C₂×#427×C₁₀₁ #27168263

order	12928
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#427×C₁₀₁

Subgroups

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

512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 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);(3,4,5,6)(7,8)(9,16)(10,14)(11,20)(12,15)(13,18)(17,21)(19,22);(7,9,11,18,12,19,10,17)(8,13,14,16,15,21,20,22);(7,10,12,11)(8,14,15,20)(9,17,19,18)(13,16,21,22)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#10859, C₁₀₁×#10860, C₁₀₁×#10866, …
maximal normal subgroup of	C₁₀₁×#10859, C₁₀₁×#10860, C₁₀₁×#10866, …
maximal quotient of	C₂×C₁₀₁×#1564, C₂×C₁₀₁×#1568, C₁₀₁×#17793, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 1641);

C2×#427×C101 #27168263