C₃₁×#2001 #16841934

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#2001

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

392 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 1984

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

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7078, C₃₁×#7086, C₃₁×#12105, …
maximal normal subgroup of	C₃₁×#7078, C₃₁×#7086, C₃₁×#12105, …
maximal quotient of	C₃₁×#9924, C₃₁×#9925, C₃₁×#9928, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 733);