C₃₁×#2640 #16842573

order	3968
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₃₁×#2640

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

500 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 200 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 — 4 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)(32,33,34,35)(36,37,40,43)(38,39,42,41)(46,50)(48,51);(32,33,34,35)(36,38,40,42)(37,41,43,39)(44,45,47,49)(46,48,50,51);(32,33)(34,35)(36,38)(37,41)(39,43)(40,42)(44,46)(45,48)(47,50)(49,51);(36,39,40,41)(37,42,43,38)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#10090, C₃₁×#10092, C₃₁×#10098, …
maximal normal subgroup of	C₃₁×#10090, C₃₁×#10092, C₃₁×#10098, …
maximal quotient of	C₃₁×#14260, C₃₁×#14261, C₃₁×#14273, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1372);