#43270034

order	28561
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₁₃² · Schur multiplier (H₂) = C₁₃

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₁₃	H(3,𝔽₁₃)
commutator, Frattini, cosocle	C₁₃²	C₁₃²
prime radical	C₁₃³	C₁₃
LCS	G ▷ C₁₃² ▷ C₁₃ ▷ C₁
UCS	C₁ ◁ C₁₃ ◁ C₁₃² ◁ G
LECS	G ▷ C₁₃² ▷ C₁₃ ▷ C₁

59 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 169
transitive degree	μ_t(G) = 169

A minimal faithful permutation representation — 2 generators on 169 points:

(1,2,4,8,16,31,51,69,86,100,115,130,144,3,5,11,19,34,54,72,89,103,118,132,145,158,7,12,25,37,59,52,73,87,104,121,137,146,159,15,26,9,20,38,61,80,96,111,126,131,147,160,30,45,22,40,62,74,92,105,122,117,134,149,162,50,65,42,47,58,76,94,109,102,120,136,151,164,68,82,21,17,35,60,79,88,106,116,133,148,161,85,84,41,36,56,77,71,91,108,124,139,153,166,99,67,48,44,64,53,70,90,107,123,138,152,165,114,49,24,39,33,57,78,95,110,125,140,154,167,129,29,28,18,32,55,75,93,101,119,135,150,163,143,14,10,23,43,63,81,97,112,127,141,155,168,157,6,13,27,46,66,83,98,113,128,142,156,169);(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|a^13=d,b^a=bc,c^a=cd

Conjugacy classes & character table

337 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28561, 8);