C₃₁×#1535 #16841468

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

Aliases

direct product

C₃₁×#1535

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×#121	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

200 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 63
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 63 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)(32,33,36,41,37,40,35,39)(34,42,43,47,44,38,45,46)(48,49)(50,56,58,63)(51,55)(52,54)(53,57,61,59)(60,62);(32,34)(33,38)(35,43)(36,45)(37,44)(39,46)(40,42)(41,47)(48,50)(49,53)(51,59)(52,58)(54,61)(55,63)(56,62)(57,60);(32,35,37,36)(33,39,40,41)(34,43,44,45)(38,46,42,47)(49,54)(53,61)(55,62)(56,63)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#8489, C₃₁×#8490, C₃₁×#8500, …
maximal normal subgroup of	C₃₁×#8489, C₃₁×#8490, C₃₁×#8500, …
maximal quotient of	C₃₁×#7594, C₃₁×#7596, C₃₁×#7599, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 267);