,31,7)(3,21,19,6)(4,32,48,9)(5,41,40,8)(13,70,28,45)(14,44,73,36)(15,52,25,58)(16,93,128,57)(17,69,27,102)(18,68,138,105)(20,54,30,59)(22,112,64,50)(23,55,84,51)(24,121,101,53)(26,108,107,104)(29,109,60,145)(33,150,178,117)(34,143,153,158)(35,79,56,78)(37,46,47,163)(38,148,62,171)(39,63,49,67)(42,127,157,98)(43,75,166,126)(61,170,66,65)(71,133,173,92)(72,129,110,100)(74,154,114,176)(76,195,116,205)(77,193,196,140)(80,185,118,168)(81,161,146,156)(82,130,120,85)(83,96,137,216)(86,136,131,191)(87,115,227,208)(88,186,243,198)(89,177,183,155)(90,167,182,184)(91,149,132,111)(94,224,235,210)(95,159,242,180)(97,192,124,141)(99,165,113,181)(103,194,125,218)(106,187,134,209)(119,139,190,202)(122,151,215,175)(123,230,204,200)(135,142,220,197)(144,189,172,164)(147,162,169,188)(152,179,213,228)(160,229,217,206)(174,241,238,203)(199,221,225,236)(201,212,240,237)(207,222,231,233)(211,232,226,219)(214,234,239,223), (2,13,14)(3,22,23)(4,33,34)(5,42,43)(15,85,86)(16,94,95)(17,76,82)(18,77,83)(19,64,84)(20,78,60)(24,122,123)(25,130,131)(26,115,119)(27,116,120)(28,73,31)(29,30,79)(35,148,162)(36,154,163)(37,155,70)(38,109,164)(39,156,165)(40,157,166)(44,176,46)(45,47,177)(48,178,153)(49,161,181)(50,61,168)(51,184,170)(55,167,65)(56,171,188)(62,145,189)(66,185,112)(68,192,186)(69,136,194)(74,199,200)(75,202,203)(80,210,211)(81,213,205)(87,91,220)(88,193,92)(89,151,221)(90,219,159)(96,217,158)(97,150,206)(98,174,106)(99,103,179)(100,214,207)(101,215,204)(102,191,218)(104,187,197)(105,141,198)(107,208,190)(108,209,142)(113,125,228)(114,225,230)(117,229,124)(118,224,226)(126,139,241)(127,238,134)(128,235,242)(129,239,231)(132,135,227)(133,243,140)(137,138,196)(143,216,160)(144,240,233)(146,152,195)(147,234,237)(169,223,212)(172,201,222)(175,236,183)(180,182,232), (1,2,3)(4,15,24)(5,16,25)(6,17,26)(7,18,27)(8,60,22)(9,64,23)(10,19,30)(11,20,31)(12,68,69)(13,73,41)(14,79,32)(21,102,108)(28,63,84)(29,67,78)(33,151,152)(34,159,160)(35,87,124)(36,88,125)(37,169,143)(38,172,146)(39,91,128)(40,92,129)(42,174,175)(43,179,180)(44,96,130)(45,97,131)(46,81,182)(47,75,183)(48,100,132)(49,101,133)(50,86,134)(51,103,135)(52,186,115)(53,187,119)(54,104,138)(55,85,139)(56,106,140)(57,190,116)(58,191,120)(59,107,105)(61,89,113)(62,150,114)(65,90,117)(66,158,118)(70,193,195)(71,197,136)(72,76,194)(74,127,163)(77,198,173)(80,126,148)(82,196,149)(83,209,110)(93,192,137)(94,225,188)(95,226,184)(98,144,170)(99,147,171)(109,220,216)(111,208,142)(112,205,227)(121,218,141)(122,155,199)(123,164,211)(145,202,243)(153,203,233)(154,212,242)(156,210,238)(157,223,229)(161,213,234)(162,201,239)(165,222,228)(166,200,241)(167,221,231)(168,219,235)(176,204,236)(177,214,232)(178,206,224)(181,217,230)(185,215,240)(189,207,237), (1,4,5)(2,15,16)(3,24,25)(6,50,51)(7,55,56)(8,61,62)(9,65,66)(10,39,48)(11,40,49)(12,70,44)(13,74,75)(14,80,81)(17,86,103)(18,85,106)(19,91,100)(20,92,101)(21,109,45)(22,113,114)(23,117,118)(26,134,135)(27,139,140)(28,143,144)(29,146,147)(30,128,132)(31,129,133)(32,148,46)(33,153,57)(34,161,53)(35,145,54)(36,112,59)(37,170,63)(38,171,67)(41,163,47)(42,93,165)(43,121,157)(52,167,177)(58,168,176)(60,89,150)(64,90,158)(68,193,96)(69,195,130)(71,154,188)(72,166,178)(73,127,183)(76,200,206)(77,199,207)(78,172,99)(79,126,182)(82,211,215