C₃₁×#1545 #16841478

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃₁×#1545

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

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

normal subgroups only

This group has 344 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) = 47
transitive degree	μ_t(G) = 992

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)(40,41)(42,44)(43,45)(46,47);(33,36)(34,38)(41,44)(42,47)(45,46);(33,36)(37,39)(41,45)(44,46)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#8383, C₃₁×#8384, C₃₁×#8385, …
maximal normal subgroup of	C₃₁×#8383, C₃₁×#8384, C₃₁×#8385, …
maximal quotient of	C₃₁×#7572, C₃₁×#7587, C₃₁×#7606, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 277);