Finite Group DatabasebetaFinite Group Databasebeta| order | 7776 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C4 |
| direct product | C3×#72213199 |
| Fitting, socle, Sylow | C35 ⋊ C4×C8 |
| derived | C34 ⋊ C3×C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #72213199 |
| commutator, hypocenter | C34 | C3×C4×C8 |
| Frattini | C1 | G |
| Fitting, socle | C35 | C4×C8 |
| prime radical | C3×C3⋊D32 | C2×C4 |
| cosocle | #10620198 | C22×C3 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C3 | |
436 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3)(5,7,8,6,9,10,12,11)(13,17,14,21,19,16,15,18);(4,12,8)(5,7,6)(9,11,10)(14,15,18,16)(17,19,21,20)126 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72213199, C3×#72213315, #72259512 |
| maximal normal subgroup of | D3×#72213199, C3×#72213315, #72259512 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,7,8,6,9,10,12,11)(13,17,14,21,19,16,15,18), (4,12,8)(5,7,6)(9,11,10)(14,15,18,16)(17,19,21,20));