Finite Group DatabasebetaFinite Group Databasebeta| order | 5832 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C2×C33 |
| direct product | S4×#6891 |
| Fitting | C22×#6891 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C33×S4 |
| commutator, cosocle | C32×A4 | C2×C33 |
| Fitting | C22×#6891 | D3 |
| socle | C22×C32 | C33×D3 |
| prime radical | C34×S4 | C3 |
| hypercenter | #6891 | S4 |
| hypocenter | A4 | C2×#6891 |
| LCS | G ▷ C32×A4 ▷ A4 | |
| UCS | C1 ◁ C32 ◁ #6891 | |
2452 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 2452 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,4,3,2)(15,18,22)(17,20,21);(2,3,4)(6,9,10)(8,12,13);(5,6,8)(7,10,12)(9,13,11)(14,18,20,16,22,17,19,15,21)255 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#10624749 |
| maximal normal subgroup of | S4×#10624749 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3,2)(15,18,22)(17,20,21), (2,3,4)(6,9,10)(8,12,13), (5,6,8)(7,10,12)(9,13,11)(14,18,20,16,22,17,19,15,21));