Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| isoclinism | isoclinic class of D5⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C3×C4 |
| direct product | C32×#87830 |
| derived, Sylow | C52 ⋊ C32×C42 |
| Hall | C32×C52 ⋊ C42 |
456 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 138 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)(8,9,10,11)(12,13,14,15);(4,5,6)(7,8,11,9,10)(13,14)(15,16)(17,18,19,20)252 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#87830, C2×C32×#87830, C3×#72229763, … |
| maximal normal subgroup of | C3⋊D3×#87830, C2×C32×#87830, C3×#72229763, … |
| maximal quotient of | C2×C32×#87830, 3−1+2×#87830 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(8,9,10,11)(12,13,14,15), (4,5,6)(7,8,11,9,10)(13,14)(15,16)(17,18,19,20));