Finite Group DatabasebetaFinite Group Databasebeta| order | 9072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| direct product | #272×GL(3,𝔽2) |
| Fitting | H(3,𝔽3) ⋊ C2×GL(3,𝔽2) |
| derived | H(3,𝔽3)×GL(3,𝔽2) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C3⋊D3×GL(3,𝔽2) |
| commutator, hypocenter | H(3,𝔽3)×GL(3,𝔽2) | C2 |
| Fitting, cosocle | H(3,𝔽3) | C2×GL(3,𝔽2) |
| socle | C3×GL(3,𝔽2) | C3⋊D3 |
| LCS | G ▷ H(3,𝔽3)×GL(3,𝔽2) | |
| UCS | C1 ◁ C3 | |
574 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 3 generators on 16 points:
(8,13)(9,15)(11,16);(2,3)(4,5)(10,12,14)(13,16,15);(1,4,7,5,2,3,6)(8,10,15)(9,12,16)(11,14,13)60 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((8,13)(9,15)(11,16), (2,3)(4,5)(10,12,14)(13,16,15), (1,4,7,5,2,3,6)(8,10,15)(9,12,16)(11,14,13));