Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×#32895⋊C3 |
| Fitting, Sylow | C2×#32895 ⋊ C3 |
5800 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 5800 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(3,4,8,12)(5,14,13,9)(6,24,19,25,18,11,17,27,7,10,20,28)(15,34,30,21,31,26,29,33,16,23,32,22);(3,20)(4,34)(5,31)(6,19,17,7)(8,18)(9,27)(10,26,25,22)(11,21,28,23)(12,33)(13,32)(14,24)(15,16,29,30);(1,2)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72236743, C2×#72236770, C2×#72236755, … |
| maximal normal subgroup of | C2×#72236743, C2×#72236770, C2×#72236755, … |
| maximal quotient of | #72342118 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,8,12)(5,14,13,9)(6,24,19,25,18,11,17,27,7,10,20,28)(15,34,30,21,31,26,29,33,16,23,32,22), (3,20)(4,34)(5,31)(6,19,17,7)(8,18)(9,27)(10,26,25,22)(11,21,28,23)(12,33)(13,32)(14,24)(15,16,29,30), (1,2));