#99×C₁₃₁ #16862106

order	4192
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

#99×C₁₃₁

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃₁	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₁₃₁
Frattini, cosocle	C₂³	C₂²×C₁₃₁
prime radical	C₂×D₄×C₁₃₁	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₁ ◁ C₂³×C₁₃₁ ◁ G

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 139
transitive degree	μ_t(G) = 1048

A minimal faithful permutation representation — 2 generators on 139 points:

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

Conjugacy classes & character table

1441 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	AΣL(1,𝔽₁₆)×C₁₃₁, #352×C₁₃₁, #353×C₁₃₁, …
maximal normal subgroup of	AΣL(1,𝔽₁₆)×C₁₃₁, #352×C₁₃₁, #353×C₁₃₁, …
maximal quotient of	#323×C₁₃₁, #324×C₁₃₁, #327×C₁₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4192, 6);

#99×C131 #16862106