Finite Group DatabasebetaFinite Group Databasebeta| order | 60060 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D30030 |
| direct product | C2×D15015 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D15015 |
| commutator, cosocle, hypocenter | C3×C5×C7×C11×C13 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5×C7×C11×C13 | C2 |
| LCS | G ▷ C3×C5×C7×C11×C13 | |
| UCS | C1 ◁ C2 | |
160 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 30030 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2)(3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41);(4,5)(7,10)(8,9)(12,17)(13,16)(14,15)(19,28)(20,27)(21,26)(22,25)(23,24)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)15018 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41), (4,5)(7,10)(8,9)(12,17)(13,16)(14,15)(19,28)(20,27)(21,26)(22,25)(23,24)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36));