C₃₁×#3335 #16843268

order	3968
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#3335

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#547	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ G

388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 47 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)(32,33,36,35)(34,38,39,37)(40,41,43,46)(42,44,47,45);(32,34)(33,37)(35,38)(36,39);(32,35,36,33)(34,38,39,37)(41,44)(42,47)(45,46);(34,39)(37,38)

Conjugacy classes & character table

1085 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#19766, C₃₁×#19768, C₃₁×#19792, …
maximal normal subgroup of	C₃₁×#19766, C₃₁×#19768, C₃₁×#19792, …
maximal quotient of	C₃₁×#22485, C₃₁×#22487, C₃₁×#22609, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2067);

C31×#3335 #16843268