Finite Group DatabasebetaFinite Group Databasebeta| order | 18816 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | C2×#72214179 |
| Fitting, socle, Sylow | C27 ⋊ #3861 |
| derived | AGL(1,𝔽8)2 ⋊ C2×C3 |
| Hall | C2×AGL(1,𝔽8)2 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72214179 |
| commutator, cosocle, hypocenter | AGL(1,𝔽8)2 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C27 | #3861 |
| LCS | G ▷ AGL(1,𝔽8)2 | |
| UCS | C1 ◁ C2 | |
608 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 608 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 98 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(3,4)(5,10,6,7,9,8)(11,12,17,15,14,16)(13,18);(12,16,15,18,17,13,14);(4,7,10,9,8,5,6)(12,14,13,17,18,15,16)64 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72214179 |
| maximal quotient of | D3×#72214179 |
| holomorph of | C2×AGL(1,𝔽8) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4)(5,10,6,7,9,8)(11,12,17,15,14,16)(13,18), (12,16,15,18,17,13,14), (4,7,10,9,8,5,6)(12,14,13,17,18,15,16));