Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#87516 |
| Fitting | C2×C4×H(5,𝔽2) ⋊ D3 |
3060 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 3060 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(6,10)(7,18,17)(8,14,15)(11,12,20,16,13,19);(1,2,3,4)(5,15,9,7)(6,16,10,11)(8,14,18,17)(12,19,13,20);(5,6)(7,13)(8,16)(9,10)(11,18)(12,15)(14,20)(17,19)92 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#72209772, C4×#11714170, C4×#11715828, … |
| maximal normal subgroup of | C4×#11714170, C4×#11715828, C4×#11714171 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((6,10)(7,18,17)(8,14,15)(11,12,20,16,13,19), (1,2,3,4)(5,15,9,7)(6,16,10,11)(8,14,18,17)(12,19,13,20), (5,6)(7,13)(8,16)(9,10)(11,18)(12,15)(14,20)(17,19));