C₃₁×#1525 #16841458

order	3968
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₁×#1525

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
prime radical	C₃₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

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

normal subgroups only

This group has 270 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) = 47
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 47 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)(41,45,46,47);(33,36)(34,38)(40,41)(42,45)(43,47)(44,46);(40,42,44,43)(41,45,46,47)

Conjugacy classes & character table

1178 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#7486, C₃₁×#7492, C₃₁×#8241, …
maximal normal subgroup of	C₃₁×#7486, C₃₁×#7492, C₃₁×#8241, …
maximal quotient of	C₃₁×#7563, C₃₁×#7569, C₃₁×#7571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 257);