C₄×C₃₁×#137 #16842878

order	3968
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₄×C₃₁×#137

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂×#41×C₃₁	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 51 points:

(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(5,6)(7,13)(8,16)(9,11)(10,14)(12,18)(15,20)(17,19);(5,7,9,15)(6,10,11,19)(8,17,18,14)(12,20,16,13);(6,11)(10,19)(12,16)(13,20)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#8953, C₃₁×#8955, C₃₁×#9096, …
maximal normal subgroup of	C₃₁×#8953, C₃₁×#8955, C₃₁×#9096, …
maximal quotient of	C₈×C₃₁×#137, C₃₁×#17870, C₃₁×#17872, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1677);

C4×C31×#137 #16842878