Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C22×#10596770 |
| Fitting, socle | C26 ⋊ #157 |
| derived | A42 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #10596770 |
| commutator, hypocenter | A42 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C26 | #157 |
| prime radical | C22×A4⋊S4 | C2 |
| cosocle | A4⋊S4 | C23 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C22 | |
1126 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 1126 conjugacy classes of subgroups — too many to draw. Only its 37 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 — 3 generators on 12 points:
(5,10,7,9)(6,12)(8,11);(3,4)(7,12,8)(9,10,11);(1,2)52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72218849, C22×#10596770⋊C2, C2×D3×#10596770 |
| maximal normal subgroup of | C22×#10596770⋊C2 |
| maximal quotient of | C2×D3×#10596770 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,10,7,9)(6,12)(8,11), (3,4)(7,12,8)(9,10,11), (1,2));