C₃₁×#3417 #16843350

order	3968
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₁×#3417

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₃₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ #41×C₃₁ ◁ C₃₁×#131 ◁ G

350 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 95
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 95 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)(32,33,38,43)(34,44,48,72)(35,40,49,63)(36,52,56,70)(37,57,54,42)(39,59,62,46)(41,66,68,55)(45,71,73,93)(47,76,58,67)(50,79,77,92)(51,82,78,65)(53,69,84,61)(60,85,86,74)(64,89,88,81)(75,94,83,90)(80,91,95,87);(32,34,38,48)(33,39,43,62)(35,45,49,73)(36,53,56,84)(37,58,54,47)(40,60,63,86)(41,67,68,76)(42,69,57,61)(44,70,72,52)(46,66,59,55)(50,80,77,95)(51,83,78,75)(64,90,88,94)(65,91,82,87)(71,92,93,79)(74,89,85,81);(32,35)(33,40)(34,45)(36,50)(37,51)(38,49)(39,60)(41,64)(42,65)(43,63)(44,71)(46,74)(47,75)(48,73)(52,79)(53,80)(54,78)(55,81)(56,77)(57,82)(58,83)(59,85)(61,87)(62,86)(66,89)(67,90)(68,88)(69,91)(70,92)(72,93)(76,94)(84,95);(35,49)(40,63)(45,73)(50,77)(51,78)(60,86)(64,88)(65,82)(71,93)(74,85)(75,83)(79,92)(80,95)(81,89)(87,91)(90,94)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#13670, C₃₁×#13672, C₃₁×#33855, …
maximal normal subgroup of	C₃₁×#13670, C₃₁×#13672, C₃₁×#33855, …
maximal quotient of	C₃₁×#33382, C₃₁×#33410, C₃₁×#33455, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2149);