Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C5 |
| direct product | C2×A4⋊#1044 |
| Fitting, socle | C23×C52 ⋊ Dic3 |
| derived | C52×A4 ⋊ C2×C4 |
| Sylow | C52 ⋊ C2×SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4⋊#1044 |
| commutator, hypocenter | C52×A4 | C2×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C52 | Dic3 |
| prime radical | C2×A4×C5⋊D5 | C2 |
| cosocle | A4×C5⋊D5 | C22 |
| LCS | G ▷ C52×A4 | |
| UCS | C1 ◁ C2 | |
330 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 36 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 — 2 generators on 16 points:
(1,2)(3,5,4)(7,8,11,9,10)(12,13,14,16,15);(5,6)(8,9,10,11)(13,14,15,16)68 conjugacy classes — view the full character table.
| maximal subgroup of | C22×A4⋊#1044, C2×#72214019, C2×#72213888, … |
| maximal normal subgroup of | C22×A4⋊#1044, C2×#72214019, C2×#72213888, … |
| maximal quotient of | C22×A4⋊#1044, #72260349 |
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)(12,13,14,16,15), (5,6)(8,9,10,11)(13,14,15,16));