Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×C7⋊#1020 |
| Fitting, socle | C24×C3×C7 ⋊ D3 |
| derived | C7×C22⋊A4 ⋊ C2×C3 |
| Sylow | C7 ⋊ C3×#1020 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C7⋊#1020 |
| commutator, cosocle, hypocenter | C7×C22⋊A4 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C7 | D3 |
| LCS | G ▷ C7×C22⋊A4 | |
| UCS | C1 ◁ C3 | |
238 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 26 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:
(2,3)(4,5)(6,7)(8,9,10)(11,13)(15,18,17,16);(1,2,4,6,7,5,3)(12,13,14)(16,17,18)102 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72214032 |
| maximal normal subgroup of | C3×#72214032 |
| maximal quotient of | C3×C7⋊#5935 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,5)(6,7)(8,9,10)(11,13)(15,18,17,16), (1,2,4,6,7,5,3)(12,13,14)(16,17,18));