Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×#72220155 |
| derived | A4×C22⋊C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72220155 |
| commutator, cosocle, hypocenter | A4×C22⋊C9 | C22×C3 |
| Frattini | C3 | C2×C3×A4⋊S4 |
| Fitting | C25×C32 | C3⋊D3 |
| socle | C25×C3 | C3×C3⋊D3 |
| LCS | G ▷ A4×C22⋊C9 | |
| UCS | C1 ◁ C2 | |
1482 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 1482 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 54 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,5,6,3,9,8,7,2,4)(10,11)(12,13,15)(16,19,18);(2,4)(3,7)(5,8)(6,9)(13,14)(18,19);(17,19,18)116 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#72220155 |
| maximal normal subgroup of | C22×#72220155 |
| maximal quotient of | C2×#72220160, C22×#72220155 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,5,6,3,9,8,7,2,4)(10,11)(12,13,15)(16,19,18), (2,4)(3,7)(5,8)(6,9)(13,14)(18,19), (17,19,18));