C₃₁×#1697 #16841630

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

Aliases

direct product

C₃₁×#1697

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃₁
prime radical	C₃₁×#459	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

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

normal subgroups only

This group has 170 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) = 47
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 47 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)(34,38)(37,39)(40,41)(42,44)(43,45)(46,47);(32,33,34,36,35,37,38,39)(41,44)(45,46);(33,37)(36,39)(41,45)(43,47)(44,46)

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#1697, C₃₁×#19142, C₃₁×#19156, …
maximal normal subgroup of	C₂×C₃₁×#1697, C₃₁×#19142, C₃₁×#19156, …
maximal quotient of	C₃₁×#8654, C₃₁×#9223, C₃₁×#9229, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 429);