Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C32 |
| direct product | C22×C3⋊D3×#157 |
| Fitting, socle | C22×C34 ⋊ C2×C4 |
| derived, Sylow | C34 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊D3×#157 |
| commutator, hypocenter | C34 | C23×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | C2×C4 |
| prime radical | C22×C3⋊D32 | C2 |
| cosocle | C32×C3⋊D3 | C24 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C22 | |
2924 conjugacy classes of subgroups; only the 294 normal subgroups are shown.
This group has 2924 conjugacy classes of subgroups — too many to draw. Only its 294 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 16 points:
(4,5,6)(7,8)(9,11,12,10);(1,2,3)(7,9,12)(8,11,10)(15,16);(2,3)(5,6);(13,14)144 conjugacy classes — view the full character table.
| maximal subgroup of | C22×C3⋊D3×PSU(3,𝔽2), C22×#1572, C22×(#157⋊D3)⋊D3, … |
| maximal normal subgroup of | C22×C3⋊D3×PSU(3,𝔽2), C22×#1572, C22×(#157⋊D3)⋊D3, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5,6)(7,8)(9,11,12,10), (1,2,3)(7,9,12)(8,11,10)(15,16), (2,3)(5,6), (13,14));