C₃₁×#3338 #16843271

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×#3338

Subgroups

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

356 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)(32,33,36,35)(34,38,39,37)(40,41)(42,49)(43,47)(44,46)(45,50)(48,52)(51,54)(53,55);(32,34)(33,37)(35,38)(36,39)(40,42)(41,45)(43,50)(44,51)(46,53)(47,49)(48,54)(52,55);(32,35,36,33)(34,38,39,37)(41,46)(44,52)(45,53)(47,48)(49,54)(51,55);(34,39)(37,38)(42,50)(45,49)(51,55)(53,54)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#12342, C₃₁×#12346, C₃₁×#12394, …
maximal normal subgroup of	C₃₁×#12342, C₃₁×#12346, C₃₁×#12394, …
maximal quotient of	C₃₁×#22487, C₃₁×#22612, D₄×C₃₁×#107, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2070);

C31×#3338 #16843271