Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×(D32⋊D3)⋊D3 |
| Fitting, socle, Sylow | C35 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | (D32⋊D3)⋊D3 |
| commutator, cosocle | C32×C3⋊D3 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C35 | C2×D4 |
| hypocenter | C34 | C2×C3×D4 |
| LCS | G ▷ C32×C3⋊D3 ▷ C34 | |
| UCS | C1 ◁ C3 | |
1646 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 1646 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 24 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(5,6)(7,8)(11,13)(12,15);(4,5,6)(8,9)(13,14);(10,11)(12,13)(14,15)135 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,6)(7,8)(11,13)(12,15), (4,5,6)(8,9)(13,14), (10,11)(12,13)(14,15));