Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | SL(2,𝔽5)⋊A4 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | S4×A5 |
| commutator, hypocenter | SL(2,𝔽5)⋊A4 | C2 |
| Fitting | Q8 | D3×A5 |
| cosocle | SL(2,𝔽3) | C2×A5 |
| LCS | G ▷ SL(2,𝔽5)⋊A4 | |
| UCS | C1 ◁ C2 | |
177 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 177 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(2,12,30,48,15)(3,4,25,49,14)(9,17,46,77,28)(10,16,45,78,27)(11,24,55,79,32)(13,18,47,80,29)(22,34,76,103,54)(23,33,75,104,53)(26,73,118,91,44)(31,69,113,92,43)(39,59,101,131,85)(40,60,102,132,86)(41,57,99,133,83)(42,58,100,134,84)(50,94,140,115,71)(51,93,139,116,70)(52,98,147,117,74)(56,95,141,120,72)(65,88,128,157,110)(66,87,127,158,109)(67,90,130,159,112)(68,89,129,160,111)(81,122,162,145,97)(82,121,161,146,96)(105,151,175,165,125)(106,152,176,166,126)(107,149,173,167,123)(108,150,174,168,124)(135,170,186,181,154)(136,169,185,182,153)(137,172,188,183,156)(138,171,187,184,155);(1,4,2)(3,8,12)(5,16,9)(6,17,10)(7,18,11)(13,19,24)(14,25,26)(15,30,31)(20,33,22)(21,34,23)(27,45,50)(28,46,51)(29,47,52)(32,55,56)(35,57,39)(36,58,40)(37,59,41)(38,60,42)(43,49,69)(44,48,73)(53,75,81)(54,76,82)(61,87,65)(62,88,66)(63,89,67)(64,90,68)(70,78,93)(71,77,94)(72,80,95)(74,79,98)(83,99,105)(84,100,106)(85,101,107)(86,102,108)(91,113,114)(92,118,119)(96,104,121)(97,103,122)(109,127,135)(110,128,136)(111,129,137)(112,130,138)(115,139,142)(116,140,143)(117,141,144)(120,147,148)(123,133,149)(124,134,150)(125,131,151)(126,132,152)(145,161,163)(146,162,164)(153,158,169)(154,157,170)(155,160,171)(156,159,172)(165,173,177)(166,174,178)(167,175,179)(168,176,180)(181,185,189)(182,186,190)(183,187,191)(184,188,192);(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);(1,5,8,6)(2,9,3,10)(4,16,12,17)(7,64,19,63)(11,68,13,67)(14,27,15,28)(18,90,24,89)(20,37,21,35)(22,41,23,39)(25,45,30,46)(26,50,31,51)(29,112,32,111)(33,59,34,57)(36,62,38,61)(40,66,42,65)(43,70,44,71)(47,130,55,129)(48,77,49,78)(52,138,56,137)(53,85,54,83)(58,88,60,87)(69,93,73,94)(72,156,74,155)(75,101,76,99)(79,160,80,159)(81,107,82,105)(84,110,86,109)(91,115,92,116)(95,172,98,171)(96,125,97,123)(100,128,102,127)(103,133,104,131)(106,136,108,135)(113,139,118,140)(114,142,119,143)(117,184,120,183)(121,151,122,149)(124,154,126,153)(132,158,134,157)(141,188,147,187)(144,192,148,191)(145,167,146,165)(150,170,152,169)(161,175,162,173)(163,179,164,177)(166,182,168,181)(174,186,176,185)(178,190,180,189);(1,20,8,21)(2,22,3,23)(4,33,12,34)(5,35,6,37)(7,36,19,38)(9,39,10,41)(11,40,13,42)(14,53,15,54)(16,57,17,59)(18,58,24,60)(25,75,30,76)(26,81,31,82)(27,83,28,85)(29,84,32,86)(43,96,44,97)(45,99,46,101)(47,100,55,102)(48,103,49,104)(50,105,51,107)(52,106,56,108)(61,63,62,64)(65,67,66,68)(69,121,73,122)(70,123,71,125)(72,124,74,126)(77,131,78,133)(79,132,80,134)(87,89,88,90)(91,145,92,146)(93,149,94,151)(95,150,98,152)(109