Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
| direct product | D3×C22⋊(C22⋊ΓL(1,𝔽8)) |
| Fitting, socle, derived | C24×C3×C7 ⋊ C2×C3 |
| Sylow | C7 ⋊ D3×C22⋊A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C24×C3×C7 | C2×C3 |
| LCS | G ▷ C24×C3×C7 | |
| UCS | C1 | |
606 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 606 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) = 252 |
A minimal faithful permutation representation — 3 generators on 18 points:
(2,3,4)(5,7,6)(9,10)(12,13,14)(16,17,18);(1,2,4,6,3,5,7)(8,9,10)(15,18)(16,17);(11,14)(12,13)120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×ΓL(1,𝔽8)×C22⋊A4, D3×D7⋊(C22⋊A4) |
| maximal normal subgroup of | D3×ΓL(1,𝔽8)×C22⋊A4, D3×D7⋊(C22⋊A4) |
| maximal quotient of | D3×C7⋊#997 |
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)(9,10)(12,13,14)(16,17,18), (1,2,4,6,3,5,7)(8,9,10)(15,18)(16,17), (11,14)(12,13));