C₂²×#104×C₁₀₁ #27168253

order	12928
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#104×C₁₀₁

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₁₀₁	D₄
commutator	C₄	C₂³×C₄×C₁₀₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₁
socle	C₂³×C₁₀₁	C₂²⋊C₄
prime radical	C₂²×#41×C₁₀₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₂³×C₄×C₁₀₁ ◁ G

856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 4 generators on 113 points:

(1,2)(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);(3,4);(5,6,8,11,9,7,12,10);(5,7)(6,9)(8,10)(11,12)

Conjugacy classes & character table

5656 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#10780, C₁₀₁×#10781, C₁₀₁×#10790, …
maximal normal subgroup of	C₁₀₁×#10780, C₁₀₁×#10781, C₁₀₁×#10790, …
maximal quotient of	C₂²×D₄⋊C₈×C₁₀₁, C₂²×Q₈⋊C₈×C₁₀₁, C₂×C₁₀₁×#1476, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 1631);

C22×#104×C101 #27168253