C₃₁×#3361 #16843294

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×#3361

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#459	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ G

362 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)(40,41,44,48)(42,45,50,54)(43,46,51,47)(49,52,55,53);(32,33,35,36)(34,38,39,37)(40,42,44,50)(41,45,48,54)(43,49,51,55)(46,52,47,53);(41,46)(43,51)(45,52)(47,48)(49,55)(53,54);(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×#19789, C₃₁×#19798, C₃₁×#19825, …
maximal normal subgroup of	C₃₁×#19789, C₃₁×#19798, C₃₁×#19825, …
maximal quotient of	C₃₁×#22519, C₃₁×#22543, C₃₁×#22739, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2093);

C31×#3361 #16843294