C₃₁×#1954 #16841887

order	3968
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₃₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₁×#1954

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂³
commutator	C₂²	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
socle	C₂³×C₃₁	C₂²×C₄
prime radical	C₂²×D₄×C₃₁	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

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

normal subgroups only

This group has 396 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) = 47
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 47 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)(37,39)(40,41,43,46,44,42,45,47);(32,33,34,35)(40,42,43,47,44,41,45,46);(36,37,38,39)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#7056, C₃₁×#7102, C₃₁×#7205, …
maximal normal subgroup of	C₃₁×#7056, C₃₁×#7102, C₃₁×#7205, …
maximal quotient of	C₃₁×#9637, C₃₁×#9638, C₃₁×#9639, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 686);