Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×A4⋊(A4⋊C8) |
| derived | A42 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | A4⋊S4 |
| commutator, hypocenter | A42 | C2×C8 |
| Frattini | C4 | C2×A4⋊S4 |
| Fitting | C25×C4 | C3⋊D3 |
| socle | C26 | C3⋊Dic3 |
| prime radical | C22×A42 | C4 |
| cosocle | C4×A42 | C22 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C2×C4 | |
1197 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 1197 conjugacy classes of subgroups — too many to draw. Only its 91 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:
(1,2)(3,6)(4,5)(7,8,10);(4,6)(8,9)(11,12,13,15,14,16,17,18);(4,5,6)(8,9,10)112 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C8×A4⋊S4, #72253047 |
| maximal normal subgroup of | C2×C8×A4⋊S4, #72253047 |
| maximal quotient of | #72253105 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,6)(4,5)(7,8,10), (4,6)(8,9)(11,12,13,15,14,16,17,18), (4,5,6)(8,9,10));