,111,110,112)(113,161,118,162)(114,163,119,164)(115,165,116,167)(117,166,120,168)(127,129,128,130)(135,137,136,138)(139,173,140,175)(141,174,147,176)(142,177,143,179)(144,178,148,180)(153,155,154,156)(157,159,158,160)(169,171,170,172)(181,183,182,184)(185,187,186,188)(189,191,190,192);(1,21,6)(2,23,10)(3,22,9)(4,34,17)(5,8,20)(7,64,62)(11,68,66)(12,33,16)(13,67,65)(14,54,28)(15,53,27)(18,90,88)(19,63,61)(24,89,87)(25,76,46)(26,82,51)(29,112,110)(30,75,45)(31,81,50)(32,111,109)(43,97,71)(44,96,70)(47,130,128)(48,104,78)(49,103,77)(52,138,136)(55,129,127)(56,137,135)(69,122,94)(72,156,154)(73,121,93)(74,155,153)(79,160,158)(80,159,157)(91,146,116)(92,145,115)(95,172,170)(98,171,169)(113,162,140)(114,164,143)(117,184,182)(118,161,139)(119,163,142)(120,183,181)(141,188,186)(144,192,190)(147,187,185)(148,191,189);(1,7,35,61,8,19,37,62)(2,11,39,65,3,13,41,66)(4,18,57,87,12,24,59,88)(5,38,21,63,6,36,20,64)(9,42,23,67,10,40,22,68)(14,29,83,109,15,32,85,110)(16,60,34,89,17,58,33,90)(25,47,99,127,30,55,101,128)(26,52,105,135,31,56,107,136)(27,86,54,111,28,84,53,112)(43,72,123,153,44,74,125,154)(45,102,76,129,46,100,75,130)(48,79,131,157,49,80,133,158)(50,108,82,137,51,106,81,138)(69,95,149,169,73,98,151,170)(70,126,97,155,71,124,96,156)(77,134,104,159,78,132,103,160)(91,117,165,181,92,120,167,182)(93,152,122,171,94,150,121,172)(113,141,173,185,118,147,175,186)(114,144,177,189,119,148,179,190)(115,168,146,183,116,166,145,184)(139,176,162,187,140,174,161,188)(142,180,164,191,143,178,163,192)37 conjugacy classes — view the full character table.
| maximal subgroup of | #72209753, #72228181, #72276932, … |
| maximal normal subgroup of | #72276932, #72277256 |
| maximal quotient of | #226×SL(2,𝔽5), #72276859 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((2,12,30,48,15)(3,4,25,49,14)(9,17,46,77,28)(10,16,45,78,27)(11,24,55,79,32)(13,18,47,80,29)(22,34,76,103,54)(23,33,75,104,53)(26,73,118,91,44)(31,69,113,92,43)(39,59,101,131,85)(40,60,102,132,86)(41,57,99,133,83)(42,58,100,134,84)(50,94,140,115,71)(51,93,139,116,70)(52,98,147,117,74)(56,95,141,120,72)(65,88,128,157,110)(66,87,127,158,109)(67,90,130,159,112)(68,89,129,160,111)(81,122,162,145,97)(82,121,161,146,96)(105,151,175,165,125)(106,152,176,166,126)(107,149,173,167,123)(108,150,174,168,124)(135,170,186,181,154)(136,169,185,182,153)(137,172,188,183,156)(138,171,187,184,155), (1,4,2)(3,8,12)(5,16,9)(6,17,10)(7,18,11)(13,19,24)(14,25,26)(15,30,31)(20,33,22)(21,34,23)(27,45,50)(28,46,51)(29,47,52)(32,55,56)(35,57,39)(36,58,40)(37,59,41)(38,60,42)(43,49,69)(44,48,73)(53,75,81)(54,76,82)(61,87,65)(62,88,66)(63,89,67)(64,90,68)(70,78,93)(71,77,94)(72,80,95)(74,79,98)(83,99,105)(84,100,106)(85,101,107)(86,102,108)(91,113,114)(92,118,119)(96,104,121)(97,103,122)(109,127,135)(110,128,136)(111,129,137)(112,130,138)(115,139,142)(116,140,143)(117,141,144)(120,147,148)(123,133,149)(124,134,150)(125,131,151)(