Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C211 · Schur multiplier (H2) = C254 |
| direct product | C2×2−1+10 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C210 |
| commutator, Frattini, cosocle | C2 | C211 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
8934820297 conjugacy classes of subgroups; only the 8933488747 normal subgroups are shown.
This group has 8934820297 conjugacy classes of subgroups — too many to draw. Only its 8933488747 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 130 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 11 generators on 130 points:
(3,4,10,16)(5,17,22,11)(6,12,23,37)(7,13,27,41)(8,14,30,44)(9,15,32,46)(18,47,51,33)(19,48,55,34)(20,49,58,35)(21,50,60,36)(24,38,61,77)(25,39,62,78)(26,40,63,79)(28,42,67,83)(29,43,68,84)(31,45,70,86)(52,87,93,71)(53,88,94,72)(54,89,95,73)(56,90,99,74)(57,91,100,75)(59,92,102,76)(64,80,103,112)(65,81,104,113)(66,82,105,114)(69,85,107,116)(96,117,121,108)(97,118,122,109)(98,119,123,110)(101,120,125,111)(106,115,126,128)(124,129,130,127);(3,5,10,22)(4,11,16,17)(6,18,23,51)(7,19,27,55)(8,20,30,58)(9,21,32,60)(12,33,37,47)(13,34,41,48)(14,35,44,49)(15,36,46,50)(24,52,61,93)(25,53,62,94)(26,54,63,95)(28,56,67,99)(29,57,68,100)(31,59,70,102)(38,71,77,87)(39,72,78,88)(40,73,79,89)(42,74,83,90)(43,75,84,91)(45,76,86,92)(64,96,103,121)(65,97,104,122)(66,98,105,123)(69,101,107,125)(80,108,112,117)(81,109,113,118)(82,110,114,119)(85,111,116,120)(106,124,126,130)(115,127,128,129);(6,23)(12,37)(18,51)(24,61)(25,62)(26,63)(33,47)(38,77)(39,78)(40,79)(52,93)(53,94)(54,95)(64,103)(65,104)(66,105)(71,87)(72,88)(73,89)(80,112)(81,113)(82,114)(96,121)(97,122)(98,123)(106,126)(108,117)(109,118)(110,119)(115,128)(124,130)(127,129);(3,6)(4,12)(5,18)(7,24)(8,25)(9,26)(10,23)(11,33)(13,38)(14,39)(15,40)(16,37)(17,47)(19,52)(20,53)(21,54)(22,51)(27,61)(28,64)(29,65)(30,62)(31,66)(32,63)(34,71)(35,72)(36,73)(41,77)(42,80)(43,81)(44,78)(45,82)(46,79)(48,87)(49,88)(50,89)(55,93)(56,96)(57,97)(58,94)(59,98)(60,95)(67,103)(68,104)(69,106)(70,105)(74,108)(75,109)(76,110)(83,112)(84,113)(85,115)(86,114)(90,117)(91,118)(92,119)(99,121)(100,122)(101,124)(102,123)(107,126)(111,127)(116,128)(120,129)(125,130);(7,27)(13,41)(19,55)(24,61)(28,67)(29,68)(34,48)(38,77)(42,83)(43,84)(52,93)(56,99)(57,100)(64,103)(65,104)(69,107)(71,87)(74,90)(75,91)(80,112)(81,113)(85,116)(96,121)(97,122)(101,125)(106,126)(108,117)(109,118)(111,120)(115,128)(124,130)(127,129);(3,7)(4,13)(5,19)(6,24)(8,28)(9,29)(10,27)(11,34)(12,38)(14,42)(15,43)(16,41)(17,48)(18,52)(20,56)(21,57)(22,55)(23,61)(25,64)(26,65)(30,67)(31,69)(32,68)(33,71)(35,74)(36,75)(37,77)(39,80)(40,81)(44,83)(45,85)(46,84)(47,87)(49,90)(50,91)(51,93)(53,96)(54,97)(58,99)(59,101)(60,100)(62,103)(63,104)(66,106)(70,107)(72,108)(73,109)(76,111)(78,112)(79,113)(82