C₃₁×#3107 #16843040

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

Aliases

direct product

C₃₁×#3107

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#41×C₃₁	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)(40,41,43,47)(42,49,50,45)(44,46,52,48)(51,54,55,53);(33,36)(37,39)(40,42,43,50)(41,45,47,49)(44,51,52,55)(46,53,48,54);(41,46)(44,52)(45,53)(47,48)(49,54)(51,55);(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×#8971, C₃₁×#8972, C₃₁×#9212, …
maximal normal subgroup of	C₃₁×#8971, C₃₁×#8972, C₃₁×#9212, …
maximal quotient of	C₃₁×#20737, C₃₁×#21114, C₃₁×#21116, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1839);

C31×#3107 #16843040