C₃₁×#1778 #16841711

order	3968
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₁×#1778

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂³×C₃₁	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 51 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)(32,33,35,34)(36,37,40,45,41,43,39,44)(38,46,48,42,49,51,47,50);(36,38,39,48,41,49,40,47)(37,42,44,51,43,50,45,46);(32,34,35,33)(37,43)(42,50)(44,45)(46,51)

Conjugacy classes & character table

1364 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#12651, C₃₁×#12678, C₃₁×#13114, …
maximal normal subgroup of	C₃₁×#12651, C₃₁×#12678, C₃₁×#13114, …
maximal quotient of	C₃₁×#8275, C₃₁×#8276, C₃₁×#8277, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 510);

C31×#1778 #16841711