#72209751

order	4032
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

derived

#13062015 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	A₄×GL(3,𝔽₂)
commutator, hypocenter	#13062015	C₃
Fitting, cosocle	Q₈	C₃×GL(3,𝔽₂)
LCS	G ▷ #13062015
UCS	C₁ ◁ C₂

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

(1,5,8,6)(2,9,3,10)(4,16,12,17)(7,20,19,21)(11,22,13,23)(14,27,15,28)(18,33,24,34)(25,45,30,46)(26,50,31,51)(29,53,32,54)(35,61,37,62)(36,63,38,64)(39,65,41,66)(40,67,42,68)(43,70,44,71)(47,75,55,76)(48,77,49,78)(52,81,56,82)(57,87,59,88)(58,89,60,90)(69,93,73,94)(72,96,74,97)(79,103,80,104)(83,109,85,110)(84,111,86,112)(91,115,92,116)(95,121,98,122)(99,127,101,128)(100,129,102,130)(105,135,107,136)(106,137,108,138)(113,139,118,140)(114,142,119,143)(117,145,120,146)(123,153,125,154)(124,155,126,156)(131,157,133,158)(132,159,134,160)(141,161,147,162)(144,163,148,164)(149,169,151,170)(150,171,152,172)(165,181,167,182)(166,183,168,184)(173,185,175,186)(174,187,176,188)(177,189,179,190)(178,191,180,192);(1,7,5)(2,11,9)(3,13,10)(4,18,16)(6,8,19)(12,24,17)(14,29,27)(15,32,28)(20,35,36)(21,37,38)(22,39,40)(23,41,42)(25,47,45)(26,52,50)(30,55,46)(31,56,51)(33,57,58)(34,59,60)(43,72,70)(44,74,71)(48,79,77)(49,80,78)(53,83,84)(54,85,86)(69,95,93)(73,98,94)(75,99,100)(76,101,102)(81,105,106)(82,107,108)(91,117,115)(92,120,116)(96,123,124)(97,125,126)(103,131,132)(104,133,134)(113,141,139)(114,144,142)(118,147,140)(119,148,143)(121,149,150)(122,151,152)(145,165,166)(146,167,168)(161,173,174)(162,175,176)(163,177,178)(164,179,180);(1,8)(2,3)(4,12)(5,6)(7,19)(9,10)(11,13)(14,15)(16,17)(18,24)(20,21)(22,23)(25,30)(26,31)(27,28)(29,32)(33,34)(35,37)(36,38)(39,41)(40,42)(43,44)(45,46)(47,55)(48,49)(50,51)(52,56)(53,54)(57,59)(58,60)(61,62)(63,64)(65,66)(67,68)(69,73)(70,71)(72,74)(75,76)(77,78)(79,80)(81,82)(83,85)(84,86)(87,88)(89,90)(91,92)(93,94)(95,98)(96,97)(99,101)(100,102)(103,104)(105,107)(106,108)(109,110)(111,112)(113,118)(114,119)(115,116)(117,120)(121,122)(123,125)(124,126)(127,128)(129,130)(131,133)(132,134)(135,136)(137,138)(139,140)(141,147)(142,143)(144,148)(145,146)(149,151)(150,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,167)(166,168)(169,170)(171,172)(173,175)(174