Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C11×C25 · Schur multiplier (H2) = C1 |
| direct product | C11×#15691084 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C11 | C25⋊C25 |
| commutator | C25 | C5×C11×C25 |
| Frattini, cosocle | C5×C25 | C52×C11 |
| prime radical | C52×C11 | 5−1+2 |
| LCS | G ▷ C25 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C11 ◁ C5×C11×C25 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 136 |
| transitive degree | μt(G) = 1375 |
A minimal faithful permutation representation — 2 generators on 136 points:
(2,5,11,25,56,7,14,28,59,76,19,35,64,34,58,47,77,100,27,63,96,116,13,33,75)(3,8,20,48,97)(6,15,36,31,57,17,39,79,73,38,45,84,66,32,67,18,26,60,74,104,46,62,82,12,29)(9,22,51,21,49)(16,40,85,87,37,42,88,72,105,81,43,65,30,68,61,93,109,70,111,107,44,94,71,78,83)(23,53,50,98,52)(41,89,86,118,115,90,123,121,119,113,92,124,122,120,69,95,117,125,106,112,91,110,108,80,114)(54,103,102,101,99)(126,127,128,129,130,131,132,133,134,135,136);(1,2,6,16,41,3,5,12,30,69,9,13,15,37,80,23,34,38,40,86,54,75,82,87,89,4,7,18,43,91,8,14,32,71,113,21,11,26,61,106,50,27,29,65,108,99,56,66,68,110,10,19,17,44,95,20,35,31,72,115,22,28,39,83,120,52,33,67,94,125,101,76,36,78,117,24,47,46,42,92,48,77,74,70,114,49,64,62,81,119,53,25,57,88,122,102,58,60,105,124,55,96,45,93,90,97,116,73,85,112,51,100,84,107,118,98,59,104,109,121,103,63,79,111,123)1639 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 27);