C₃₁×#1984 #16841917

order	3968
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₃₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃₁×#1984

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂³
commutator	C₂²	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

292 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 160 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)(44,45,47,50,48,46,49,51);(32,33,34,35)(37,40)(41,42)(44,46,47,51,48,45,49,50);(36,38,39,43)(37,41,40,42)

Conjugacy classes & character table

1736 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#7057, C₃₁×#7128, C₃₁×#7206, …
maximal normal subgroup of	C₃₁×#7057, C₃₁×#7128, C₃₁×#7206, …
maximal quotient of	C₃₁×#9864, C₃₁×#9865, C₃₁×#9866, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 716);