Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C4×#87547 |
| Fitting, Sylow | C4×H(7,𝔽2) ⋊ C3 |
19754 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 19754 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,10)(7,18,15,16,13,12,14,20,8,11,19,17);(1,3,4,2)(7,14)(11,16)(13,19)(18,20);(1,2,4,3)(5,13)(6,20)(7,8)(9,19)(10,18)(11,17)(12,16)(14,15)140 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#72209831, C4×#11715839, C4×#11715828, … |
| maximal normal subgroup of | C4×#11715839, C4×#11715828, C4×#60399⋊C3, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6,9,10)(7,18,15,16,13,12,14,20,8,11,19,17), (1,3,4,2)(7,14)(11,16)(13,19)(18,20), (1,2,4,3)(5,13)(6,20)(7,8)(9,19)(10,18)(11,17)(12,16)(14,15));