Finite Group DatabasebetaFinite Group Databasebeta| order | 6048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | AGL(1,𝔽7)×C3⋊GL(2,𝔽3) |
| Fitting | C3×C7×Q8 ⋊ C2×C3×D3 |
| derived | C3×C7×SL(2,𝔽3) ⋊ C22×C3 |
| Sylow | C7 ⋊ C2×C3×C3⋊GL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AGL(1,𝔽7)×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×C7×SL(2,𝔽3) | C22×C3 |
| Fitting | C3×C7×Q8 | C2×C3×D3 |
| socle | C2×C3×C7 | C2×C3×S4 |
| LCS | G ▷ C3×C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
696 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(1,7,3)(2,5,4)(10,13,14,11,12,15);(2,5)(3,7)(6,8)(9,10,12,15,14,13,11)(17,18);(2,6,3)(5,7,8)(16,17,18)105 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×SO+(4,𝔽3) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,7,3)(2,5,4)(10,13,14,11,12,15), (2,5)(3,7)(6,8)(9,10,12,15,14,13,11)(17,18), (2,6,3)(5,7,8)(16,17,18));