C₂×#322×C₁₀₁ #27166802

order	12928
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈×C₁₀₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×#322×C₁₀₁

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₁₀₁	C₂²
commutator	C₂	C₂×C₄×C₈×C₁₀₁
Frattini, cosocle	C₄²	C₂³×C₁₀₁
socle, prime radical	C₂³×C₁₀₁	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 12928

A minimal faithful permutation representation — 3 generators on 119 points:

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

Conjugacy classes & character table

8080 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#7137, C₁₀₁×#7139, C₁₀₁×#7145, …
maximal normal subgroup of	C₁₀₁×#7137, C₁₀₁×#7139, C₁₀₁×#7145, …
maximal quotient of	C₄×#322×C₁₀₁, C₂×C₁₀₁×#1273, C₁₀₁×#8131, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 180);

C2×#322×C101 #27166802