126,132,152)(145,161,163)(146,162,164)(153,158,169)(154,157,170)(155,160,171)(156,159,172)(165,173,177)(166,174,178)(167,175,179)(168,176,180)(181,185,189)(182,186,190)(183,187,191)(184,188,192), (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), (1,5,8,6)(2,9,3,10)(4,16,12,17)(7,64,19,63)(11,68,13,67)(14,27,15,28)(18,90,24,89)(20,37,21,35)(22,41,23,39)(25,45,30,46)(26,50,31,51)(29,112,32,111)(33,59,34,57)(36,62,38,61)(40,66,42,65)(43,70,44,71)(47,130,55,129)(48,77,49,78)(52,138,56,137)(53,85,54,83)(58,88,60,87)(69,93,73,94)(72,156,74,155)(75,101,76,99)(79,160,80,159)(81,107,82,105)(84,110,86,109)(91,115,92,116)(95,172,98,171)(96,125,97,123)(100,128,102,127)(103,133,104,131)(106,136,108,135)(113,139,118,140)(114,142,119,143)(117,184,120,183)(121,151,122,149)(124,154,126,153)(132,158,134,157)(141,188,147,187)(144,192,148,191)(145,167,146,165)(150,170,152,169)(161,175,162,173)(163,179,164,177)(166,182,168,181)(174,186,176,185)(178,190,180,189), (1,20,8,21)(2,22,3,23)(4,33,12,34)(5,35,6,37)(7,36,19,38)(9,39,10,41)(11,40,13,42)(14,53,15,54)(16,57,17,59)(18,58,24,60)(25,75,30,76)(26,81,31,82)(27,83,28,85)(29,84,32,86)(43,96,44,97)(45,99,46,101)(47,100,55,102)(48,103,49,104)(50,105,51,107)(52,106,56,108)(61,63,62,64)(65,67,66,68)(69,121,73,122)(70,123,71,125)(72,124,74,126)(77,131,78,133)(79,132,80,134)(87,89,88,90)(91,145,92,146)(93,149,94,151)(95,150,98,152)(109,111,110,112)(113,161,118,162)(114,163,119,164)(115,165,116,167)(117,166,120,168)(127,129,128,130)(135,137,136,138)(139,173,140,175)(141,174,147,176)(142,177,143,179)(144,178,148,180)(153,155,154,156)(157,159,158,160)(169,171,170,172)(181,183,182,184)(185,187,186,188)(189,191,190,192), (1,21,6)(2,23,10)(3,22,9)(4,34,17)(5,8,20)(7,64,62)(11,68,66)(12,33,16)(13,67,65)(14,54,28)(15,53,27)(18,90,88)(19,63,61)(24,89,87)(25,76,46)(26,82,51)(29,112,110)(30,75,45)(31,81,50)(32,111,109)(43,97,71)(44,96,70)(47,130,128)(48,104,78)(49,103,77)(52,138,136)(55,129,127)(56,137,135)(69,122,94)(72,156,154)(73,121,93)(74,155,153)(79,160,158)(80,159,157)(91,146,116)(92,145,115)(95,172,170)(98,171,169)(113,162,140)(114,164,143)(117,184,182)(118,161,139)(119,163,142)(120,183,181)(141,188,186)(144,192,190)(147,187,185)(148,191,189), (1,7,35,61,8,19,37,62)(2,11,39,65,3,13,41,66)(4,18,57,87,12,24,59,88)(5,38,21,63,6,36,20,64)(9,42,23,67,10,40,22,68)(14,29,83,109,15,32,85,110)(16,60,34,89,17,58,33,90)(25,47,99,127,30,55,101,128)(26,52,105,135,31,56,107,136)(27,86,54,111,28,84,53,112)(43,72,123,153,44,74,125,154)(45,102,76,129,46,100,75,130)(48,79,131,157,49,80,133,158)(50,108,82,137,51,106,81,138)(69,95,149,169,73,98,151,170)(70,126,97,155,71,124,96,156)(77,134,104,159,78,132,103,160)(91,117,165,181,92,120,167,182)(93,152,122,171,94,150,121,172)(113,141,173,185,118,147,175,186)(114,144,177,189,119,148,179,190)(115,168,146,183,116,166,145,184)(139,176,162,187,140,174,161,188)(142,180,164,191,143,178,163,192));