Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| direct product | D4×A6 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2 | C22×A6 |
| commutator, socle | C2×A6 | C22 |
| Fitting, hypercenter | D4 | A6 |
| hypocenter | A6 | D4 |
| LCS | G ▷ C2×A6 ▷ A6 | |
| UCS | C1 ◁ C2 ◁ D4 | |
352 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,3)(2,4)(5,9,8)(6,7,10);(3,4)(5,8)(6,7,9,10)35 conjugacy classes — view the full character table.
| maximal subgroup of | S4×A6 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3)(2,4)(5,9,8)(6,7,10), (3,4)(5,8)(6,7,9,10));