Finite Group DatabasebetaFinite Group Databasebeta| order | 4800 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C2×C4 |
| direct product | A4×AGL(1,𝔽5)2 |
| Fitting, socle, derived | C22×C52 ⋊ C3×C42 |
| Hall | C22×AGL(1,𝔽5)2 ⋊ C3 |
| Sylow | C52 ⋊ C42×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C22×C52 | C3×C42 |
| prime radical | D52×A4 | C22 |
| cosocle | C22×D52 | C22×C3 |
| LCS | G ▷ C22×C52 | |
| UCS | C1 | |
551 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,4,3)(6,7,8,9)(10,11,14,13);(1,4)(2,3)(5,9,8,6,7)(11,12,14,13)100 conjugacy classes — view the full character table.
| maximal subgroup of | #72258468 |
| maximal quotient of | #72258468 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4,3)(6,7,8,9)(10,11,14,13), (1,4)(2,3)(5,9,8,6,7)(11,12,14,13));