C₃₁×#1806 #16841739

order	3968
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₁×#1806

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×C₂²⋊C₄
prime radical	C₂×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 992

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

Conjugacy classes & character table

1178 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#8420, C₃₁×#8421, C₃₁×#8422, …
maximal normal subgroup of	C₃₁×#8420, C₃₁×#8421, C₃₁×#8422, …
maximal quotient of	C₃₁×#8563, C₃₁×#8564, C₃₁×#8565, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 538);

C31×#1806 #16841739