#125×C₁₃₁ #16862132

order	4192
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

#125×C₁₃₁

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₁₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ G

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 147
transitive degree	μ_t(G) = 4192

A minimal faithful permutation representation — 3 generators on 147 points:

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

Conjugacy classes & character table

1834 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#330×C₁₃₁, #478×C₁₃₁, #479×C₁₃₁, …
maximal normal subgroup of	#330×C₁₃₁, #478×C₁₃₁, #479×C₁₃₁, …
maximal quotient of	#382×C₁₃₁, #387×C₁₃₁, #389×C₁₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4192, 32);

#125×C131 #16862132