Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C92 · Schur multiplier (H2) = C22×C9 |
| direct product | C92×#102 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C92 | D4 |
| commutator | C4 | C2×C4×C92 |
| Frattini, cosocle | C2×C32×C4 | C22×C32 |
| socle | C22×C32 | C32×D4 |
| prime radical | C2×C32×D4 | C2×C32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C92 ◁ C2×C4×C92 ◁ G | |
575 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 575 conjugacy classes of subgroups — too many to draw. Only its 322 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1296 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,4,3)(6,9)(7,10)(11,12)(13,14,15,16,17,18,19,20,21);(1,3,4,2)(5,6,10,12,8,11,7,9)(22,23,24,25,26,27,28,29,30)1134 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | #102×C9⋊D9, #72254568 |
| maximal subgroup of | C92×#442, #102×C9⋊D9, #72254568 |
| maximal normal subgroup of | C92×#442, #102×C9⋊D9, #72254568 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,3)(6,9)(7,10)(11,12)(13,14,15,16,17,18,19,20,21), (1,3,4,2)(5,6,10,12,8,11,7,9)(22,23,24,25,26,27,28,29,30));