C₃₁×#3296 #16843229

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×#3296

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#546	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ G

408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 992

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₃₁×#3296, C₃₁×#61124, C₃₁×#61156, …
maximal normal subgroup of	C₂×C₃₁×#3296, C₃₁×#61124, C₃₁×#61156, …
maximal quotient of	C₃₁×#22414, C₃₁×#22486, C₃₁×#22570, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2028);

C31×#3296 #16843229