C₃₁×#2027 #16841960

order	3968
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₁×#2027

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

282 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 992

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

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7430, C₃₁×#12208, C₃₁×#12209, …
maximal normal subgroup of	C₃₁×#7430, C₃₁×#12208, C₃₁×#12209, …
maximal quotient of	C₃₁×#10340, C₃₁×#10343, C₃₁×#10344, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 759);