,115)(86,116)(88,117)(89,118)(92,120)(94,121)(95,122)(98,124)(102,125)(105,126)(110,127)(114,128)(119,129)(123,130);(8,30)(14,44)(20,58)(25,62)(28,67)(31,70)(35,49)(39,78)(42,83)(45,86)(53,94)(56,99)(59,102)(64,103)(66,105)(69,107)(72,88)(74,90)(76,92)(80,112)(82,114)(85,116)(96,121)(98,123)(101,125)(106,126)(108,117)(110,119)(111,120)(115,128)(124,130)(127,129);(3,8)(4,14)(5,20)(6,25)(7,28)(9,31)(10,30)(11,35)(12,39)(13,42)(15,45)(16,44)(17,49)(18,53)(19,56)(21,59)(22,58)(23,62)(24,64)(26,66)(27,67)(29,69)(32,70)(33,72)(34,74)(36,76)(37,78)(38,80)(40,82)(41,83)(43,85)(46,86)(47,88)(48,90)(50,92)(51,94)(52,96)(54,98)(55,99)(57,101)(60,102)(61,103)(63,105)(65,106)(68,107)(71,108)(73,110)(75,111)(77,112)(79,114)(81,115)(84,116)(87,117)(89,119)(91,120)(93,121)(95,123)(97,124)(100,125)(104,126)(109,127)(113,128)(118,129)(122,130);(9,32)(15,46)(21,60)(26,63)(29,68)(31,70)(36,50)(40,79)(43,84)(45,86)(54,95)(57,100)(59,102)(65,104)(66,105)(69,107)(73,89)(75,91)(76,92)(81,113)(82,114)(85,116)(97,122)(98,123)(101,125)(106,126)(109,118)(110,119)(111,120)(115,128)(124,130)(127,129);(3,9)(4,15)(5,21)(6,26)(7,29)(8,31)(10,32)(11,36)(12,40)(13,43)(14,45)(16,46)(17,50)(18,54)(19,57)(20,59)(22,60)(23,63)(24,65)(25,66)(27,68)(28,69)(30,70)(33,73)(34,75)(35,76)(37,79)(38,81)(39,82)(41,84)(42,85)(44,86)(47,89)(48,91)(49,92)(51,95)(52,97)(53,98)(55,100)(56,101)(58,102)(61,104)(62,105)(64,106)(67,107)(71,109)(72,110)(74,111)(77,113)(78,114)(80,115)(83,116)(87,118)(88,119)(90,120)(93,122)(94,123)(96,124)(99,125)(103,126)(108,127)(112,128)(117,129)(121,130);(1,2)2050 conjugacy classes. The character table is too large to display.
| maximal subgroup of | 2−1+12, C2×#72274291, C22×2−1+10 |
| maximal normal subgroup of | 2−1+12, C2×#72274291, C22×2−1+10 |
| maximal quotient of | C22×2−1+10 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,10,16)(5,17,22,11)(6,12,23,37)(7,13,27,41)(8,14,30,44)(9,15,32,46)(18,47,51,33)(19,48,55,34)(20,49,58,35)(21,50,60,36)(24,38,61,77)(25,39,62,78)(26,40,63,79)(28,42,67,83)(29,43,68,84)(31,45,70,86)(52,87,93,71)(53,88,94,72)(54,89,95,73)(56,90,99,74)(57,91,100,75)(59,92,102,76)(64,80,103,112)(65,81,104,113)(66,82,105,114)(69,85,107,116)(96,117,121,108)(97,118,122,109)(98,119,123,110)(101,120,125,111)(106,115,126,128)(124,129,130,127), (3,5,10,22)(4,11,16,17)(6,18,23,51)(7,19,27,55)(8,20,30,58)(9,21,32,60)(12,33,37,47)(13,34,41,48)(14,35,44,49)(15,36,46,50)(24,52,61,93)(25,53,62,94)(26,54,63,95)(28,56,67,99)(29,57,68,100)(31,59,70,102)(38,71,77,87)(39,72,78,88)(40,73,79,89)(42,74,83,90)(43,75,84,91)(45,76,86,92)(64,96,103,121)(65,97,104,122)(66,98,105,123)(69,101,107,125)(80,108,112,117)(81,109,113,118)(82,110,114,119)(85,111,116,120)(106,124,126,130)(115,127,128,129), (6,23)(12,37)(18,51)(24,61)(25,62)(26,63)(33,47)(38,77)(39,78)(40,79)(52,93)(53,94)(54,