Finite Group DatabasebetaFinite Group Databasebeta| order | 4800 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C22⋊D5×C5⋊S4 |
| Fitting | C24×C52 ⋊ C2×D3 |
| Sylow | C52 ⋊ D4×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D5×C5⋊S4 |
| commutator, cosocle | C2×C52×A4 | C23 |
| Fitting | C24×C52 | C2×D3 |
| socle | C23×C52 | C22×D3 |
| hypercenter | C22 | D5×C5⋊S4 |
| hypocenter | C52×A4 | C2×D4 |
| LCS | G ▷ C2×C52×A4 ▷ C52×A4 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1060 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 18 points:
(2,4)(5,7,8)(9,10,12,13,11)(14,15,17,18,16);(1,2)(3,4)(6,7)(10,11)(12,13)(15,16)(17,18);(6,7)(15,16)(17,18)169 conjugacy classes — view the full character table.
| maximal subgroup of | C5⋊S42, #72260259 |
| maximal normal subgroup of | #72260259 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4)(5,7,8)(9,10,12,13,11)(14,15,17,18,16), (1,2)(3,4)(6,7)(10,11)(12,13)(15,16)(17,18), (6,7)(15,16)(17,18));