Finite Group DatabasebetaFinite Group Databasebeta| order | 4800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#72214019 |
| Fitting, socle | C23×C52 ⋊ C4×D3 |
| derived | C52×A4 ⋊ C22×C4 |
| Sylow | C52 ⋊ C2×C4×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72214019 |
| commutator, hypocenter | C52×A4 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C52 | C4×D3 |
| prime radical | C2×(C5⋊S4)⋊D5 | C2 |
| cosocle | A4×C5⋊D5 | C23 |
| LCS | G ▷ C52×A4 | |
| UCS | C1 ◁ C2 | |
1064 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,5,4)(7,8,11,9,10)(12,13,16,15,14);(8,9,10,11)(13,15,14,16);(5,6)(13,14)(15,16)76 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#72214019 |
| maximal normal subgroup of | C22×#72214019 |
| maximal quotient of | C22×#72214019 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,5,4)(7,8,11,9,10)(12,13,16,15,14), (8,9,10,11)(13,15,14,16), (5,6)(13,14)(15,16));