Finite Group DatabasebetaFinite Group Databasebeta| order | 7200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#72214019 |
| Fitting, socle | C22×C3×C52 ⋊ C4×D3 |
| derived | C52×A4 ⋊ C2×C3×C4 |
| Sylow | C52 ⋊ C3×C4×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #72214019 |
| commutator, hypocenter | C52×A4 | C2×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C52 | C4×D3 |
| prime radical | C3×(C5⋊S4)⋊D5 | C2 |
| cosocle | A4×C5⋊D5 | C22×C3 |
| LCS | G ▷ C52×A4 | |
| UCS | C1 ◁ C3 | |
572 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(1,3,4)(5,6,7)(9,10,11,12)(13,14,15,16);(2,3)(8,9,12,10,11)(14,15)(16,17)114 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((1,3,4)(5,6,7)(9,10,11,12)(13,14,15,16), (2,3)(8,9,12,10,11)(14,15)(16,17));