Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C9×A4⋊S4 |
| Fitting | C24×C9 ⋊ C3⋊D3 |
| derived | A42 ⋊ C2×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | A4⋊S4 |
| commutator, hypocenter | A42 | C2×C9 |
| Frattini | C3 | C3×A4⋊S4 |
| Fitting | C24×C9 | C3⋊D3 |
| socle | C24×C3 | C3×C3⋊D3 |
| prime radical | C3×A4⋊S4 | C3 |
| cosocle | C3×A42 | C2×C3 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C9 | |
361 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 361 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,4,3,2)(5,8,6,7)(9,13,17,11,16,12,14,10,15);(6,7,8);(2,3,4)(6,7,8)126 conjugacy classes — view the full character table.
| maximal subgroup of | 3−1+2×A4⋊S4 |
| maximal normal subgroup of | 3−1+2×A4⋊S4 |
| maximal quotient of | #72259316 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3,2)(5,8,6,7)(9,13,17,11,16,12,14,10,15), (6,7,8), (2,3,4)(6,7,8));