C₃₁×#3021 #16842954

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

Aliases

direct product

C₃₁×#3021

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

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

normal subgroups only

This group has 736 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)(33,38)(35,36)(39,46)(40,43)(41,44)(45,47);(32,33)(34,39)(35,38)(36,43)(37,40)(41,45)(42,47)(44,46);(32,34)(33,39)(35,41)(36,44)(37,42)(38,45)(40,47)(43,46);(32,35,37,36)(33,38,40,43)(34,41,42,44)(39,45,47,46)

Conjugacy classes & character table

899 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#13353, C₃₁×#13354, C₃₁×#13360, …
maximal normal subgroup of	C₃₁×#13353, C₃₁×#13354, C₃₁×#13360, …
maximal quotient of	C₃₁×#18135, C₃₁×#18141, C₃₁×#18145, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1753);