)(83,210,217)(84,169,98)(87,202,104)(88,205,107)(94,197,212)(95,208,201)(97,102,220)(105,125,227)(108,216,131)(110,156,181)(111,162,184)(115,231,232)(116,152,233)(119,160,234)(120,235,236)(122,237,198)(123,240,196)(124,243,138)(136,242,225)(137,228,175)(141,229,180)(142,239,226)(149,164,185)(151,203,190)(155,189,173)(159,213,187)(174,192,222)(179,218,223)(186,221,214)(191,219,204)(194,241,224)(209,238,230), (1,6,7)(2,17,18)(3,26,27)(4,35,36)(5,44,45)(8,52,57)(9,53,58)(10,54,21)(11,12,59)(13,76,77)(14,82,83)(15,87,88)(16,96,97)(19,104,102)(20,68,107)(22,115,116)(23,119,120)(24,124,125)(25,130,131)(28,136,141)(29,137,142)(30,138,108)(31,69,105)(32,149,110)(33,148,154)(34,162,163)(37,167,156)(38,166,168)(39,70,55)(40,50,109)(41,72,173)(42,176,47)(43,46,177)(48,51,112)(49,145,56)(60,186,190)(61,164,157)(62,188,181)(63,71,121)(64,187,191)(65,165,155)(66,178,184)(67,93,111)(73,194,198)(74,159,201)(75,174,204)(78,192,208)(79,196,209)(80,212,151)(81,214,179)(84,197,218)(85,91,193)(86,220,92)(89,211,223)(90,222,199)(94,217,150)(95,158,206)(98,215,203)(99,213,207)(100,103,205)(101,202,106)(113,123,229)(114,225,230)(117,228,122)(118,224,226)(126,242,152)(127,160,239)(128,195,139)(129,134,216)(132,135,227)(133,243,140)(143,231,238)(144,240,233)(146,241,235)(147,234,237)(153,170,185)(161,189,171)(169,221,210)(172,200,219)(175,236,183)(180,182,232), (1,8,9)(2,14,13)(3,28,29)(4,37,38)(5,46,47)(6,52,53)(7,57,58)(10,32,67)(11,63,41)(12,71,72)(15,89,90)(16,98,99)(17,82,76)(18,83,77)(19,84,64)(20,60,78)(21,110,111)(22,73,30)(23,31,79)(24,126,127)(25,114,118)(26,136,137)(27,141,142)(33,155,109)(34,70,164)(35,167,166)(36,156,168)(39,61,163)(40,148,65)(42,44,177)(43,176,45)(48,170,66)(49,62,171)(50,154,165)(51,185,178)(54,149,93)(55,157,162)(56,181,189)(59,121,173)(68,186,192)(69,196,119)(74,92,169)(75,100,150)(80,172,91)(81,158,101)(85,151,219)(86,221,159)(87,211,222)(88,223,199)(94,174,103)(95,106,179)(96,215,213)(97,203,207)(102,218,191)(104,197,187)(105,209,120)(107,190,208)(108,116,198)(112,153,184)(113,146,129)(115,194,138)(117,128,143)(122,139,238)(123,241,134)(124,242,160)(125,152,239)(130,225,224)(131,230,226)(132,182,147)(133,144,183)(135,232,234)(140,233,236)(145,188,161)(175,243,240)(180,237,227)(193,212,200)(195,231,228)(201,220,210)(202,214,206)(204,205,217)(216,229,235), (1,10,11)(2,19,20)(3,30,31)(4,39,40)(5,48,49)(6,54,12)(7,21,59)(8,32,63)(9,67,41)(13,64,78)(14,84,60)(15,91,92)(16,100,101)(17,104,68)(18,102,107)(22,79,28)(23,29,73)(24,128,129)(25,132,133)(26,138,69)(27,108,105)(33,156,157)(34,165,166)(35,70,50)(36,55,109)(37,61,148)(38,163,65)(42,178,161)(43,153,181)(44,51,145)(45,112,56)(46,170,62)(47,66,171)(52,149,71)(53,93,72)(57,110,121)(58,111,173)(74,90,172)(75,158,99)(76,187,192)(77,191,208)(80,169,89)(81,98,150)(82,197,186)(83,218,190)(85,220,88)(86,87,193)(94,214,215)(95,207,204)(96,103,202)(97,205,106)(113,126,143)(114,182,144)(115,196,136)(116,209,141)(117,146,127)(118,147,183)(119,137,194)(120,142,198)(122,235,239)(123,242,231)(124,195,134)(125,139,216)(130,135,243)(131,227,140)(151,210,223)(152,238,229)(154,167,164)(155,168,162)(159,222,200)(160,228,241)(174,206,213)(175,224,234)(176,184,189)(177,185,188)(179,203,217)(180,233,230)(199,219,201)(211,212,221)(225,232,240)(226,237,236));