Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C11 · Schur multiplier (H2) = C5 |
| direct product | C11×#15691093 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C11 | H(3,𝔽5)⋊C5 |
| commutator, Frattini, cosocle | C53 | C52×C11 |
| prime radical | C5×C11×H(3,𝔽5) | C5 |
| LCS | G ▷ C53 ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C11 ◁ C52×C11 ◁ C53×C11 ◁ G | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 136 |
| transitive degree | μt(G) = 1375 |
A minimal faithful permutation representation — 2 generators on 136 points:
(1,2,5,13,32,31,23,15,39,71,4,7,16,34,62,60,8,22,49,82,12,21,40,35,66)(3,6,17,41,72,29,47,14,33,63,11,20,37,70,94,10,19,38,65,61,30,48,43,74,91)(9,24,42,75,101,26,56,36,67,95,27,18,44,73,90,28,51,69,64,92,58,46,77,97,89)(25,52,68,98,115,57,45,78,93,111,55,80,100,114,112,59,81,104,102,110,54,50,76,96,116)(53,83,108,113,121,87,84,109,120,125,85,79,103,119,124,88,107,99,117,122,86,106,105,118,123)(126,127,128,129,130,131,132,133,134,135,136);(1,3,9,25,53)(2,6,18,45,79)(4,11,27,55,85)(5,14,36,68,99)(7,20,46,81,106)(8,19,24,50,83)(10,28,59,88,60)(12,30,58,54,86)(13,33,64,93,113)(15,37,44,78,105)(16,38,69,100,108)(17,42,76,103,40)(21,48,56,52,84)(22,43,77,104,109)(23,47,51,80,107)(26,57,87,31,29)(32,61,89,110,121)(34,65,75,102,119)(35,41,73,98,118)(39,70,97,114,120)(49,74,67,96,117)(62,72,95,115,124)(63,90,111,122,82)(66,94,92,112,123)(71,91,101,116,125)583 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 36);