Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C42⋊C3×#126⋊C3 |
| Fitting, Sylow | C42×#126 ⋊ C32 |
| derived | C44 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C44 | C2×C32 |
| Frattini, socle | C24 | C2×A42 |
| Fitting | C42×#126 | C32 |
| LCS | G ▷ C44 | |
| UCS | C1 | |
676 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,5)(3,8,12)(4,7,11)(6,10,9)(13,27,26,28,23,21)(14,16,20)(15,19,22,17,18,25);(1,3,4,9)(2,7)(5,12,11,10)(6,8)(14,21,23)(15,22,20)(16,26,25)(17,27,24)(18,19,28)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72258729, #72258804, #126⋊C32 |
| maximal normal subgroup of | #72258729, #72258804, #126⋊C32 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,5)(3,8,12)(4,7,11)(6,10,9)(13,27,26,28,23,21)(14,16,20)(15,19,22,17,18,25), (1,3,4,9)(2,7)(5,12,11,10)(6,8)(14,21,23)(15,22,20)(16,26,25)(17,27,24)(18,19,28));