C₃₁×#3076 #16843009

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

Aliases

direct product

C₃₁×#3076

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

388 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)(32,33,35,37)(34,38,39,36)(40,41)(42,49)(43,47)(44,52)(45,48)(46,50)(51,54)(53,55);(32,34,35,39)(33,36,37,38);(40,42)(41,46)(43,50)(44,53)(45,51)(47,54)(48,55)(49,52);(40,43,45,44)(41,47,48,52)(42,50,51,53)(46,54,55,49)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#8993, C₃₁×#8994, C₃₁×#9207, …
maximal normal subgroup of	C₃₁×#8993, C₃₁×#8994, C₃₁×#9207, …
maximal quotient of	C₃₁×#20645, C₃₁×#20872, C₃₁×#21032, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1808);

C31×#3076 #16843009