,176)(177,179)(178,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192);(1,4,8,12)(2,30,3,25)(5,16,6,17)(7,18,19,24)(9,46,10,45)(11,55,13,47)(14,69,15,73)(20,33,21,34)(22,76,23,75)(26,48,31,49)(27,93,28,94)(29,95,32,98)(35,57,37,59)(36,58,38,60)(39,101,41,99)(40,102,42,100)(43,118,44,113)(50,77,51,78)(52,79,56,80)(53,121,54,122)(61,87,62,88)(63,89,64,90)(65,128,66,127)(67,130,68,129)(70,140,71,139)(72,147,74,141)(81,103,82,104)(83,149,85,151)(84,150,86,152)(91,119,92,114)(96,162,97,161)(105,131,107,133)(106,132,108,134)(109,169,110,170)(111,171,112,172)(115,143,116,142)(117,148,120,144)(123,175,125,173)(124,176,126,174)(135,157,136,158)(137,159,138,160)(145,164,146,163)(153,186,154,185)(155,188,156,187)(165,179,167,177)(166,180,168,178)(181,190,182,189)(183,192,184,191);(1,91,8,92)(2,113,3,118)(4,114,12,119)(5,115,6,116)(7,117,19,120)(9,139,10,140)(11,141,13,147)(14,49,15,48)(16,142,17,143)(18,144,24,148)(20,145,21,146)(22,161,23,162)(25,43,30,44)(26,69,31,73)(27,78,28,77)(29,80,32,79)(33,163,34,164)(35,165,37,167)(36,166,38,168)(39,173,41,175)(40,174,42,176)(45,70,46,71)(47,72,55,74)(50,93,51,94)(52,95,56,98)(53,104,54,103)(57,177,59,179)(58,178,60,180)(61,181,62,182)(63,183,64,184)(65,185,66,186)(67,187,68,188)(75,96,76,97)(81,121,82,122)(83,133,85,131)(84,134,86,132)(87,189,88,190)(89,191,90,192)(99,123,101,125)(100,124,102,126)(105,149,107,151)(106,150,108,152)(109,158,110,157)(111,160,112,159)(127,153,128,154)(129,155,130,156)(135,169,136,170)(137,171,138,172);(1,43,73)(2,31,12)(3,26,4)(5,70,94)(6,71,93)(7,72,98)(8,44,69)(9,51,17)(10,50,16)(11,56,24)(13,52,18)(14,119,30)(15,114,25)(19,74,95)(20,96,122)(21,97,121)(22,82,34)(23,81,33)(27,143,46)(28,142,45)(29,148,55)(32,144,47)(35,123,151)(36,124,152)(37,125,149)(38,126,150)(39,107,59)(40,108,60)(41,105,57)(42,106,58)(48,91,118)(49,92,113)(53,164,76)(54,163,75)(61,153,170)(62,154,169)(63,155,172)(64,156,171)(65,136,88)(66,135,87)(67,138,90)(68,137,89)(77,115,140)(78,116,139)(79,117,147)(80,120,141)(83,179,101)(84,180,102)(85,177,99)(86,178,100)(103,145,162)(104,146,161)(109,190,128)(110,189,127)(111,192,130)(112,191,129)(131,165,175)(132,166,176)(133,167,173)(134,168,174)(157,181,186)(158,182,185)(159,183,188)(160,184,187)

