Finite Group DatabasebetaFinite Group Databasebeta| order | 11520 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C3 |
| direct product | C2×AGL(1,𝔽5)×A4⋊S4 |
| Fitting, socle | C25×C5 ⋊ C4×C3⋊D3 |
| derived | C5×A42 ⋊ C22×C4 |
| Sylow | C5 ⋊ C2×C4×A4⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | AGL(1,𝔽5)×A4⋊S4 |
| commutator, hypocenter | C5×A42 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C25×C5 | C4×C3⋊D3 |
| prime radical | C2×D5×A4⋊S4 | C2 |
| cosocle | D5×A42 | C23 |
| LCS | G ▷ C5×A42 | |
| UCS | C1 ◁ C2 | |
6210 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 6210 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 90 |
A minimal faithful permutation representation — 3 generators on 15 points:
(2,3,4,5)(8,11)(9,10)(12,15,14);(1,2,5,3,4)(9,11)(14,15);(6,7)(9,10,11)(13,14,15)140 conjugacy classes — view the full character table.
| holomorph of | C2×C5×A4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4,5)(8,11)(9,10)(12,15,14), (1,2,5,3,4)(9,11)(14,15), (6,7)(9,10,11)(13,14,15));