Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×Dic3×#1043 |
| derived, Hall | C3×C52 ⋊ C2×C42 |
| Sylow | C52 ⋊ C2×C4×Dic3 |
| Sylow | C3 ⋊ C2×C4×#1043 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×#1043 |
| commutator, hypocenter | C3×C52 | C2×C42 |
| Frattini | C2 | C2×D3×#1043 |
| Fitting, socle | C22×C3×C52 | C2×C4 |
| prime radical | C22×C3×C5⋊D5 | C22 |
| cosocle | C2×C3×C5⋊D5 | C23 |
| LCS | G ▷ C3×C52 | |
| UCS | C1 ◁ C22 | |
864 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 600 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(9,10,13,11,12)(14,15);(5,6,7,8)(10,11,12,13);(2,3)(4,5,8,6,7)(16,17,18,19)120 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(9,10,13,11,12)(14,15), (5,6,7,8)(10,11,12,13), (2,3)(4,5,8,6,7)(16,17,18,19));