C₃₁×#1998 #16841931

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₁×#1998

Subgroups

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

228 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 112 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)(34,38)(35,36)(37,39)(40,41,44,47,43,46,48,54)(42,45,50,53,49,52,51,55);(33,36)(37,38)(40,42,48,50,43,49,44,51)(41,45,47,55,46,52,54,53);(32,34,35,39)(33,37,36,38)(41,46)(45,52)(47,54)(53,55)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#8485, C₃₁×#8486, C₃₁×#10497, …
maximal normal subgroup of	C₃₁×#8485, C₃₁×#8486, C₃₁×#10497, …
maximal quotient of	C₃₁×#9880, C₃₁×#9881, C₃₁×#9892, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 730);