Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C32×C4 |
| direct product | C3×#1572 |
| Fitting, socle, Sylow | C35 ⋊ C42 |
| derived | C34 ⋊ C3×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #1572 |
| commutator, hypocenter | C34 | C3×C42 |
| Frattini | C1 | G |
| Fitting, socle | C35 | C42 |
| prime radical | C3×C3⋊D32 | C22 |
| cosocle | C3⋊D32 | C22×C3 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C3 | |
600 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 48 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3)(4,5)(6,7,8,9)(10,14,12,11)(13,15);(5,7,9)(10,11)(12,14,15,13)108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72213199, D3×#1572, #72228480, … |
| maximal normal subgroup of | C3×#72213199, D3×#1572, #72228480, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,5)(6,7,8,9)(10,14,12,11)(13,15), (5,7,9)(10,11)(12,14,15,13));