C₃₁×#3019 #16842952

order	3968
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃₁×#3019

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

762 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 212 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)(35,43)(36,38)(37,41)(39,45)(42,47)(44,46);(32,34)(33,37)(35,44)(36,42)(38,46)(39,47)(40,43)(41,45);(35,45)(39,43)(40,47)(41,44);(33,38)(37,46)(39,43)(40,47)

Conjugacy classes & character table

899 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#23872, C₃₁×#23873, C₃₁×#23892, …
maximal normal subgroup of	C₃₁×#23872, C₃₁×#23873, C₃₁×#23892, …
maximal quotient of	C₃₁×#18132, C₃₁×#18133, C₃₁×#18137, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1751);