D₁₆×C₁₃₁ #16862118

order	4192
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₃₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

D₁₆×C₁₃₁

Subgroups

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

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 147
transitive degree	μ_t(G) = 2096

A minimal faithful permutation representation — 2 generators on 147 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₃₂×C₁₃₁, SD₆₄×C₁₃₁, C₂×D₁₆×C₁₃₁, …
maximal normal subgroup of	D₃₂×C₁₃₁, SD₆₄×C₁₃₁, C₂×D₁₆×C₁₃₁, …
maximal quotient of	#357×C₁₃₁, #366×C₁₃₁, D₃₂×C₁₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4192, 18);

D16×C131 #16862118