Finite Group DatabasebetaFinite Group Databasebeta| order | 10800 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22 |
| direct product | C3×A52 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C3 | A52 |
| commutator, hypocenter | A52 | C3 |
| Frattini | C1 | G |
| LCS | G ▷ A52 | |
| UCS | C1 ◁ C3 | |
302 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 302 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) = 13 |
| transitive degree | μt(G) = 75 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3)(4,5,8,6,7)(9,13,10,11,12);(5,8,6)(9,10,13,11,12)75 conjugacy classes — view the full character table.
| derived subgroup of | D3×A52, #72247654, D3×#72219548 |
| socle of | C3×#72219548, D3×A52, #72247654, … |
| maximal subgroup of | C3×#72219548, D3×A52, #72247654 |
| maximal normal subgroup of | C3×#72219548, D3×A52, #72247654 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,5,8,6,7)(9,13,10,11,12), (5,8,6)(9,10,13,11,12));