C₃₁×#1544 #16841477

order	3968
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#1544

Subgroups

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

202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 63 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)(32,33,35,39,36,38,44,46)(34,40,41,47,42,37,43,45)(48,49)(50,57)(51,55)(52,60)(53,56)(54,58)(59,62)(61,63);(32,34,36,42)(33,37,38,40)(35,43,44,41)(39,47,46,45)(48,50,53,59)(49,54,56,63)(51,58,52,61)(55,62,60,57);(48,51,53,52)(49,55,56,60)(50,58,59,61)(54,62,63,57)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#8242, C₃₁×#8244, C₃₁×#8246, …
maximal normal subgroup of	C₃₁×#8242, C₃₁×#8244, C₃₁×#8246, …
maximal quotient of	C₃₁×#7564, C₃₁×#7568, C₃₁×#7646, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 276);

C31×#1544 #16841477