C₃₁×#1702 #16841635

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃₁×#1702

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃₁
prime radical	C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

162 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 64 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)(33,36)(34,37)(38,39)(40,41)(42,48)(43,45)(44,52)(46,50)(47,51)(49,55)(53,54);(32,33,37,39,35,38,34,36)(41,45)(46,54)(47,52)(48,55);(33,38)(36,39)(40,42,43,49)(41,46,45,54)(44,53,51,50)(47,55,52,48)

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#1702, C₃₁×#19155, C₃₁×#19157, …
maximal normal subgroup of	C₂×C₃₁×#1702, C₃₁×#19155, C₃₁×#19157, …
maximal quotient of	C₃₁×#8657, C₃₁×#9226, C₃₁×#9230, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 434);