C₃₁×#2227 #16842160

order	3968
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃₁×#2227

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ C₄×C₈×C₃₁ ◁ G

130 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 3968

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)(32,33,35,37)(34,38,39,36)(41,45)(42,48)(44,51)(46,54)(47,50)(49,55)(52,53);(32,34,35,39)(33,36,37,38)(40,41,43,45)(42,46,49,53)(44,47,50,51)(48,52,55,54);(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	C₃₁×#12373, C₃₁×#12374, C₃₁×#12379, …
maximal normal subgroup of	C₃₁×#12373, C₃₁×#12374, C₃₁×#12379, …
maximal quotient of	C₃₁×#12812, C₃₁×#12936, C₃₁×#13112, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 959);