Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | D3×C22⋊AΓL(1,𝔽8) |
| Fitting, socle | C25×C3 ⋊ C2×ΓL(1,𝔽8) |
| derived | C22×C3×AGL(1,𝔽8) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C3×AGL(1,𝔽8) | C2×C3 |
| Fitting, socle | C25×C3 | C2×ΓL(1,𝔽8) |
| LCS | G ▷ C22×C3×AGL(1,𝔽8) | |
| UCS | C1 | |
470 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 470 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 15 points:
(2,3,4)(5,6,10)(7,12,11)(14,15);(1,4)(2,3)(6,9,10,8,12,7,11)(13,14,15)48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×C22⋊AΓL(1,𝔽8) |
| maximal normal subgroup of | C2×D3×C22⋊AΓL(1,𝔽8) |
| maximal quotient of | C2×D3×C22⋊AΓL(1,𝔽8) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4)(5,6,10)(7,12,11)(14,15), (1,4)(2,3)(6,9,10,8,12,7,11)(13,14,15));