Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#11925429 |
| Hall | C3×C52 ⋊ C22×D4 |
| Sylow | C52 ⋊ C22×D12 |
| Sylow | C3 ⋊ C2×#87886 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×D52 |
| commutator, cosocle | C2×C3×C52 | C24 |
| Frattini | C2 | C2×D3×D52 |
| Fitting, socle | C22×C3×C52 | C23 |
| hypocenter | C3×C52 | C22×D4 |
| LCS | G ▷ C2×C3×C52 ▷ C3×C52 | |
| UCS | C1 ◁ C22 | |
1212 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 1212 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) = 120 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(11,12)(13,14)(15,16)(17,18);(2,3)(7,9)(10,11,13,14,12);(4,5);(2,3)(6,7)(8,9)(16,17)(18,19)156 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72224600, C2×#72224578, C2×D3×#87886, … |
| maximal normal subgroup of | C2×#72224578, C2×D3×#87886, C2×#72224577, … |
| maximal quotient of | C2×#72224600, #72230924, C2×#72229316 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(11,12)(13,14)(15,16)(17,18), (2,3)(7,9)(10,11,13,14,12), (4,5), (2,3)(6,7)(8,9)(16,17)(18,19));