Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| direct product | Dic3×SL(2,𝔽8) |
| derived | C3×SL(2,𝔽8) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×SL(2,𝔽8) |
| commutator, hypocenter | C3×SL(2,𝔽8) | C4 |
| Fitting, cosocle | C2×C3 | C2×SL(2,𝔽8) |
| socle, prime radical | C2×C3×SL(2,𝔽8) | C2 |
| LCS | G ▷ C3×SL(2,𝔽8) | |
| UCS | C1 ◁ C2 | |
110 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 10 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) = 108 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3)(4,6,5,7)(8,10,16,11,14,13,9);(1,2)(4,7,5,6)(8,13,9,15,12,10,16)54 conjugacy classes — view the full character table.
| maximal subgroup of | SL(2,𝔽5)×SL(2,𝔽8) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,6,5,7)(8,10,16,11,14,13,9), (1,2)(4,7,5,6)(8,13,9,15,12,10,16));