Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×C8×#1020 |
| Fitting | C25×C8 ⋊ D3 |
| derived | C22⋊A4 ⋊ C22×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | #1020 |
| commutator, hypocenter | C22⋊A4 | C22×C8 |
| Frattini | C4 | C22×#1020 |
| Fitting | C25×C8 | D3 |
| socle | C26 | C4×D3 |
| prime radical | C22×#1020 | C4 |
| cosocle | C4×C22⋊A4 | C23 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2×C8 | |
3961 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 3961 conjugacy classes of subgroups — too many to draw. Only its 93 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) = 48 |
A minimal faithful permutation representation — 3 generators on 18 points:
(3,6,5)(8,9,10)(11,12,13,15,14,16,17,18);(1,2)(7,8)(9,10);(4,6)(8,9)160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C8×A4⋊S4, #97×#1020, C22×C8×#1020, … |
| maximal normal subgroup of | #97×#1020, C22×C8×#1020, #72342683, … |
| maximal quotient of | C22×C8×#1020, C2×#72319766, #72341738 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,6,5)(8,9,10)(11,12,13,15,14,16,17,18), (1,2)(7,8)(9,10), (4,6)(8,9));