Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C11×C25 · Schur multiplier (H2) = C52 |
| direct product | C5×C11×C125⋊C5 |
| unambiguous semidirect product | C125⋊(C52×C11) |
| subgroup | G/H | |
|---|---|---|
| center | C5×C11×C25 | C52 |
| commutator | C5 | C52×C11×C25 |
| Frattini, cosocle | C25 | C53×C11 |
| socle | C52×C11 | C5×C25 |
| prime radical | C53×C11 | C25 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C11×C25 ◁ G | |
272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 141 |
| transitive degree | μt(G) = 6875 |
A minimal faithful permutation representation — 3 generators on 141 points:
(1,2,3,4,5)(131,132,133,134,135,136,137,138,139,140,141);(6,7,10,17,31,8,12,19,33,53,14,24,37,56,77,27,44,61,80,99,47,69,85,102,117,9,13,20,34,52,15,25,38,57,76,28,45,62,81,98,48,70,86,103,116,72,93,107,120,127,16,26,39,32,54,29,46,63,55,78,49,71,87,79,100,73,94,108,101,118,95,113,123,119,128,30,22,18,35,58,50,42,36,59,82,74,67,60,83,104,96,91,84,105,121,114,112,106,122,129,51,11,21,40,64,75,23,41,65,88,97,43,66,89,109,115,68,90,110,124,126,92,111,125,130);(7,11,22,26,13)(10,18,20,21,39)(12,23,42,46,25)(17,32,40,34,35)(19,36,38,41,63)(24,43,67,71,45)(31,52,54,58,64)(33,55,65,57,59)(37,60,62,66,87)(44,68,91,94,70)(53,76,78,82,88)(56,79,89,81,83)(61,84,86,90,108)(69,92,112,113,93)(77,98,100,104,109)(80,101,110,103,105)(85,106,107,111,123)(99,116,118,121,124)(102,119,125,120,122)(117,127,128,129,130)7975 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 59);