Finite Group DatabasebetaFinite Group Databasebeta| order | 17125 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C137 · Schur multiplier (H2) = C52 |
| direct product | H(3,𝔽5)×C137 |
| unambiguous semidirect product | C52⋊(C5×C137) |
| unambiguous semidirect product | (C52×C137)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C137 | C52 |
| commutator, Frattini, cosocle | C5 | C52×C137 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C137 ◁ G | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 162 |
| transitive degree | μt(G) = 3425 |
A minimal faithful permutation representation — 2 generators on 162 points:
(1,2,4,8,15)(3,6,10,17,22)(5,11,20,25,21)(7,13,19,16,23)(9,18,24,14,12)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162);(2,5,12,13,6)(4,9,10,11,19)(8,16,20,17,18)(15,22,23,24,25)3973 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(17125, 3);