Finite Group DatabasebetaFinite Group Databasebeta| order | 6912 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#72215293 |
| Fitting | C26×C3 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×S42 |
| commutator, cosocle | C2×A42 | C23×C3 |
| Frattini | C2 | C2×C3×S42 |
| Fitting | C26×C3 | D32 |
| socle | C25×C3 | C2×D32 |
| hypercenter | C22×C3 | S42 |
| hypocenter | A42 | C2×C3×D4 |
| LCS | G ▷ C2×A42 ▷ A42 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
5705 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 5705 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 15 points:
(5,6,8,7)(9,12,11,10)(13,14,15);(2,4)(6,7,8);(1,2,3,4)(11,12)192 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72215293 |
| maximal normal subgroup of | D3×#72215293 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6,8,7)(9,12,11,10)(13,14,15), (2,4)(6,7,8), (1,2,3,4)(11,12));