Finite Group DatabasebetaFinite Group Databasebeta| order | 1500 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C52 · Schur multiplier (H2) = C52 |
| direct product | Dic3×H(3,𝔽5) |
| unambiguous semidirect product | C3⋊(C4×H(3,𝔽5)) |
| unambiguous semidirect product | (C52×Dic3)⋊C5 |
| Hall | C3×H(3,𝔽5) ⋊ C4 |
| Sylow | C3 ⋊ C4×H(3,𝔽5) |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 300 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,4)(5,6,7)(8,9,11,15,22)(10,13,17,24,29)(12,18,27,32,28)(14,20,26,23,30)(16,25,31,21,19);(1,3,2,4)(6,7)(9,12,19,20,13)(11,16,17,18,26)(15,23,27,24,25)(22,29,30,31,32)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1500, 23);