Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C11×C25 · Schur multiplier (H2) = C5 |
| direct product | C11×5−1+2⋊C25 |
| unambiguous semidirect product | 5−1+2⋊(C11×C25) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C11 | H(3,𝔽5) |
| commutator | C52 | C5×C11×C25 |
| Frattini, cosocle | C53 | C52×C11 |
| prime radical | C53×C11 | C52 |
| LCS | G ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C52×C11 ◁ C53×C11 ◁ G | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 6875 |
A minimal faithful permutation representation — 2 generators on 61 points:
(1,2,4,7,11,3,5,8,12,16,6,9,13,17,20,10,14,18,21,23,15,19,22,24,25)(26,27,30,37,45,29,28,33,38,46,36,35,41,43,49,44,42,31,39,47,50,32,34,40,48)(51,52,53,54,55,56,57,58,59,60,61);(26,28,34,37,45,29,35,30,38,46,36,42,33,43,49,44,32,41,39,47,50,27,31,40,48)2695 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 23);