C₃₁×#1669 #16841602

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#1669

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃₁
prime radical	C₂×D₄×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

172 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 72 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)(34,36)(35,38)(37,39)(40,41,44,47)(42,48,51,55)(43,52,49,46)(45,53,54,50);(40,42,49,53,44,51,43,50)(41,45,52,55,47,54,46,48);(33,36)(34,39)(37,38)

Conjugacy classes & character table

899 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#12320, C₃₁×#12321, C₃₁×#12322, …
maximal normal subgroup of	C₃₁×#12320, C₃₁×#12321, C₃₁×#12322, …
maximal quotient of	C₃₁×#8644, C₃₁×#9219, C₃₁×#9561, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 401);