Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#31077⋊C3 |
| Fitting, Sylow | C2×#31077 ⋊ C3 |
| derived | #31077 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #5464 |
| commutator, cosocle, hypocenter | #31077 | C2×C3 |
| Frattini | C22×C4 | C2×C22⋊A4 |
| Fitting | C2×#31077 | C3 |
| socle | C22 | #2679⋊C3 |
| LCS | G ▷ #31077 | |
| UCS | C1 ◁ C2×C4 | |
1311 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 1311 conjugacy classes of subgroups — too many to draw. Only its 27 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 — 3 generators on 66 points:
(1,2)(4,11,21)(5,15,36)(6,22,7)(12,37,53)(13,33,62)(14,16,19)(17,48,64)(18,41,66)(20,31,35)(23,58,27)(24,54,42)(25,56,29)(26,57,30)(28,32,55)(34,51,39)(38,60,43)(40,65,52)(44,46,61)(45,63,49)(47,50,59);(3,22)(4,39)(5,49)(6,7)(8,56)(9,55)(10,57)(11,37)(12,13)(14,59)(15,48)(16,58)(17,18)(19,61)(20,60)(21,62)(23,44)(24,52)(25,29)(26,30)(27,50)(28,32)(31,54)(33,34)(35,65)(36,66)(38,42)(40,43)(41,45)(46,47)(51,53)(63,64);(3,35,9,36,10,16,8,11)(4,21,14,19,15,5,31,20)(6,46,25,63,26,38,32,33)(7,34,29,47,30,64,28,42)(12,17,40,50,41,51,23,24)(13,52,43,18,45,27,44,53)(22,37,55,65,57,66,56,58)(39,60,59,62,48,61,54,49)44 conjugacy classes — view the full character table.
| maximal subgroup of | #72267405, #72267958 |
| maximal quotient of | #72267382 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,11,21)(5,15,36)(6,22,7)(12,37,53)(13,33,62)(14,16,19)(17,48,64)(18,41,66)(20,31,35)(23,58,27)(24,54,42)(25,56,29)(26,57,30)(28,32,55)(34,51,39)(38,60,43)(40,65,52)(44,46,61)(45,63,49)(47,50,59), (3,22)(4,39)(5,49)(6,7)(8,56)(9,55)(10,57)(11,37)(12,13)(14,59)(15,48)(16,58)(17,18)(19,61)(20,60)(21,62)(23,44)(24,52)(25,29)(26,30)(27,50)(28,32)(31,54)(33,34)(35,65)(36,66)(38,42)(40,43)(41,45)(46,47)(51,53)(63,64), (3,35,9,36,10,16,8,11)(4,21,14,19,15,5,31,20)(6,46,25,63,26,38,32,33)(7,34,29,47,30,64,28,42)(12,17,40,50,41,51,23,24)(13,52,43,18,45,27,44,53)(22,37,55,65,57,66,56,58)(39,60,59,62,48,61,54,49));