C₃₁×#3026 #16842959

order	3968
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃₁×#3026

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂⁴×C₃₁ ◁ G

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

normal subgroups only

This group has 800 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)(34,37)(35,38)(36,39)(40,46)(41,44)(42,45)(43,47);(32,34)(33,37)(35,40)(36,41)(38,43)(39,44)(42,47)(45,46);(32,35)(33,38)(34,40)(36,42)(37,43)(39,45)(41,47)(44,46);(33,39)(37,44)(38,45)(43,46)

Conjugacy classes & character table

899 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#23895, C₃₁×#23896, C₃₁×#23996, …
maximal normal subgroup of	C₃₁×#23895, C₃₁×#23896, C₃₁×#23996, …
maximal quotient of	C₃₁×#18284, C₃₁×#18285, C₃₁×#18286, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1758);