Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
| direct product | C22×S42 |
| Fitting, socle | C26 ⋊ D32 |
| derived | A42 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | S42 |
| commutator, cosocle, hypocenter | A42 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C26 | D32 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C22 | |
9563 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 9563 conjugacy classes of subgroups — too many to draw. Only its 151 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) = 36 |
A minimal faithful permutation representation — 4 generators on 12 points:
(3,4)(5,6,8)(9,12,10);(1,2);(6,7)(11,12);(11,12)100 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72215293, C22×#10596770⋊C2, C2×D3×S42, … |
| maximal normal subgroup of | C2×#72215293, C22×#10596770⋊C2, D4×S42, … |
| maximal quotient of | C2×#72215280, #72228062, #72228310, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(5,6,8)(9,12,10), (1,2), (6,7)(11,12), (11,12));