C₄⋊Q₈×C₁₀₁ #15692144

order	3232
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₄⋊Q₈×C₁₀₁
unambiguous semidirect product	C₄⋊(Q₈×C₁₀₁)

Subgroups

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

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 3 generators on 113 points:

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

Conjugacy classes & character table

1414 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#332×C₁₀₁, #333×C₁₀₁, #356×C₁₀₁, …
maximal normal subgroup of	#332×C₁₀₁, #333×C₁₀₁, #356×C₁₀₁, …
maximal quotient of	#384×C₁₀₁, #389×C₁₀₁, #391×C₁₀₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3232, 35);

C4⋊Q8×C101 #15692144