Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C92 · Schur multiplier (H2) = C23×C9 |
| direct product | C2×C92×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C92 | D4 |
| commutator | C4 | C23×C92 |
| Frattini, cosocle | C32×C4 | C23×C32 |
| socle | C22×C32 | C32×D4 |
| prime radical | C2×C32×D8 | C32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C92 ◁ C2×C4×C92 ◁ G | |
874 conjugacy classes of subgroups; only the 506 normal subgroups are shown.
This group has 874 conjugacy classes of subgroups — too many to draw. Only its 506 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) = 1296 |
A minimal faithful permutation representation — 3 generators on 28 points:
(4,7)(5,8)(9,10)(11,12,13,14,15,16,17,18,19);(3,4,8,10,6,9,5,7)(20,21,22,23,24,25,26,27,28);(1,2)1134 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | C2×D8×C9⋊D9, #72244280, C2×#72244273 |
| maximal subgroup of | C2×D8×C9⋊D9, C92×#442, #72244280, … |
| maximal normal subgroup of | C2×D8×C9⋊D9, C92×#442, #72244280, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,7)(5,8)(9,10)(11,12,13,14,15,16,17,18,19), (3,4,8,10,6,9,5,7)(20,21,22,23,24,25,26,27,28), (1,2));