Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×C8×#120⋊C3 |
| Fitting, Sylow | C2×C8×#120 ⋊ C3 |
| derived | C24 ⋊ C22×C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | #120⋊C3 |
| commutator, hypocenter | C24 | C22×C3×C8 |
| Frattini | C22×C4 | C23×A4 |
| Fitting | C2×C8×#120 | C3 |
| socle | C24 | C2×C4×A4 |
| prime radical | C22×#120⋊C3 | C4 |
| cosocle | C24×C4 | C23×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2×C8 | |
2330 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 2330 conjugacy classes of subgroups — too many to draw. Only its 125 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) = 96 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(5,6,9)(7,10,8);(4,10)(7,8)(11,12,13,15,14,16,17,18);(3,4)(5,8)(6,7)(9,10)160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C8×#1020⋊C2, C2×#72342670, C2×#72342675 |
| maximal normal subgroup of | C2×C8×#1020⋊C2, C2×#72342670, C2×#72342675 |
| maximal quotient of | #72342027 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(5,6,9)(7,10,8), (4,10)(7,8)(11,12,13,15,14,16,17,18), (3,4)(5,8)(6,7)(9,10));