Finite Group DatabasebetaFinite Group Databasebeta| order | 7776 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22×C32 |
| direct product | C3×#72213207 |
| derived | (C22⋊A4)⋊C9 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #72213207 |
| commutator, cosocle, hypocenter | (C22⋊A4)⋊C9 | C2×C32 |
| Frattini | C3 | C32×A4⋊S4 |
| Fitting | C24×C33 | C3⋊D3 |
| socle | C24×C32 | C3×C3⋊D3 |
| LCS | G ▷ (C22⋊A4)⋊C9 | |
| UCS | C1 ◁ C3 | |
1618 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 1618 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,4,3,2)(5,6,8,7)(9,10,11)(13,15)(14,19)(16,17)(18,20);(6,8,7)(12,18,20,14,16,15,19,13,17);(2,3,4)(6,8,7)150 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3,2)(5,6,8,7)(9,10,11)(13,15)(14,19)(16,17)(18,20), (6,8,7)(12,18,20,14,16,15,19,13,17), (2,3,4)(6,8,7));