Finite Group DatabasebetaFinite Group Databasebeta| order | 7200 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C3 |
| direct product | Dic5×A6 |
| derived | C5×A6 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×A6 |
| commutator, hypocenter | C5×A6 | C4 |
| Fitting, cosocle | C2×C5 | C2×A6 |
| socle, prime radical | C2×C5×A6 | C2 |
| LCS | G ▷ C5×A6 | |
| UCS | C1 ◁ C2 | |
200 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 10 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) = 120 |
A minimal faithful permutation representation — 2 generators on 15 points:
(5,6,7,8,9)(10,11)(12,15);(1,3,2,4)(5,7)(8,9)(10,13,11,14,15)56 conjugacy classes — view the full character table.
| maximal subgroup of | SL(2,𝔽5)×A6 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6,7,8,9)(10,11)(12,15), (1,3,2,4)(5,7)(8,9)(10,13,11,14,15));