Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26×C5 |
| direct product | C23×C3×#1044 |
| Fitting, socle | C23×C3×C52 ⋊ C4 |
| derived, Sylow | C52 ⋊ C23×C3×C4 |
| Hall | C3×C52 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | #1044 |
| commutator, hypocenter | C52 | C23×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C52 | C4 |
| prime radical | C23×C3×C5⋊D5 | C2 |
| cosocle | C5⋊D5 | C24×C3 |
| LCS | G ▷ C52 | |
| UCS | C1 ◁ C23×C3 | |
1212 conjugacy classes of subgroups; only the 332 normal subgroups are shown.
This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 332 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) = 240 |
A minimal faithful permutation representation — 4 generators on 19 points:
(5,6)(7,8,9)(10,11,14,12,13)(15,16,17,19,18);(3,4);(1,2);(11,12,13,14)(16,17,18,19)240 conjugacy classes — view the full character table.
| maximal subgroup of | C23×C3×D5⋊AGL(1,𝔽5) |
| maximal normal subgroup of | C23×C3×D5⋊AGL(1,𝔽5) |
| maximal quotient of | C3×#13107767 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(7,8,9)(10,11,14,12,13)(15,16,17,19,18), (3,4), (1,2), (11,12,13,14)(16,17,18,19));