Finite Group DatabasebetaFinite Group Databasebeta| order | 14400 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | A42×#1043 |
| Fitting, socle, derived | C24×C52 ⋊ C32×C4 |
| Hall | C24×#1043 ⋊ C32 |
| Sylow | C52 ⋊ C4×A42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C24×C52 | C32×C4 |
| prime radical | A42×C5⋊D5 | C2 |
| cosocle | C24×C5⋊D5 | C2×C32 |
| LCS | G ▷ C24×C52 | |
| UCS | C1 | |
1624 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 300 |
A minimal faithful permutation representation — 3 generators on 18 points:
(2,3,4,5)(7,8,9,10)(11,14)(12,13)(15,18,16);(6,7,10,8,9)(12,14,13)(16,18,17);(1,2,5,3,4)160 conjugacy classes — view the full character table.
| maximal subgroup of | #1043×A4⋊S4, A4×#72214019, A42×D5⋊AGL(1,𝔽5) |
| maximal normal subgroup of | #1043×A4⋊S4, A4×#72214019, A42×D5⋊AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4,5)(7,8,9,10)(11,14)(12,13)(15,18,16), (6,7,10,8,9)(12,14,13)(16,18,17), (1,2,5,3,4));