C₄×C₄⋊Q₈×C₁₀₁ #27167661

order	12928
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₄×C₄⋊Q₈×C₁₀₁

Subgroups

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

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 12928

A minimal faithful permutation representation — 4 generators on 117 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101)(102,103,104,105);(107,109)(110,111,113,115)(112,116,117,114);(110,112,113,117)(111,114,115,116);(106,107,108,109)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#7570, C₁₀₁×#7571, C₁₀₁×#7588, …
maximal normal subgroup of	C₁₀₁×#7570, C₁₀₁×#7571, C₁₀₁×#7588, …
maximal quotient of	C₄×#389×C₁₀₁, C₄×#391×C₁₀₁, C₄×#384×C₁₀₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 1039);

C4×C4⋊Q8×C101 #27167661