Finite Group DatabasebetaFinite Group Databasebeta| order | 60900 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D30450 |
| direct product | C2×D15225 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D15225 |
| commutator, cosocle, hypocenter | C3×C7×C25×C29 | C22 |
| Frattini | C5 | C2×D3045 |
| Fitting | C2×C3×C7×C25×C29 | C2 |
| socle | C2×C3×C5×C7×C29 | D5 |
| LCS | G ▷ C3×C7×C25×C29 | |
| UCS | C1 ◁ C2 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 30450 |
A minimal faithful permutation representation — 2 generators on 66 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)(38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66);(4,5)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(32,37)(33,36)(34,35)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)15228 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)(38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66), (4,5)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(32,37)(33,36)(34,35)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53));