C₃₁×#1819 #16841752

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#1819

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₃₁
Frattini, cosocle	C₂⁴	C₂³×C₃₁
prime radical	C₂⁵×C₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂⁵×C₃₁ ◁ G

504 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 112 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 — 3 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)(34,40,42,38)(35,37)(36,39)(41,46,47,44)(43,45);(33,37)(38,44)(39,45)(40,46);(32,34,35,41)(33,38,37,44)(36,42,43,47)(39,40,45,46)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7213, C₃₁×#7349, C₃₁×#8535, …
maximal normal subgroup of	C₃₁×#7213, C₃₁×#7349, C₃₁×#8535, …
maximal quotient of	C₃₁×#8698, C₃₁×#8700, C₃₁×#8702, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 551);