C₃₁×#2163 #16842096

order	3968
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#2163

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	C₂³×C₃₁	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ G

152 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 55
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 55 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)(40,41,42,46,43,47,48,53,44,45,49,51,50,52,54,55);(33,36)(37,39);(32,34,35,38)(33,37,36,39)(41,45)(46,51)(47,52)(53,55)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#7143, C₃₁×#7144, C₃₁×#7146, …
maximal normal subgroup of	C₃₁×#7143, C₃₁×#7144, C₃₁×#7146, …
maximal quotient of	C₃₁×#12591, C₃₁×#12599, C₃₁×#12653, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 895);