C₃₁×#1776 #16841709

order	3968
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃₁×#1776

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₄
commutator	C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

272 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 3968

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

Conjugacy classes & character table

1364 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#12650, C₃₁×#12677, C₃₁×#13119, …
maximal normal subgroup of	C₃₁×#12650, C₃₁×#12677, C₃₁×#13119, …
maximal quotient of	C₃₁×#8275, C₃₁×#8276, C₃₁×#8279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 508);