C₃₁×#3108 #16843041

order	3968
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#3108

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₃₁×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 55 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)(34,37)(35,36)(38,39)(40,41)(42,48)(43,45)(44,52)(46,50)(47,51)(49,55)(53,54);(33,36)(37,39)(41,45)(46,54)(47,52)(48,55);(40,42,43,49)(41,46,45,54)(44,53,51,50)(47,55,52,48);(32,34,35,38)(33,37,36,39)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#9051, C₃₁×#17911, C₃₁×#17915, …
maximal normal subgroup of	C₃₁×#9051, C₃₁×#17911, C₃₁×#17915, …
maximal quotient of	C₃₁×#20735, C₃₁×#21116, C₃₁×#21120, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1840);

C31×#3108 #16843041