C₃₁×#1372 #16841305

order	3968
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₃₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃₁×#1372

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	D₈
commutator	C₈	C₂×C₈×C₃₁
Frattini, cosocle	C₄×C₈	C₂²×C₃₁
socle, prime radical	C₂²×C₃₁	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄²×C₃₁ ◁ C₄×C₈×C₃₁ ◁ G

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 3968

A minimal faithful permutation representation — 2 generators on 55 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)(32,33,34,36,35,37,38,39)(41,45)(42,49)(43,50)(46,51)(47,55)(48,54)(52,53);(40,41,46,55,43,47,42,45,44,48,49,53,50,52,51,54)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#7385, C₃₁×#7386, C₃₁×#7446, …
maximal normal subgroup of	C₃₁×#7385, C₃₁×#7386, C₃₁×#7446, …
maximal quotient of	C₃₁×#7101, C₃₁×#7154, C₃₁×#7463, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 104);

C31×#1372 #16841305