C₁₁₃×Q₈⋊C₁₆ #28386133

order	14464
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₁₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₁₁₃×Q₈⋊C₁₆
unambiguous semidirect product	Q₈⋊(C₁₆×C₁₁₃)

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₁₃	D₄
commutator	C₄	C₂×C₁₆×C₁₁₃
Frattini, cosocle	C₄×C₈	C₂²×C₁₁₃
socle, prime radical	C₂²×C₁₁₃	C₂²⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₁₃ ◁ C₄×C₈×C₁₁₃ ◁ G

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 137
transitive degree	μ_t(G) = 14464

A minimal faithful permutation representation — 2 generators on 137 points:

(2,5)(3,7)(6,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,2,4,6)(3,8,7,5)

Conjugacy classes & character table

6328 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₁₃×Q₈⋊C₁₆, C₁₁₃×#11606, C₁₁₃×#11607, …
maximal normal subgroup of	C₂×C₁₁₃×Q₈⋊C₁₆, C₁₁₃×#11606, C₁₁₃×#11607, …
maximal quotient of	C₁₁₃×#7118, C₁₁₃×#7128, C₁₁₃×#7167, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14464, 69);

C113×Q8⋊C16 #28386133