Q₈×C₁₆×C₃₁ #16842115

order	3968
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

Q₈×C₁₆×C₃₁

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₃₁	C₂²
commutator	C₂	C₂²×C₁₆×C₃₁
Frattini, cosocle	C₂×C₈	C₂³×C₃₁
socle, prime radical	C₂²×C₃₁	C₂²×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₃₁ ◁ G

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 3968

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8);(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)

Conjugacy classes & character table

2480 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×Q₈⋊C₃₂, C₁₆×SD₁₆×C₃₁, C₃₁×#12050, …
maximal normal subgroup of	C₃₁×Q₈⋊C₃₂, C₁₆×SD₁₆×C₃₁, C₃₁×#12050, …
maximal quotient of	C₁₆×C₄⋊C₄×C₃₁, C₃₁×#13104, C₃₁×#13159, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 914);

Q8×C16×C31 #16842115