Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C8×A42 |
| Fitting, Sylow | C25×C8 ⋊ C32 |
| derived | C24 ⋊ C2×C32×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | A42 |
| commutator, hypocenter | C24 | C2×C32×C8 |
| Frattini | C4 | C22×A42 |
| Fitting | C25×C8 | C32 |
| socle | C26 | C32×C4 |
| prime radical | C22×A42 | C4 |
| cosocle | C24×C4 | C22×C32 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2×C8 | |
1481 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 1481 conjugacy classes of subgroups — too many to draw. Only its 121 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 — 2 generators on 18 points:
(3,6)(4,5)(7,10,9)(11,12,13,15,14,16,17,18);(1,2)(4,6,5)(8,9,10)256 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C8×A4⋊S4, A42×#97, #72253047, … |
| maximal normal subgroup of | C2×C8×A4⋊S4, A42×#97, #72253047, … |
| maximal quotient of | A42×M32 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6)(4,5)(7,10,9)(11,12,13,15,14,16,17,18), (1,2)(4,6,5)(8,9,10));