Finite Group DatabasebetaFinite Group Databasebeta| order | 14400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×#72219548 |
| socle | C2×A52 ⋊ C2 |
| derived | A52 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2 | #72219548 |
| commutator, cosocle, hypocenter | A52 | C22 |
| Frattini | C1 | G |
| socle | C2×A52 | C2 |
| LCS | G ▷ A52 | |
| UCS | C1 ◁ C2 | |
895 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 895 conjugacy classes of subgroups — too many to draw. Only its 11 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) = 50 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2)(3,5,4)(8,11,12,10,9);(1,2)(3,4)(5,6,7)(8,10,9,12)52 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72219548, C22×#72219548 |
| maximal normal subgroup of | C22×#72219548 |
| maximal quotient of | D3×#72219548, C2×SO+(4,𝔽5), C22×#72219548 |
| holomorph of | C2×A5 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,5,4)(8,11,12,10,9), (1,2)(3,4)(5,6,7)(8,10,9,12));