C₃₁×#2820 #16842753

order	3968
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₁×#2820

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

464 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 184 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)(36,37,39,41)(38,40,42,43)(45,48)(49,50);(33,35)(38,42)(40,43)(44,45,47,48)(46,49,51,50);(32,33)(34,35)(36,37)(38,40)(39,41)(42,43)(44,45)(46,50)(47,48)(49,51);(36,38)(37,40)(39,42)(41,43)(44,46,47,51)(45,49,48,50)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#10079, C₃₁×#10082, C₃₁×#10083, …
maximal normal subgroup of	C₃₁×#10079, C₃₁×#10082, C₃₁×#10083, …
maximal quotient of	C₃₁×#14624, C₃₁×#14626, C₃₁×#14628, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1552);