Finite Group DatabasebetaFinite Group Databasebeta| order | 23040 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C2×D3×#11757163⋊C2 |
| Fitting, socle | C25×C3 ⋊ C2×S5 |
| derived | C3×#11757163 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×#11757163⋊C2 |
| commutator, cosocle, hypocenter | C3×#11757163 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C25×C3 | C2×S5 |
| LCS | G ▷ C3×#11757163 | |
| UCS | C1 ◁ C2 | |
At least 6298 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has at least 6298 conjugacy classes of subgroups — too many to draw. Only its 35 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) = 30 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,4,7,3,6,10,5)(8,9)(11,13,12);(1,10,3,4)(2,5,8)(6,7,9)(11,12);(1,3)(2,6)(4,10)(5,7)(8,9)108 conjugacy classes — view the full character table.
| automorphism group of | #5968 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,7,3,6,10,5)(8,9)(11,13,12), (1,10,3,4)(2,5,8)(6,7,9)(11,12), (1,3)(2,6)(4,10)(5,7)(8,9));