Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C3×ΓL(1,𝔽8)×#1020 |
| Fitting, socle | C24×C3×C7 ⋊ C3×D3 |
| derived | C7×C22⋊A4 ⋊ C2×C32 |
| Sylow | C7 ⋊ C32×#1020 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(1,𝔽8)×#1020 |
| commutator, cosocle, hypocenter | C7×C22⋊A4 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C7 | C3×D3 |
| LCS | G ▷ C7×C22⋊A4 | |
| UCS | C1 ◁ C3 | |
784 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,4,6,3,5,7)(8,10)(12,15,14,13)(16,17,18);(2,3,4)(5,7,6)(9,10,11)(13,14,15)150 conjugacy classes — view the full character table.
| maximal quotient of | C3×ΓL(1,𝔽8)×#5935 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,6,3,5,7)(8,10)(12,15,14,13)(16,17,18), (2,3,4)(5,7,6)(9,10,11)(13,14,15));