Finite Group DatabasebetaFinite Group Databasebeta| order | 56448 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | D3×#72214179 |
| Fitting, socle | C26×C3 ⋊ C2×#3861 |
| derived | C3×AGL(1,𝔽8)2 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×AGL(1,𝔽8)2 | C2×C3 |
| Fitting, socle | C26×C3 | C2×#3861 |
| LCS | G ▷ C3×AGL(1,𝔽8)2 | |
| UCS | C1 | |
1298 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 1298 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 3 generators on 19 points:
(2,3)(4,8,6)(5,11,9)(12,13,18)(14,19,17);(1,2,3)(13,16,18,15,17,19,14);(5,8,7,11,6,9,10)(13,14,19,17,15,18,16)96 conjugacy classes — view the full character table.
| holomorph of | C3×AGL(1,𝔽8) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,8,6)(5,11,9)(12,13,18)(14,19,17), (1,2,3)(13,16,18,15,17,19,14), (5,8,7,11,6,9,10)(13,14,19,17,15,18,16));