Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C32×C4 |
| direct product | #157×C3⋊#157 |
| Fitting, socle, derived, Sylow | C35 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C35 | C42 |
| prime radical, cosocle | C3×C3⋊D32 | C22 |
| LCS | G ▷ C35 | |
| UCS | C1 | |
600 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 36 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:
(2,3)(4,5)(6,8,9,7)(10,15,12,11)(13,14);(1,2,3)(5,8,7)(10,11)(12,15,13,14)72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1572, #72228547 |
| maximal normal subgroup of | D3×#1572, #72228547 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,5)(6,8,9,7)(10,15,12,11)(13,14), (1,2,3)(5,8,7)(10,11)(12,15,13,14));