Finite Group DatabasebetaFinite Group Databasebeta| order | 4800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24×C5 |
| direct product | C22×A4×#1044 |
| Fitting, socle | C24×C52 ⋊ C3×C4 |
| derived | C22×C52 ⋊ C22×C3×C4 |
| Hall | C24×#1044 ⋊ C3 |
| Sylow | C52 ⋊ C22×C4×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4×#1044 |
| commutator, hypocenter | C22×C52 | C22×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C24×C52 | C3×C4 |
| prime radical | C22×A4×C5⋊D5 | C2 |
| cosocle | C22×C5⋊D5 | C23×C3 |
| LCS | G ▷ C22×C52 | |
| UCS | C1 ◁ C22 | |
1624 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 126 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:
(3,4)(6,8,7)(9,10,13,11,12)(14,15,16,18,17);(1,2)(5,8)(6,7);(10,11,12,13)(15,16,17,18)160 conjugacy classes — view the full character table.
| maximal subgroup of | A42×#1044, C22×A4×D5⋊AGL(1,𝔽5) |
| maximal normal subgroup of | A42×#1044, C22×A4×D5⋊AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(6,8,7)(9,10,13,11,12)(14,15,16,18,17), (1,2)(5,8)(6,7), (10,11,12,13)(15,16,17,18));