C₂×C₃₁×M₆₄ #16842190

order	3968
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×C₃₁×M₆₄

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₃₁	C₂²
commutator	C₂	C₂²×C₁₆×C₃₁
Frattini, cosocle	C₁₆	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₁₆
prime radical	C₂³×C₃₁	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₃₁ ◁ G

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 65
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 65 points:

(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65);(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);(4,20)(6,22)(8,24)(10,26)(12,28)(14,30)(16,32)(18,34)

Conjugacy classes & character table

2480 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#7466, C₃₁×#7467, C₃₁×#7468, …
maximal normal subgroup of	C₃₁×#7466, C₃₁×#7467, C₃₁×#7468, …
maximal quotient of	C₂×C₃₁×#1397, C₄×C₃₁×M₆₄, C₂×C₃₁×C₂²⋊C₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 989);

C2×C31×M64 #16842190