Q₃₂×C₁₃₁ #16862120

order	4192
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₃₁ · Schur multiplier (H₂) = C₁

Aliases

direct product

Q₃₂×C₁₃₁

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂×C₁₃₁	D₈
commutator, Frattini, cosocle	C₈	C₂²×C₁₃₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₁ ◁ C₄×C₁₃₁ ◁ C₈×C₁₃₁ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 163
transitive degree	μ_t(G) = 4192

A minimal faithful permutation representation — 2 generators on 163 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	SD₆₄×C₁₃₁, Q₆₄×C₁₃₁, C₂×Q₃₂×C₁₃₁, …
maximal normal subgroup of	SD₆₄×C₁₃₁, Q₆₄×C₁₃₁, C₂×Q₃₂×C₁₃₁, …
maximal quotient of	#358×C₁₃₁, #366×C₁₃₁, C₂×Q₃₂×C₁₃₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4192, 20);

Q32×C131 #16862120