Conjugacy classes & character table

39 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209753, #72221415
maximal normal subgroup of	#72221415
maximal quotient of	SL(2,𝔽₃)×SL(2,𝔽₇)

GAP

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

gap> G := Group((1,5,8,6)(2,9,3,10)(4,16,12,17)(7,20,19,21)(11,22,13,23)(14,27,15,28)(18,33,24,34)(25,45,30,46)(26,50,31,51)(29,53,32,54)(35,61,37,62)(36,63,38,64)(39,65,41,66)(40,67,42,68)(43,70,44,71)(47,75,55,76)(48,77,49,78)(52,81,56,82)(57,87,59,88)(58,89,60,90)(69,93,73,94)(72,96,74,97)(79,103,80,104)(83,109,85,110)(84,111,86,112)(91,115,92,116)(95,121,98,122)(99,127,101,128)(100,129,102,130)(105,135,107,136)(106,137,108,138)(113,139,118,140)(114,142,119,143)(117,145,120,146)(123,153,125,154)(124,155,126,156)(131,157,133,158)(132,159,134,160)(141,161,147,162)(144,163,148,164)(149,169,151,170)(150,171,152,172)(165,181,167,182)(166,183,168,184)(173,185,175,186)(174,187,176,188)(177,189,179,190)(178,191,180,192), (1,7,5)(2,11,9)(3,13,10)(4,18,16)(6,8,19)(12,24,17)(14,29,27)(15,32,28)(20,35,36)(21,37,38)(22,39,40)(23,41,42)(25,47,45)(26,52,50)(30,55,46)(31,56,51)(33,57,58)(34,59,60)(43,72,70)(44,74,71)(48,79,77)(49,80,78)(53,83,84)(54,85,86)(69,95,93)(73,98,94)(75,99,100)(76,101,102)(81,105,106)(82,107,108)(91,117,115)(92,120,116)(96,123,124)(97,125,126)(103,131,132)(104,133,134)(113,141,139)(114,144,142)(118,147,140)(119,148,143)(121,149,150)(122,151,152)(145,165,166)(146,167,168)(161,173,174)(162,175,176)(163,177,178)(164,179,180), (1,8)(2,3)(4,12)(5,6)(7,19)(9,10)(11,13)(14,15)(16,17)(18,24)(20,21)(22,23)(25,30)(26,31)(27,28)(29,32)(33,34)(35,37)(36,38)(39,41)(40,42)(43,44)(45,46)(47,55)(48,49)(50,51)(52,56)(53,54)(57,59)(58,60)(61,62)(63,64)(65,66)(67,68)(69,73)(70,71)(72,74)(75,76)(77,78)(79,80)(81,82)(83,85)(84,86)(87,88)(89,90)(91,92)(93,94)(95,98)(96,97)(99,101)(100,102)(103,104)(105,107)(106,108)(109,110)(111,112)(113,118)(114,119)(115,116)(117,120)(121,122)(123,125)(124,126)(127,128)(129,130)(131,133)(132,134)(135,136)(137,138)(139,140)(141,147)(142,143)(144,148)(145,146)(149,151)(150,152)(153,154)(155,156)(157,158)(159,160)(161,162)(163,164)(165,167)(166,168)(169,170)(171,172)(173,175)(174,176)(177,179)(178,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192), (1,4,8,12)(2,30,3,25)(5,16,6,17)(7,18,19,24)(9,46,10,45)(11,55,13,47)(14,69,15,73)(20,33,21,34)(22,76,23,75)(26,48,31,49)(27,93,28,94)(29,95,32,98)(35,57,37,59)(36,58,38,60)(39,101,41,99)(40,102,42,100)(43,118,44,113)(50,77,51,78)(52,79,56,80)(53,121,54,122)(61,87,62,88)(63,89,64,90)(65,128,66,127)(67,130,68,129)(70,140,71,139)(72,147,74,141)(81,103,82,104)(83,149,85,151)(84,150,86,152)(91,119,92,114)(96,162,97,161)(105,131,107,133)(106,132,108,134)(109,169,110,170)(111,171,112,172)(115,143,116,142)(117,148,120,144)(123,175,125,173)(124,176,126,174)(135,157,136,158)(137,159,138,160)(145,164,146,163)(153,186,154,185)(155,188,156,187)(165,179,167,177)(166,180,168,178)(181,190,182,189)(183,192,184,191), (1,91,8,92)(2,113,3,118)(4,114,12,119)(5,115,6,116)(7,117,19,120)(9,139,10,140)(11,141,13,147)(14,49,15,48)(16,142,17,143)(18,144,24,148)(20,145,21,146)(22,161,23,162)(25,43,30,44)(26,69,31,73)(27,78,28,77)(29,80,32,79)(33,163,34,164)(35,165,37,167)(36,166,38,168)(39,173,41,175)(40,174,42,176)(45,70,46,71)(47,72,55,74)(50,93,51,94)(52,95,56,98)(53,104,54,103)(57,177,59,179)(58,178,60,180)(61,181,62,182)(63,183,64,184)(65,185,66,186)(67,187,68,188)(75,96,76,97)(81,121,82,122)(83,133,85,131)(84,134,86,132)(87,189,88,190)(89,191,90,192)(99,123,101,125)(100,124,102,126)(105,149,107,151)(106,150,108,152)(109,158,110,157)(111,160,112,159)(127,153,128,154)(129,155,130,156)(135,169,136,170)(137,171,138,172), (1,43,73)(2,31,12)(3,26,4)(5,70,94)(6,71,93)(7,72,98)(8,44,69)(9,51,17)(10,50,16)(11,56,24)(13,52,18)(14,119,30)(15,114,25)(19,74,95)(20,96,122)(21,97,121)(22,82,34)(23,81,33)(27,143,46)(28,142,45)(29,148,55)(32,144,47)(35,123,151)(36,124,152)(37,125,149)(38,126,150)(39,107,59)(40,108,60)(41,105,57)(42,106,58)(48,91,118)(49,92,113)(53,164,76)(54,163,75)(61,153,170)(62,154,169)(63,155,172)(64,156,171)(65,136,88)(66,135,87)(67,138,90)(68,137,89)(77,115,140)(78,116,139)(79,117,147)(80,120,141)(83,179,101)(84,180,102)(85,177,99)(86,178,100)(103,145,162)(104,146,161)(109,190,128)(110,189,127)(111,192,130)(112,191,129)(131,165,175)(132,166,176)(133,167,173)(134,168,174)(157,181,186)(158,182,185)(159,183,188)(160,184,187));