C₃₁×#1510 #16841443

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

Aliases

direct product

C₃₁×#1510

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₂×C₃₁×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

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

normal subgroups only

This group has 326 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) = 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)(34,38,39,36)(35,37)(40,41)(42,44)(43,45)(46,47);(32,34)(33,36)(35,39)(37,38)(41,44)(42,47)(45,46);(41,45)(44,46)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#8487, C₃₁×#8516, C₃₁×#8517, …
maximal normal subgroup of	C₃₁×#8487, C₃₁×#8516, C₃₁×#8517, …
maximal quotient of	C₃₁×#7592, C₃₁×#7593, C₃₁×#7599, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 242);