C₃₁×#1913 #16841846

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#1913

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₁	#386
commutator	C₂³	C₂²×C₄×C₃₁
Frattini, cosocle	C₂⁴	C₂³×C₃₁
prime radical	C₃₁×#534	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂⁴×C₃₁ ◁ G

380 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 108 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)(34,37,41,45)(35,42,43,38)(36,39)(40,44)(46,47);(32,34)(33,37,39,45)(35,40)(36,41)(38,44,42,47)(43,46);(32,35)(33,38)(34,40)(36,43)(37,44)(39,42)(41,46)(45,47)

Conjugacy classes & character table

899 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#12753, C₃₁×#12754, C₃₁×#12767, …
maximal normal subgroup of	C₃₁×#12753, C₃₁×#12754, C₃₁×#12767, …
maximal quotient of	C₃₁×#9107, C₃₁×#9108, C₃₁×#9109, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 645);