Finite Group DatabasebetaFinite Group Databasebeta| order | 34375 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C11 · Schur multiplier (H2) = C5 |
| direct product | C11×#15691062 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C52×C11 | H(3,𝔽5) |
| commutator, Frattini, cosocle | C53 | C52×C11 |
| prime radical | C5×C11×H(3,𝔽5) | C5 |
| LCS | G ▷ C53 ▷ C52 ▷ C1 | |
| UCS | C1 ◁ C52×C11 ◁ C53×C11 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 6875 |
A minimal faithful permutation representation — 2 generators on 161 points:
(2,4,8,14,6)(5,10,9,13,12)(11,16,20,17,19)(26,27,30,38,56)(28,34,48,72,84)(29,33,41,59,85)(31,42,57,86,36)(32,45,71,101,83)(35,51,75,105,113)(37,47,64,58,89)(39,60,90,82,70)(40,63,95,55,46)(43,67,87,116,81)(44,66,88,115,54)(49,61,93,114,53)(50,62,92,119,112)(52,78,97,121,133)(65,91,120,132,77)(68,98,117,136,131)(69,73,102,118,111)(74,96,122,135,110)(76,103,129,134,80)(79,108,125,141,145)(94,123,138,144,100)(99,128,137,149,150)(104,126,140,148,143)(106,124,142,147,130)(107,127,139,146,109)(151,152,153,154,155,156,157,158,159,160,161);(1,2,5,11,18)(3,6,13,19,24)(4,9,16,22,21)(7,14,10,17,23)(8,12,20,25,15)(26,28,35,52,79,29,36,53,80,109,37,54,81,110,130,55,82,111,131,143,83,112,132,144,150)(27,31,43,68,99,33,44,69,100,108,47,70,77,78,107,46,50,51,76,106,32,34,49,74,104)(30,39,61,94,124,41,62,67,97,126,64,48,73,103,128,40,42,65,96,125,45,66,75,98,127)(38,57,87,117,137,59,88,102,123,141,58,60,91,121,139,63,92,105,129,142,71,72,93,122,140)(56,84,113,133,145,85,86,114,134,146,89,115,116,135,147,95,90,118,136,148,101,119,120,138,149)1639 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34375, 5);