Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C4 |
| direct product | C2×#72210818 |
| Fitting, Sylow | C2×#147497 ⋊ C3 |
| derived | #13834 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #10935⋊C3 |
| commutator, cosocle, hypocenter | #13834 | C22×C3 |
| Frattini | C23×Q8 | C22×A4 |
| Fitting | C2×#147497 | C3 |
| socle | C24 | B(4,𝔽2)⋊C3 |
| hypercenter | C23 | #1437⋊C3 |
| LCS | G ▷ #13834 | |
| UCS | C1 ◁ C22 ◁ C23 | |
4207 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 4207 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 192 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,5,4,6,10)(3,8,11,9,7,12)(15,23,26)(16,25,24)(18,27,19)(20,21,28);(1,9,4,3)(2,7,6,8)(5,10)(11,12)(13,21,16,22,17,27,23,14)(15,19,25,18,24,28,26,20)72 conjugacy classes — view the full character table.
| maximal subgroup of | #72267024, #72267061, #72267520, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,5,4,6,10)(3,8,11,9,7,12)(15,23,26)(16,25,24)(18,27,19)(20,21,28), (1,9,4,3)(2,7,6,8)(5,10)(11,12)(13,21,16,22,17,27,23,14)(15,19,25,18,24,28,26,20));