C₂×C₃₁×#368 #16842093

order	3968
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×C₃₁×#368

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	D₈
commutator	C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	#107
prime radical	C₂³×C₃₁	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ G

144 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);(3,4,7,12,9,13,8,23)(5,14,17,29,19,30,18,33)(6,20,21,27,22,34,15,11)(10,24,25,32,26,31,28,16);(3,5,15,31,8,18,22,32,9,19,21,24,7,17,6,16)(4,10,20,29,12,25,27,30,13,26,34,33,23,28,11,14)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#7472, C₃₁×#7473, C₃₁×#7474, …
maximal normal subgroup of	C₃₁×#7472, C₃₁×#7473, C₃₁×#7474, …
maximal quotient of	C₂×C₃₁×#1371, C₂×C₃₁×#1372, C₃₁×#12008, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 892);

C2×C31×#368 #16842093