C₃₁×#1985 #16841918

order	3968
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₁×#1985

Subgroups

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

280 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 136 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)(33,35)(36,37)(38,42)(39,40)(41,43)(44,45,49,50,48,51,47,46);(32,33,34,35)(37,40)(41,42)(44,46,47,51,48,50,49,45);(36,38,39,43)(37,41,40,42)(44,47,48,49)(45,50,51,46)

Conjugacy classes & character table

1364 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#7061, C₃₁×#7133, C₃₁×#7284, …
maximal normal subgroup of	C₃₁×#7061, C₃₁×#7133, C₃₁×#7284, …
maximal quotient of	C₃₁×#9867, C₃₁×#9868, C₃₁×#9869, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 717);