C₃₁×#2028 #16841961

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

Aliases

direct product

C₃₁×#2028

Subgroups

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

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

normal subgroups only

This group has 246 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) = 55
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 3 generators on 55 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)(42,48)(43,51)(44,47)(45,49)(46,52)(50,55)(53,54);(32,34,35,39)(33,36,37,38)(43,52)(45,54)(46,51)(49,53);(40,42,44,50)(41,45,47,54)(43,53,52,49)(46,55,51,48)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7431, C₃₁×#12210, C₃₁×#12212, …
maximal normal subgroup of	C₃₁×#7431, C₃₁×#12210, C₃₁×#12212, …
maximal quotient of	C₃₁×#10341, C₃₁×#10342, C₃₁×#10345, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 760);