C₃₁×#2061 #16841994

order	3968
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#2061

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

230 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 92 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)(32,33,36,38)(34,39,35,37)(40,41,43,47)(42,49,50,45)(44,46,52,48)(51,55,53,54);(32,34,36,35)(33,37,38,39)(40,42,43,50)(41,45,47,49)(44,53,52,51)(46,55,48,54);(32,35,36,34)(33,37,38,39)(41,46)(44,52)(45,54)(47,48)(49,55)(51,53)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7442, C₃₁×#12235, C₃₁×#12237, …
maximal normal subgroup of	C₃₁×#7442, C₃₁×#12235, C₃₁×#12237, …
maximal quotient of	C₃₁×#10568, C₃₁×#10569, C₃₁×#10571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 793);