Finite Group DatabasebetaFinite Group Databasebeta| order | 7200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | A4×#10597182 |
| Fitting, socle, derived | C22×C3×C52 ⋊ C2×C3×C4 |
| Sylow | C52 ⋊ C4×D3×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C22×C3×C52 | C2×C3×C4 |
| prime radical | A4×D15⋊D5 | C2 |
| cosocle | C22×C3×C5⋊D5 | C22×C3 |
| LCS | G ▷ C22×C3×C52 | |
| UCS | C1 | |
652 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3)(5,6,7)(9,10,11,12)(13,14,15,16);(2,3)(4,7)(5,6)(8,9,12,10,11)(14,15)(16,17)96 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#72214019 |
| maximal quotient of | A4×#72214019 |
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)(9,10,11,12)(13,14,15,16), (2,3)(4,7)(5,6)(8,9,12,10,11)(14,15)(16,17));