C₂×C₂²⋊C₄×C₁₀₁ #15692131

order	3232
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×C₂²⋊C₄×C₁₀₁

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₀₁	C₂²
commutator	C₂	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂²	C₂³×C₁₀₁
prime radical	C₂⁴×C₁₀₁	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₁ ◁ G

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 1616

A minimal faithful permutation representation — 3 generators on 111 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);(3,4,6,8)(5,7,9,10);(4,7)(8,10)

Conjugacy classes & character table

2020 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#323×C₁₀₁, #342×C₁₀₁, C₄×C₂²⋊C₄×C₁₀₁, …
maximal normal subgroup of	#323×C₁₀₁, #342×C₁₀₁, C₄×C₂²⋊C₄×C₁₀₁, …
maximal quotient of	C₂×#95×C₁₀₁, C₄×C₂²⋊C₄×C₁₀₁, #379×C₁₀₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3232, 22);

C2×C22⋊C4×C101 #15692131