95)(64,103)(65,104)(66,105)(71,87)(72,88)(73,89)(80,112)(81,113)(82,114)(96,121)(97,122)(98,123)(106,126)(108,117)(109,118)(110,119)(115,128)(124,130)(127,129), (3,6)(4,12)(5,18)(7,24)(8,25)(9,26)(10,23)(11,33)(13,38)(14,39)(15,40)(16,37)(17,47)(19,52)(20,53)(21,54)(22,51)(27,61)(28,64)(29,65)(30,62)(31,66)(32,63)(34,71)(35,72)(36,73)(41,77)(42,80)(43,81)(44,78)(45,82)(46,79)(48,87)(49,88)(50,89)(55,93)(56,96)(57,97)(58,94)(59,98)(60,95)(67,103)(68,104)(69,106)(70,105)(74,108)(75,109)(76,110)(83,112)(84,113)(85,115)(86,114)(90,117)(91,118)(92,119)(99,121)(100,122)(101,124)(102,123)(107,126)(111,127)(116,128)(120,129)(125,130), (7,27)(13,41)(19,55)(24,61)(28,67)(29,68)(34,48)(38,77)(42,83)(43,84)(52,93)(56,99)(57,100)(64,103)(65,104)(69,107)(71,87)(74,90)(75,91)(80,112)(81,113)(85,116)(96,121)(97,122)(101,125)(106,126)(108,117)(109,118)(111,120)(115,128)(124,130)(127,129), (3,7)(4,13)(5,19)(6,24)(8,28)(9,29)(10,27)(11,34)(12,38)(14,42)(15,43)(16,41)(17,48)(18,52)(20,56)(21,57)(22,55)(23,61)(25,64)(26,65)(30,67)(31,69)(32,68)(33,71)(35,74)(36,75)(37,77)(39,80)(40,81)(44,83)(45,85)(46,84)(47,87)(49,90)(50,91)(51,93)(53,96)(54,97)(58,99)(59,101)(60,100)(62,103)(63,104)(66,106)(70,107)(72,108)(73,109)(76,111)(78,112)(79,113)(82,115)(86,116)(88,117)(89,118)(92,120)(94,121)(95,122)(98,124)(102,125)(105,126)(110,127)(114,128)(119,129)(123,130), (8,30)(14,44)(20,58)(25,62)(28,67)(31,70)(35,49)(39,78)(42,83)(45,86)(53,94)(56,99)(59,102)(64,103)(66,105)(69,107)(72,88)(74,90)(76,92)(80,112)(82,114)(85,116)(96,121)(98,123)(101,125)(106,126)(108,117)(110,119)(111,120)(115,128)(124,130)(127,129), (3,8)(4,14)(5,20)(6,25)(7,28)(9,31)(10,30)(11,35)(12,39)(13,42)(15,45)(16,44)(17,49)(18,53)(19,56)(21,59)(22,58)(23,62)(24,64)(26,66)(27,67)(29,69)(32,70)(33,72)(34,74)(36,76)(37,78)(38,80)(40,82)(41,83)(43,85)(46,86)(47,88)(48,90)(50,92)(51,94)(52,96)(54,98)(55,99)(57,101)(60,102)(61,103)(63,105)(65,106)(68,107)(71,108)(73,110)(75,111)(77,112)(79,114)(81,115)(84,116)(87,117)(89,119)(91,120)(93,121)(95,123)(97,124)(100,125)(104,126)(109,127)(113,128)(118,129)(122,130), (9,32)(15,46)(21,60)(26,63)(29,68)(31,70)(36,50)(40,79)(43,84)(45,86)(54,95)(57,100)(59,102)(65,104)(66,105)(69,107)(73,89)(75,91)(76,92)(81,113)(82,114)(85,116)(97,122)(98,123)(101,125)(106,126)(109,118)(110,119)(111,120)(115,128)(124,130)(127,129), (3,9)(4,15)(5,21)(6,26)(7,29)(8,31)(10,32)(11,36)(12,40)(13,43)(14,45)(16,46)(17,50)(18,54)(19,57)(20,59)(22,60)(23,63)(24,65)(25,66)(27,68)(28,69)(30,70)(33,73)(34,75)(35,76)(37,79)(38,81)(39,82)(41,84)(42,85)(44,86)(47,89)(48,91)(49,92)(51,95)(52,97)(53,98)(55,100)(56,101)(58,102)(61,104)(62,105)(64,106)(67,107)(71,109)(72,110)(74,111)(77,113)(78,114)(80,115)(83,116)(87,118)(88,119)(90,120)(93,122)(94,123)(96,124)(99,125)(103,126)(108,127)(112,128)(117,129)(121,130), (1,2));