C₃₁×#1652 #16841585

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#1652

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃₁
prime radical	C₂×D₄×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

200 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 68 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)(33,37)(34,35)(38,39)(40,41,45,49)(42,50,53,54)(43,46,44,48)(47,51,55,52);(32,33,36,38)(34,37,35,39)(41,46)(43,44)(47,54)(48,49)(50,55)(51,52);(32,34,36,35)(33,37,38,39)(40,42,45,53)(41,47,49,55)(43,52,44,51)(46,54,48,50)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#1652, C₃₁×#18503, C₃₁×#18521, …
maximal normal subgroup of	C₂×C₃₁×#1652, C₃₁×#18503, C₃₁×#18521, …
maximal quotient of	C₃₁×#7775, C₃₁×#7776, C₃₁×#7783, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 384);