Finite Group DatabasebetaFinite Group Databasebeta| order | 7680 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
| direct product | C22×#11757163⋊C2 |
| Fitting, socle | C26 ⋊ S5 |
| derived | #11757163 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #11757163⋊C2 |
| commutator, cosocle, hypocenter | #11757163 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C26 | S5 |
| LCS | G ▷ #11757163 | |
| UCS | C1 ◁ C22 | |
6382 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 6382 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 20 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2)(3,10,4,6,5,8)(7,11);(3,11,12,8,5,6,7,9,4,10);(1,2)(3,11,5,8)(4,6,7,10)72 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72218858, C2×D3×#11757163⋊C2 |
| maximal quotient of | C2×D3×#11757163⋊C2 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,10,4,6,5,8)(7,11), (3,11,12,8,5,6,7,9,4,10), (1,2)(3,11,5,8)(4,6,7,10));