Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C53×C11 · Schur multiplier (H2) = C56 |
| direct product | C11×C52⋊H(3,𝔽5) |
| unambiguous semidirect product | C52⋊(C11×H(3,𝔽5)) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C11 | C53 |
| commutator, Frattini, cosocle | C52 | C53×C11 |
| LCS | G ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C11 ◁ G | |
1282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 1375 |
A minimal faithful permutation representation — 3 generators on 61 points:
(1,2,4,8,15)(3,6,10,17,22)(5,11,20,25,21)(7,13,19,16,23)(9,18,24,14,12)(26,27,29,33,40)(28,31,35,42,47)(30,36,45,50,46)(32,38,44,41,48)(34,43,49,39,37)(51,52,53,54,55,56,57,58,59,60,61);(2,5,12,13,6)(4,9,10,11,19)(8,16,20,17,18)(15,22,23,24,25);(27,30,37,38,31)(29,34,35,36,44)(33,41,45,42,43)(40,47,48,49,50)2695 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 45);