C₃₁×#1557 #16841490

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×#1557

Subgroups

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

246 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 92 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)(33,37)(34,35)(38,39)(40,41,43,45,44,46,42,47);(32,33)(34,37)(35,39)(36,38)(41,45)(42,43)(46,47);(32,34,36,35)(33,37,38,39)(41,46)(45,47)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#8483, C₃₁×#8486, C₃₁×#8491, …
maximal normal subgroup of	C₃₁×#8483, C₃₁×#8486, C₃₁×#8491, …
maximal quotient of	C₃₁×#7594, C₃₁×#7599, C₃₁×#7628, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 289);