C₃₁×#2226 #16842159

order	3968
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#2226

Subgroups

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

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

normal subgroups only

This group has 142 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) = 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)(32,33,35,37)(34,38,39,36)(40,41,44,48)(42,45,51,54)(43,52,50,47)(46,55,53,49);(32,34,35,39)(33,36,37,38);(41,45)(42,49)(43,50)(46,47)(48,54)(51,55)(52,53)

Conjugacy classes & character table

899 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#12372, C₃₁×#12375, C₃₁×#12378, …
maximal normal subgroup of	C₃₁×#12372, C₃₁×#12375, C₃₁×#12378, …
maximal quotient of	C₃₁×#12811, C₃₁×#12937, C₃₁×#13111, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 958);