C₂×C₄⋊C₄×C₁₃₁ #16862123

order	4192
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₄⋊C₄×C₁₃₁

Subgroups

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

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 4192

A minimal faithful permutation representation — 3 generators on 141 points:

(1,2)(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);(4,6)(7,8,9,10);(3,4,5,6)

Conjugacy classes & character table

2620 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#324×C₁₃₁, #340×C₁₃₁, #343×C₁₃₁, …
maximal normal subgroup of	#324×C₁₃₁, #340×C₁₃₁, #343×C₁₃₁, …
maximal quotient of	C₂×#95×C₁₃₁, C₄×C₄⋊C₄×C₁₃₁, #380×C₁₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4192, 23);

C2×C4⋊C4×C131 #16862123