C₃₁×#2004 #16841937

order	3968
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#2004

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₃₁×#535	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

394 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 92 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)(32,33)(34,35)(36,37)(38,44)(39,47)(40,43)(41,45)(42,49)(46,51)(48,50);(36,38,40,46)(37,41,43,50)(39,48,49,45)(42,51,47,44);(33,35)(39,49)(41,50)(42,47)(45,48)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#2004, C₃₁×#23399, C₃₁×#23401, …
maximal normal subgroup of	C₂×C₃₁×#2004, C₃₁×#23399, C₃₁×#23401, …
maximal quotient of	C₃₁×#9916, C₃₁×#9918, C₃₁×#9924, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 736);