C₃₁×#3249 #16843182

order	3968
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#3249

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂³×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

348 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#10030, C₃₁×#10031, C₃₁×#10195, …
maximal normal subgroup of	C₃₁×#10030, C₃₁×#10031, C₃₁×#10195, …
maximal quotient of	C₃₁×#21885, C₃₁×#21961, C₃₁×#22113, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1981);