Finite Group DatabasebetaFinite Group Databasebeta| order | 6336 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C3 |
| direct product | D11×A4⋊S4 |
| Fitting, socle | C24×C11 ⋊ C2×C3⋊D3 |
| derived | C11×A42 ⋊ C22 |
| Sylow | C11 ⋊ C2×A4⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C11×A42 | C22 |
| Fitting, socle | C24×C11 | C2×C3⋊D3 |
| LCS | G ▷ C11×A42 | |
| UCS | C1 | |
674 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 674 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 132 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,4)(2,3)(5,8,7)(10,11)(12,13)(14,15)(16,17)(18,19);(2,4)(7,8)(9,10,12,14,16,18,19,17,15,13,11);(2,3,4)(6,7,8)98 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4)(2,3)(5,8,7)(10,11)(12,13)(14,15)(16,17)(18,19), (2,4)(7,8)(9,10,12,14,16,18,19,17,15,13,11), (2,3,4)(6,7,8));