C₃₁×#1674 #16841607

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

Aliases

direct product

C₃₁×#1674

Subgroups

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

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

normal subgroups only

This group has 156 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) = 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)(33,36)(34,37)(38,39)(40,41,44,47)(42,48,50,55)(43,51,53,46)(45,52,54,49);(32,33,37,39,35,38,34,36);(40,42,44,50)(41,45,47,54)(43,52,53,49)(46,55,51,48)

Conjugacy classes & character table

899 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#12380, C₃₁×#12381, C₃₁×#12382, …
maximal normal subgroup of	C₃₁×#12380, C₃₁×#12381, C₃₁×#12382, …
maximal quotient of	C₃₁×#8645, C₃₁×#9218, C₃₁×#9566, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 406);