C₃₁×#1820 #16841753

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#1820

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₃₁
Frattini, cosocle	C₂⁴	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

300 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 992

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)(34,37)(35,36)(38,39)(40,41)(42,45)(43,50,51,46)(44,47)(48,54,55,52)(49,53);(33,36)(37,39)(40,42,44,49)(41,45,47,53)(43,48,51,55)(46,52,50,54);(32,34,35,38)(33,37,36,39)(40,43)(41,46)(42,48)(44,51)(45,52)(47,50)(49,55)(53,54)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7214, C₃₁×#7348, C₃₁×#8463, …
maximal normal subgroup of	C₃₁×#7214, C₃₁×#7348, C₃₁×#8463, …
maximal quotient of	C₃₁×#8699, C₃₁×#8701, C₃₁×#8703, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 552);