Finite Group DatabasebetaFinite Group Databasebeta| order | 28561 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C132 · Schur multiplier (H2) = C132 |
| unambiguous semidirect product | H(3,𝔽13)⋊C13 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C13 | H(3,𝔽13) |
| commutator, Frattini, cosocle | C132 | C132 |
| LCS | G ▷ C132 ▷ C13 ▷ C1 | |
| UCS | C1 ◁ C13 ◁ C132 ◁ G | |
| LECS | G ▷ C132 ▷ C13 ▷ C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 169 |
A minimal faithful permutation representation — 2 generators on 169 points:
(1,2,4,8,16,31,51,69,86,100,115,130,144)(3,5,11,19,34,54,72,89,103,118,132,145,158)(6,13,27,46,66,83,98,113,128,142,156,169,157)(7,12,25,37,59,52,73,87,104,121,137,146,159)(9,20,38,61,80,96,111,126,131,147,160,15,26)(10,23,43,63,81,97,112,127,141,155,168,143,14)(17,35,60,79,88,106,116,133,148,161,68,82,21)(18,32,55,75,93,101,119,135,150,163,129,29,28)(22,40,62,74,92,105,122,117,134,149,162,30,45)(24,39,33,57,78,95,110,125,140,154,167,114,49)(36,56,77,71,91,108,124,139,153,166,85,84,41)(42,47,58,76,94,109,102,120,136,151,164,50,65)(44,64,53,70,90,107,123,138,152,165,99,67,48);(2,5,12,26,45,65,82,84,67,49,29,14,6)(4,9,21,24,13,25,42,48,10,11,22,41,28)(8,17,27,47,23,40,18,20,39,37,44,19,36)(16,32,56,62,34,43,64,58,59,46,33,35,38)(31,52,74,60,53,55,66,54,61,76,77,57,63)(51,70,80,81,79,72,78,92,83,71,73,75,94)(69,87,105,88,90,93,98,89,96,109,91,95,97)(86,101,108,122,103,112,107,102,104,113,110,106,111)(100,116,128,120,127,117,119,126,125,121,123,118,124)(115,131,133,140,142,137,136,138,141,132,134,139,135)(130,145,146,147,149,151,148,153,152,154,150,155,156)Polycyclic presentation:
13^4|b^a=bc,c^a=cd337 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(28561, 7);