C₃₁×#2234 #16842167

order	3968
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃₁×#2234

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₁
prime radical	C₃₁×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ G

142 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 60 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)(41,45)(42,48)(44,51)(46,54)(47,50)(49,55)(52,53);(33,36)(37,39)(40,41,43,45)(42,46,49,53)(44,47,50,51)(48,52,55,54);(32,34,35,38)(33,37,36,39)(40,42,43,49)(41,46,45,53)(44,52,50,54)(47,55,51,48)

Conjugacy classes & character table

899 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#2234, C₃₁×#33645, C₃₁×#33647, …
maximal normal subgroup of	C₂×C₃₁×#2234, C₃₁×#33645, C₃₁×#33647, …
maximal quotient of	C₃₁×#12868, C₃₁×#12921, C₃₁×#12992, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 966);