C₃₁×#1848 #16841781

order	3968
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₁×#1848

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂³×C₃₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1984

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)(36,37,40,43,41,44,39,49)(38,45,47,50,48,51,46,42);(36,38,39,46,41,48,40,47)(37,42,43,45,44,50,49,51);(32,33,34,35)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#7270, C₃₁×#7272, C₃₁×#7273, …
maximal normal subgroup of	C₃₁×#7270, C₃₁×#7272, C₃₁×#7273, …
maximal quotient of	C₃₁×#8805, C₃₁×#8806, C₃₁×#8807, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 580);

C31×#1848 #16841781