C₃₁×#2101 #16842034

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#2101

Subgroups

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

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

normal subgroups only

This group has 198 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) = 3968

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)(42,44)(43,45)(46,47)(48,49,51,52)(50,53,55,54);(32,34,36,35)(33,37,38,39)(40,42,43,46)(41,44,45,47)(48,50,51,55)(49,53,52,54);(32,35,36,34)(33,37,38,39)(41,45)(44,47)(49,52)(50,54)(53,55)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#2101, C₃₁×#23744, C₃₁×#23746, …
maximal normal subgroup of	C₂×C₃₁×#2101, C₃₁×#23744, C₃₁×#23746, …
maximal quotient of	C₃₁×#10689, C₃₁×#11208, C₃₁×#11268, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 833);