Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | D3×SL(2,𝔽3)×AGL(1,𝔽7) |
| Fitting, derived | C3×C7×Q8 ⋊ C22×C32 |
| Sylow | C7 ⋊ C2×C3×D3×SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×A4×AGL(1,𝔽7) |
| commutator, Fitting, cosocle, hypocenter | C3×C7×Q8 | C22×C32 |
| socle | C2×C3×C7 | C22×C3×A4 |
| LCS | G ▷ C3×C7×Q8 | |
| UCS | C1 ◁ C2 | |
776 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 776 conjugacy classes of subgroups — too many to draw. Only its 97 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:
(1,2,3)(4,5)(6,8)(9,10)(12,15,13)(16,17,18);(2,3)(5,7,8)(6,9,10)(11,17,14,15)(12,13,16,18)147 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((1,2,3)(4,5)(6,8)(9,10)(12,15,13)(16,17,18), (2,3)(5,7,8)(6,9,10)(11,17,14,15)(12,13,16,18));