C₂²×C₃₁×C₄⋊Q₈ #16843374

order	3968
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃₁ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×C₃₁×C₄⋊Q₈

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃₁	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

1624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 3968

A minimal faithful permutation representation — 5 generators on 47 points:

(1,2)(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);(3,4);(6,8)(9,10,13,16)(11,15,12,14);(9,11,13,12)(10,14,16,15);(5,6,7,8)(9,12,13,11)(10,15,16,14)

Conjugacy classes & character table

1736 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#8340, C₃₁×#8342, C₃₁×#8551, …
maximal normal subgroup of	C₃₁×#8340, C₃₁×#8342, C₃₁×#8551, …
maximal quotient of	C₂²×C₃₁×#384, C₂²×C₃₁×#389, C₂²×C₃₁×#391, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2173);

C22×C31×C4⋊Q8 #16843374