C₂×#431×C₁₀₁ #27168271

order	12928
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈×C₁₀₁ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#431×C₁₀₁

Subgroups

	subgroup	G/H
center	C₂×C₄²×C₁₀₁	C₂²
commutator	C₂	C₂³×C₈×C₁₀₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₁
socle	C₂³×C₁₀₁	C₂²×C₄
prime radical	C₂⁴×C₁₀₁	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 4 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);(3,4,5,7,6,8,9,10)(11,12)(13,16)(14,15)(17,18);(12,15)(16,18);(11,13,14,17)(12,16,15,18)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#7848, C₁₀₁×#7849, C₁₀₁×#7850, …
maximal normal subgroup of	C₁₀₁×#7848, C₁₀₁×#7849, C₁₀₁×#7850, …
maximal quotient of	C₂×C₁₀₁×#1744, C₂×C₄×C₂²⋊C₈×C₁₀₁, C₂×C₄×C₄⋊C₈×C₁₀₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 1649);

C2×#431×C101 #27168271