Finite Group DatabasebetaFinite Group Databasebeta| order | 7776 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3×C3⋊AGL(2,𝔽3) |
| Fitting, socle | C2×C34 ⋊ GL(2,𝔽3) |
| derived | C3×PSU(3,𝔽2)⋊C3 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊AGL(2,𝔽3) |
| commutator, cosocle, hypocenter | C3×PSU(3,𝔽2)⋊C3 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C34 | GL(2,𝔽3) |
| LCS | G ▷ C3×PSU(3,𝔽2)⋊C3 | |
| UCS | C1 ◁ C2×C3 | |
1286 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 1286 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) = 17 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3)(7,12,14,10,9,8)(11,13)(16,17);(5,6)(8,9)(10,15)(11,14);(4,5,6)126 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×D3×AGL(2,𝔽3), C2×D3×C3⋊AGL(2,𝔽3), C3×#72259824, … |
| maximal normal subgroup of | C2×C3×D3×AGL(2,𝔽3), C2×D3×C3⋊AGL(2,𝔽3), C3×#72259824, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(7,12,14,10,9,8)(11,13)(16,17), (5,6)(8,9)(10,15)(11,14), (4,5,6));