Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C211 · Schur multiplier (H2) = C254 |
| direct product | C2×H(11,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C210 |
| commutator, Frattini, cosocle | C2 | C211 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
8936255565 conjugacy classes of subgroups; only the 8933488747 normal subgroups are shown.
This group has 8936255565 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) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 11 generators on 66 points:
(4,10)(11,25)(12,26)(13,27)(14,28)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(51,60)(52,61)(53,62)(54,63)(64,66);(3,4)(5,11)(6,12)(7,13)(8,14)(9,10)(15,25)(16,29)(17,30)(18,31)(19,26)(20,32)(21,33)(22,27)(23,34)(24,28)(35,45)(36,46)(37,47)(38,51)(39,52)(40,53)(41,48)(42,49)(43,54)(44,50)(55,60)(56,61)(57,62)(58,64)(59,63)(65,66);(5,15)(11,25)(16,35)(17,36)(18,37)(29,45)(30,46)(31,47)(38,55)(39,56)(40,57)(51,60)(52,61)(53,62)(58,65)(64,66);(3,5)(4,11)(6,16)(7,17)(8,18)(9,15)(10,25)(12,29)(13,30)(14,31)(19,35)(20,38)(21,39)(22,36)(23,40)(24,37)(26,45)(27,46)(28,47)(32,51)(33,52)(34,53)(41,55)(42,56)(43,58)(44,57)(48,60)(49,61)(50,62)(54,64)(59,65)(63,66);(6,19)(12,26)(16,35)(20,41)(21,42)(29,45)(32,48)(33,49)(38,55)(39,56)(43,59)(51,60)(52,61)(54,63)(58,65)(64,66);(3,6)(4,12)(5,16)(7,20)(8,21)(9,19)(10,26)(11,29)(13,32)(14,33)(15,35)(17,38)(18,39)(22,41)(23,43)(24,42)(25,45)(27,48)(28,49)(30,51)(31,52)(34,54)(36,55)(37,56)(40,58)(44,59)(46,60)(47,61)(50,63)(53,64)(57,65)(62,66);(7,22)(13,27)(17,36)(20,41)(23,44)(30,46)(32,48)(34,50)(38,55)(40,57)(43,59)(51,60)(53,62)(54,63)(58,65)(64,66);(3,7)(4,13)(5,17)(6,20)(8,23)(9,22)(10,27)(11,30)(12,32)(14,34)(15,36)(16,38)(18,40)(19,41)(21,43)(24,44)(25,46)(26,48)(28,50)(29,51)(31,53)(33,54)(35,55)(37,57)(39,58)(42,59)(45,60)(47,62)(49,63)(52,64)(56,65)(61,66);(8,24)(14,28)(18,37)(21,42)(23,44)(31,47)(33,49)(34,50)(39,56)(40,57)(43,59)(52,61)(53,62)(54,63)(58,65)(64,66);(3,8)(4,14)(5,18)(6,21)(7,23)(9,24)(10,28)(11,31)(12,33)(13,34)(15,37)(16,39)(17,40)(19,42)(20,43)(22,44)(25,47)(26,49)(27,50)(29,52)(30,53)(32,54)(35,56)(36,57)(38,58)(41,59)(45,61)(46,62)(48,63)(51,64)(55,65)(60,66);(1,2)2050 conjugacy classes. The character table is too large to display.
| maximal subgroup of | H(13,𝔽2), C22×H(11,𝔽2), C2×#72274291 |
| maximal normal subgroup of | H(13,𝔽2), C22×H(11,𝔽2), C2×#72274291 |
| maximal quotient of | C22×H(11,𝔽2) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,10)(11,25)(12,26)(13,27)(14,28)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(51,60)(52,61)(53,62)(54,63)(64,66), (3,4)(5,11)(6,12)(7,13)(8,14)(9,10)(15,25)(16,29)(17,30)(18,31)(19,26)(20,32)(21,33)(22,27)(23,34)(24,28)(35,45)(36,46)(37,47)(38,51)(39,52)(40,53)(41,48)(42,49)(43,54)(44,50)(55,60)(56,61)(57,62)(58,64)(59,63)(65,66), (5,15)(11,25)(16,35)(17,36)(18,37)(29,45)(30,46)(31,47)(38,55)(39,56)(40,57)(51,60)(52,61)(53,62)(58,65)(64,66), (3,5)(4,11)(6,16)(7,17)(8,18)(9,15)(10,25)(12,29)(13,30)(14,31)(19,35)(20,38)(21,39)(22,36)(23,40)(24,37)(26,45)(27,46)(28,47)(32,51)(33,52)(34,53)(41,55)(42,56)(43,58)(44,57)(48,60)(49,61)(50,62)(54,64)(59,65)(63,66), (6,19)(12,26)(16,35)(20,41)(21,42)(29,45)(32,48)(33,49)(38,55)(39,56)(43,59)(51,60)(52,61)(54,63)(58,65)(64,66), (3,6)(4,12)(5,16)(7,20)(8,21)(9,19)(10,26)(11,29)(13,32)(14,33)(15,35)(17,38)(18,39)(22,41)(23,43)(24,42)(25,45)(27,48)(28,49)(30,51)(31,52)(34,54)(36,55)(37,56)(40,58)(44,59)(46,60)(47,61)(50,63)(53,64)(57,65)(62,66), (7,22)(13,27)(17,36)(20,41)(23,44)(30,46)(32,48)(34,50)(38,55)(40,57)(43,59)(51,60)(53,62)(54,63)(58,65)(64,66), (3,7)(4,13)(5,17)(6,20)(8,23)(9,22)(10,27)(11,30)(12,32)(14,34)(15,36)(16,38)(18,40)(19,41)(21,43)(24,44)(25,46)(26,48)(28,50)(29,51)(31,53)(33,54)(35,55)(37,57)(39,58)(42,59)(45,60)(47,62)(49,63)(52,64)(56,65)(61,66), (8,24)(14,28)(18,37)(21,42)(23,44)(31,47)(33,49)(34,50)(39,56)(40,57)(43,59)(52,61)(53,62)(54,63)(58,65)(64,66), (3,8)(4,14)(5,18)(6,21)(7,23)(9,24)(10,28)(11,31)(12,33)(13,34)(15,37)(16,39)(17,40)(19,42)(20,43)(22,44)(25,47)(26,49)(27,50)(29,52)(30,53)(32,54)(35,56)(36,57)(38,58)(41,59)(45,61)(46,62)(48,63)(51,64)(55,65)(60,66), (1,2));