C₃₁×#2208 #16842141

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

Aliases

direct product

C₃₁×#2208

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

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

normal subgroups only

This group has 166 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) = 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)(33,35)(36,37,40,44)(38,41,47,50)(39,48,46,43)(42,51,49,45);(32,33,34,35);(37,41)(38,45)(39,46)(42,43)(44,50)(47,51)(48,49)

Conjugacy classes & character table

899 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#12320, C₃₁×#12323, C₃₁×#12326, …
maximal normal subgroup of	C₃₁×#12320, C₃₁×#12323, C₃₁×#12326, …
maximal quotient of	C₃₁×#12420, C₃₁×#12426, C₃₁×#12447, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 940);