C₃₁×#2040 #16841973

order	3968
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₁×#2040

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₃₁×#537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

290 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 51
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 3 generators on 51 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)(33,34)(36,37)(38,46)(39,44)(40,43)(41,45)(42,48)(47,51)(49,50);(32,33)(34,35)(36,38)(37,42)(39,48)(40,49)(41,47)(43,51)(44,46)(45,50);(32,34,35,33)(37,43)(38,47)(40,41)(44,45)(48,49)(50,51)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7416, C₃₁×#12141, C₃₁×#12143, …
maximal normal subgroup of	C₃₁×#7416, C₃₁×#12141, C₃₁×#12143, …
maximal quotient of	C₃₁×#10403, C₃₁×#10404, C₃₁×#10407, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 772);