C₃₁×#2169 #16842102

order	3968
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#2169

Subgroups

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂³
commutator	C₂²	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂²×C₈
prime radical	C₂×D₄×C₃₁	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ G

162 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 51
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 51 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)(34,35)(36,37,40,44,41,38,45,47,39,42,48,50,49,46,43,51);(33,35)(37,42)(38,46)(44,50)(47,51);(36,38,40,47,41,42,45,50,39,46,48,51,49,37,43,44)

Conjugacy classes & character table

1736 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#7380, C₃₁×#12038, C₃₁×#12039, …
maximal normal subgroup of	C₃₁×#7380, C₃₁×#12038, C₃₁×#12039, …
maximal quotient of	C₃₁×#12039, C₃₁×#12051, C₃₁×#12054, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 901);