Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C4×#10596770 |
| Fitting | C24×C4 ⋊ #157 |
| derived | A42 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #10596770 |
| commutator, hypocenter | A42 | C42 |
| Frattini | C2 | C2×#10596770 |
| Fitting | C24×C4 | #157 |
| socle | C25 | C2×#157 |
| prime radical, cosocle | C2×A4⋊S4 | C22 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C4 | |
437 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 21 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 — 2 generators on 12 points:
(1,2,3,4)(6,9,11)(7,12,8);(5,6)(7,9)(8,11,12,10)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#10596770⋊C2 |
| maximal normal subgroup of | C4×#10596770⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4)(6,9,11)(7,12,8), (5,6)(7,9)(8,11,12,10));