C₃₁×#2084 #16842017

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#2084

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

230 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 3968

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)(38,43)(39,41)(40,42)(45,48)(49,50);(36,38,40,39)(37,41,42,43)(44,45,46,48)(47,49,51,50);(33,35)(36,39,40,38)(37,41,42,43)(45,49)(47,51)(48,50)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7791, C₃₁×#7793, C₃₁×#7798, …
maximal normal subgroup of	C₃₁×#7791, C₃₁×#7793, C₃₁×#7798, …
maximal quotient of	C₃₁×#10651, C₃₁×#10652, C₃₁×#10654, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 816);