Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| isoclinism | isoclinic class of #87831 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#87832 |
| Hall | C32×C52 ⋊ C4⋊C4 |
| Sylow | C52 ⋊ C32×C4⋊C4 |
408 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 180 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3)(8,10)(12,13,14,15)(16,17,18,19);(4,5,6)(7,8,9,10)(11,12,15,13,14)(17,18)(19,20)198 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#11717375, C3⋊D3×#87832, #72231474, … |
| maximal normal subgroup of | C32×#11717375, C3⋊D3×#87832, #72231474, … |
| maximal quotient of | C2×C32×#87832, 3−1+2×#87832 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(8,10)(12,13,14,15)(16,17,18,19), (4,5,6)(7,8,9,10)(11,12,15,13,14)(17,18)(19,20));