Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25×C3 |
| direct product | C2×C8×A4⋊S4 |
| Fitting | C25×C8 ⋊ C3⋊D3 |
| derived | A42 ⋊ C22×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | A4⋊S4 |
| commutator, hypocenter | A42 | C22×C8 |
| Frattini | C4 | C22×A4⋊S4 |
| Fitting | C25×C8 | C3⋊D3 |
| socle | C26 | C4×C3⋊D3 |
| prime radical | C22×A4⋊S4 | C4 |
| cosocle | C4×A42 | C23 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C2×C8 | |
4536 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 4536 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 18 points:
(3,6)(4,5)(7,8,10)(11,12,13,15,14,16,17,18);(1,2)(4,5,6)(8,9,10);(4,6)(8,9)224 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6)(4,5)(7,8,10)(11,12,13,15,14,16,17,18), (1,2)(4,5,6)(8,9,10), (4,6)(8,9));