Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#15691070 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C11 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C53 | C52×C11 |
| prime radical | C53×C11 | C52 |
| LCS | G ▷ C53 ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C11 ◁ C53×C11 ◁ G | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 6875 |
A minimal faithful permutation representation — 2 generators on 161 points:
(1,2,6,7,19,4,9,18,20,12,5,10,3,11,23,15,22,13,24,25,16,8,14,21,17)(26,27,30,38,56,37,47,64,58,89,83,32,45,71,101,29,33,41,59,85,55,46,40,63,95)(28,34,48,72,84,54,44,66,88,115,112,50,62,92,119,36,31,42,57,86,82,70,39,60,90)(35,51,75,105,113,81,43,67,87,116,132,77,65,91,120,53,49,61,93,114,111,69,73,102,118)(52,78,97,121,133,110,74,96,122,135,144,100,94,123,138,80,76,103,129,134,131,68,98,117,136)(79,108,125,141,145,130,106,124,142,147,150,99,128,137,149,109,107,127,139,146,143,104,126,140,148)(151,152,153,154,155,156,157,158,159,160,161);(1,3,12,10,11,5,14,25,8,21,16,18,19,9,20,4,13,23,22,24,15,6,17,2,7)(26,28,35,52,79,37,54,81,110,130,83,112,132,144,150,29,36,53,80,109,55,82,111,131,143)(27,31,43,68,99,47,70,77,78,107,32,34,49,74,104,33,44,69,100,108,46,50,51,76,106)(30,39,61,94,124,64,48,73,103,128,45,66,75,98,127,41,62,67,97,126,40,42,65,96,125)(38,57,87,117,137,58,60,91,121,139,71,72,93,122,140,59,88,102,123,141,63,92,105,129,142)(56,84,113,133,145,89,115,116,135,147,101,119,120,138,149,85,86,114,134,146,95,90,118,136,148)1639 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 13);