C₃₁×#1736 #16841669

order	3968
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₁×#1736

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₄
commutator	C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

448 conjugacy classes of subgroups.

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#10742, C₃₁×#10749, C₃₁×#10750, …
maximal normal subgroup of	C₃₁×#10742, C₃₁×#10749, C₃₁×#10750, …
maximal quotient of	C₃₁×#7873, C₃₁×#7876, C₃₁×#7879, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 468);

C31×#1736 #16841669