C₃₁×#3365 #16843298

order	3968
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#3365

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ G

348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 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)(33,36)(37,38)(40,41,45,48)(42,46,44,52)(43,50,51,54)(47,53,55,49);(32,33,35,36)(34,38,39,37)(40,42,45,44)(41,46,48,52)(43,49,51,53)(47,54,55,50);(40,43,45,51)(41,47,48,55)(42,49,44,53)(46,54,52,50);(32,34,35,39)(33,37,36,38)

Conjugacy classes & character table

1085 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#19833, C₃₁×#19834, C₃₁×#19837, …
maximal normal subgroup of	C₃₁×#19833, C₃₁×#19834, C₃₁×#19837, …
maximal quotient of	C₃₁×#22521, C₃₁×#22543, C₃₁×#22741, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2097);

C31×#3365 #16843298