C₃₁×#1556 #16841489

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₁×#1556

Subgroups

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

228 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 96 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)(32,33,35,37,36,38,34,39)(40,41,43,47,44,46,52,54)(42,48,49,55,50,45,51,53);(33,37)(34,35)(38,39)(40,42,44,50)(41,45,46,48)(43,51,52,49)(47,55,54,53);(33,38)(37,39)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#8499, C₃₁×#8523, C₃₁×#8525, …
maximal normal subgroup of	C₃₁×#8499, C₃₁×#8523, C₃₁×#8525, …
maximal quotient of	C₃₁×#7593, C₃₁×#7604, C₃₁×#7627, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 288);