C₃₁×#3138 #16843071

order	3968
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₁×#3138

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

484 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 992

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

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#9424, C₃₁×#9425, C₃₁×#9438, …
maximal normal subgroup of	C₃₁×#9424, C₃₁×#9425, C₃₁×#9438, …
maximal quotient of	C₃₁×#20764, C₃₁×#21095, C₃₁×#21203, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1870);