C₂×#341×C₁₀₁ #27167091

order	12928
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₀₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#341×C₁₀₁

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₁₀₁	D₄
commutator	C₄	C₂×C₄²×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
socle	C₂³×C₁₀₁	C₄⋊C₄
prime radical	C₂⁴×C₁₀₁	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₂³×C₄×C₁₀₁ ◁ G

392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 3 generators on 123 points:

(1,2)(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);(7,8,10,15,12,16,11,14)(9,17,18,13,20,21,19,22);(3,4,5,6)(7,9,11,19,12,20,10,18)(8,13,15,21,16,22,14,17)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#7259, C₁₀₁×#7261, C₁₀₁×#7263, …
maximal normal subgroup of	C₁₀₁×#7259, C₁₀₁×#7261, C₁₀₁×#7263, …
maximal quotient of	C₂×C₁₀₁×#1277, C₁₀₁×#7872, C₁₀₁×#7874, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 469);

C2×#341×C101 #27167091