C₃₁×#2983 #16842916

order	3968
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#2983

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂³×C₄
prime radical	C₂³×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ G

356 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 268 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) = 55
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 55 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)(42,48)(43,46)(44,47)(45,50)(49,54)(51,53)(52,55);(40,42,44,50)(41,45,47,48)(43,51,52,49)(46,54,55,53);(33,36)(37,39)(40,43,44,52)(41,46,47,55)(42,49,50,51)(45,53,48,54);(41,47)(45,48)(46,55)(53,54)

Conjugacy classes & character table

1364 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#8838, C₃₁×#8865, C₃₁×#8962, …
maximal normal subgroup of	C₃₁×#8838, C₃₁×#8865, C₃₁×#8962, …
maximal quotient of	C₃₁×#20534, C₃₁×#20536, C₃₁×#20570, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1715);