C₂×C₃₁×#417 #16842828

order	3968
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×C₃₁×#417

Subgroups

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

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 45 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#10774, C₃₁×#10781, C₃₁×#10784, …
maximal normal subgroup of	C₃₁×#10774, C₃₁×#10781, C₃₁×#10784, …
maximal quotient of	C₂×C₃₁×#1477, C₂×C₃₁×#1489, C₃₁×#17451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1627);

C2×C31×#417 #16842828