Finite Group DatabasebetaFinite Group Databasebeta| order | 7200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
| direct product | C2×A52 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2 | A52 |
| commutator, hypocenter | A52 | C2 |
| Frattini | C1 | G |
| LCS | G ▷ A52 | |
| UCS | C1 ◁ C2 | |
353 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 8 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)(3,7,4,5,6)(8,10,11,12,9);(3,5,7)(8,9,12,10,11)50 conjugacy classes — view the full character table.
| socle of | C2×#72219548, C2×A5×S5 |
| maximal subgroup of | C2×#72219548, D3×A52, C22×A52, … |
| maximal normal subgroup of | C2×#72219548, C22×A52, C2×A5×S5 |
| maximal quotient of | D3×A52, C2×Ω+(4,𝔽5), C22×A52, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,7,4,5,6)(8,10,11,12,9), (3,5,7)(8,9,12,10,11));