Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C11 · Schur multiplier (H2) = C5 |
| direct product | C11×#15691095 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C11 | H(3,𝔽5)⋊C5 |
| commutator, Frattini, cosocle | C53 | C52×C11 |
| prime radical | C11×H(3,𝔽5)⋊C5 | C5 |
| LCS | G ▷ C53 ▷ C52 ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C11 ◁ C52×C11 ◁ C53×C11 ◁ G | |
74 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)(3,6,17,41,72)(4,7,16,34,62)(8,22,49,82,60)(9,24,42,75,101)(10,19,38,65,61)(11,20,37,70,94)(12,21,40,35,66)(14,33,63,29,47)(15,39,71,31,23)(18,44,73,90,27)(25,52,68,98,115)(26,56,36,67,95)(28,51,69,64,92)(30,48,43,74,91)(45,78,93,111,57)(46,77,97,89,58)(50,76,96,116,54)(53,83,108,113,121)(55,80,100,114,112)(59,81,104,102,110)(79,103,119,124,85)(84,109,120,125,87)(86,106,105,118,123)(88,107,99,117,122)(126,127,128,129,130,131,132,133,134,135,136);(1,3,9,25,53,4,11,27,55,85,12,30,58,54,86,31,29,26,57,87,60,10,28,59,88)(2,6,18,45,79,7,20,46,81,106,21,48,56,52,84,23,47,51,80,107,8,19,24,50,83)(5,14,36,68,99,16,38,69,100,108,40,17,42,76,103,15,37,44,78,105,22,43,77,104,109)(13,33,64,93,113,34,65,75,102,119,35,41,73,98,118,39,70,97,114,120,49,74,67,96,117)(32,61,89,110,121,62,72,95,115,124,66,94,92,112,123,71,91,101,116,125,82,63,90,111,122)583 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 38);