Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#21484⋊C3 |
| Fitting, Sylow | C2×#21484 ⋊ C3 |
1688 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 1688 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(4,9)(6,21,7,17,18,20)(10,22)(11,26,12)(13,32,14,25,30,27)(16,28,19)(23,33,24)(29,34,31);(3,28,5,34,8,26,15,33)(4,32,10,18,9,30,22,21)(6,19,25,24,17,12,13,31)(7,11,27,29,20,16,14,23);(1,2)92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72236799, C2×#72236787, C2×#72236829, … |
| maximal normal subgroup of | C2×#72236799, C2×#72236787, C2×#72236829, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,9)(6,21,7,17,18,20)(10,22)(11,26,12)(13,32,14,25,30,27)(16,28,19)(23,33,24)(29,34,31), (3,28,5,34,8,26,15,33)(4,32,10,18,9,30,22,21)(6,19,25,24,17,12,13,31)(7,11,27,29,20,16,14,23), (1,2));