Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | D33×S4 |
| Fitting, socle | C22×C33 ⋊ C23×D3 |
| derived | C33×A4 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33×A4 | C24 |
| Fitting, socle | C22×C33 | C23×D3 |
| LCS | G ▷ C33×A4 | |
| UCS | C1 | |
7030 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 7030 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,4,3)(6,7)(8,9,10)(11,12);(5,6,7)(9,10);(2,4);(12,13)135 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#72210357 |
| maximal quotient of | D3×#72210357 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,3)(6,7)(8,9,10)(11,12), (5,6,7)(9,10), (2,4), (12,13));