C₃₁×#538 #14520719

order	1984
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#538

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ G

250 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 47
transitive degree	μ_t(G) = 496

A minimal faithful permutation representation — 4 generators on 47 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)(34,40)(35,39)(36,38)(37,41)(42,44)(43,46)(45,47);(32,34,36,42)(33,37,38,45)(35,41,43,47)(39,40,46,44);(33,38)(34,41)(37,44)(39,46)(40,45)(42,47);(33,39)(37,40)(38,46)(44,45)

Conjugacy classes & character table

682 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#16844657, #16844691, #16844725, …
maximal subgroup of	C₃₁×#1600, C₃₁×#1606, C₃₁×#1612, …
maximal normal subgroup of	C₃₁×#1600, C₃₁×#1606, C₃₁×#1612, …
maximal quotient of	C₃₁×#2312, C₃₁×#2318, C₃₁×#2321, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1984, 219);