Finite Group DatabasebetaFinite Group Databasebeta| order | 3024 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×ΓL(1,𝔽8)×C3⋊S4 |
| Fitting, socle | C23×C3×C7 ⋊ C3×D3 |
| derived | C3×C7×A4 ⋊ C22×C3 |
| Sylow | C7 ⋊ C2×C3×C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×C7×A4 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C7 | C3×D3 |
| LCS | G ▷ C3×C7×A4 | |
| UCS | C1 ◁ C2 | |
416 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 126 |
A minimal faithful permutation representation — 3 generators on 16 points:
(3,4,6,5)(8,9,10)(11,13,12)(15,16);(1,2)(4,6,5)(7,8,10,12,9,11,13);(4,6,5)(14,15,16)90 conjugacy classes — view the full character table.
| maximal subgroup of | C2×ΓL(1,𝔽8)×A4⋊S4, C2×AGL(1,𝔽7)×C3⋊S4, ΓL(1,𝔽8)×C3⋊#5935 |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×C3⋊S4 |
| maximal quotient of | C2×ΓL(1,𝔽8)×A4⋊S4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,6,5)(8,9,10)(11,13,12)(15,16), (1,2)(4,6,5)(7,8,10,12,9,11,13), (4,6,5)(14,15,16));