Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | ΓL(1,𝔽8)×C3⋊#1020 |
| Fitting, socle | C24×C3×C7 ⋊ C3×D3 |
| derived | C3×C7×C22⋊A4 ⋊ C2×C3 |
| Sylow | C7 ⋊ C3×C3⋊#1020 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C7×C22⋊A4 | C2×C3 |
| Fitting, socle | C24×C3×C7 | C3×D3 |
| LCS | G ▷ C3×C7×C22⋊A4 | |
| UCS | C1 | |
688 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 45 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 — 3 generators on 18 points:
(2,3,4)(5,7,6)(8,10)(12,15,14,13)(17,18);(1,2,4,6,3,5,7)(16,17,18);(9,10,11)(13,14,15)90 conjugacy classes — view the full character table.
| maximal quotient of | ΓL(1,𝔽8)×C3⋊#5935 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4)(5,7,6)(8,10)(12,15,14,13)(17,18), (1,2,4,6,3,5,7)(16,17,18), (9,10,11)(13,14,15));