Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C22×#10596770⋊C2 |
| Fitting, socle | C26 ⋊ GO+(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #10596770⋊C2 |
| commutator, cosocle | A4⋊S4 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C26 | GO+(4,𝔽2) |
| hypocenter | A42 | C22×D4 |
| LCS | G ▷ A4⋊S4 ▷ A42 | |
| UCS | C1 ◁ C22 | |
7087 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 7087 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 4 generators on 12 points:
(5,10,12,9,8,6)(7,11);(10,11);(3,4);(1,2)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72218858, C2×D3×#10596770⋊C2 |
| maximal quotient of | C2×D3×#10596770⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,10,12,9,8,6)(7,11), (10,11), (3,4), (1,2));