C₃₁×#1533 #16841466

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₁×#1533

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₄²	C₂³×C₃₁
prime radical	C₂²×D₄×C₃₁	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

248 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 88 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)(42,48,50,55)(43,47)(44,46)(45,49,53,51)(52,54);(33,36)(34,38)(40,42)(41,45)(43,51)(44,50)(46,53)(47,55)(48,54)(49,52);(41,46)(45,53)(47,54)(48,55)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#7481, C₃₁×#7495, C₃₁×#8522, …
maximal normal subgroup of	C₃₁×#7481, C₃₁×#7495, C₃₁×#8522, …
maximal quotient of	C₃₁×#7592, C₃₁×#7595, C₃₁×#7601, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 265);