C₃₁×#2912 #16842845

order	3968
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₁×#2912

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator	C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ G

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

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 344 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) = 51
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 51 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)(32,33,34,35)(37,41)(39,46)(42,47)(43,48)(45,50)(49,51);(36,37,39,41,40,43,46,48)(38,42,45,47,44,49,50,51);(36,38)(37,42)(39,45)(40,44)(41,47)(43,49)(46,50)(48,51);(38,44)(42,49)(45,50)(47,51)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#12873, C₃₁×#12874, C₃₁×#12934, …
maximal normal subgroup of	C₃₁×#12873, C₃₁×#12874, C₃₁×#12934, …
maximal quotient of	C₃₁×#17788, C₃₁×#17798, C₃₁×#17811, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1644);