C₄×#108×C₁₀₁ #27167131

order	12928
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₀₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₄×#108×C₁₀₁

Subgroups

	subgroup	G/H
center	C₄²×C₁₀₁	D₄
commutator	C₄	C₂×C₄²×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
socle	C₂²×C₁₀₁	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₁	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 3 generators on 121 points:

(1,2,3,4)(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);(5,6,9,12,10,13,8,18)(7,14,16,19,17,20,15,11);(5,7,8,15,10,17,9,16)(6,11,12,14,13,19,18,20)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#7172, C₁₀₁×#7173, C₁₀₁×#7176, …
maximal normal subgroup of	C₁₀₁×#7172, C₁₀₁×#7173, C₁₀₁×#7176, …
maximal quotient of	C₄×#334×C₁₀₁, C₄×#335×C₁₀₁, C₁₀₁×#8273, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 509);

C4×#108×C101 #27167131