C₃₁×#2222 #16842155

order	3968
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#2222

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₁	#466
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₁
prime radical	#41×C₃₁	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#131 ◁ G

140 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 3 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)(32,33,39,45)(34,46,52,74)(35,53,56,70)(36,57,59,68)(37,44,38,43)(40,61,65,49)(41,60,67,55)(42,58,69,47)(48,77,80,94)(50,73,51,72)(54,85,86,66)(62,89,90,95)(63,81,64,75)(71,82,93,79)(76,84,87,92)(78,83,88,91);(32,34,47,75,38,51,59,61,39,52,58,81,37,50,36,49)(33,40,57,73,44,64,69,74,45,65,68,72,43,63,42,46)(35,48,76,95,55,79,86,88,56,80,87,89,60,82,54,78)(41,62,84,77,53,83,85,93,67,90,92,94,70,91,66,71);(32,35)(33,41)(34,48)(36,54)(37,60)(38,55)(39,56)(40,62)(42,66)(43,70)(44,53)(45,67)(46,71)(47,76)(49,78)(50,82)(51,79)(52,80)(57,84)(58,87)(59,86)(61,88)(63,91)(64,83)(65,90)(68,92)(69,85)(72,94)(73,77)(74,93)(75,95)(81,89)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#13671, C₃₁×#13672, C₃₁×#13701, …
maximal normal subgroup of	C₃₁×#13671, C₃₁×#13672, C₃₁×#13701, …
maximal quotient of	C₃₁×#12431, C₃₁×#12434, C₃₁×#12440, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 954);