C₂×#72221849 #72244264

order	13122
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁵ · Schur multiplier (H₂) = C₃¹¹

Aliases

direct product

C₂×#72221849

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₃×H(3,𝔽₃)²
commutator, Frattini, cosocle	C₃³	C₂×C₃⁵
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃⁴ ◁ G

29148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 5 generators on 83 points:

(1,2)(3,4,8)(5,9,21)(6,11,23)(7,12,22)(10,28,20)(13,24,40)(14,27,43)(15,25,42)(16,26,41)(17,30,44)(18,31,46)(19,29,45)(32,48,61)(33,49,60)(34,52,64)(35,50,63)(36,51,65)(37,47,62)(38,54,66)(39,53,67)(55,69,76)(56,68,77)(57,70,75)(58,72,79)(59,71,78)(73,81,82)(74,80,83);(3,5,13)(4,9,24)(6,15,32)(7,14,33)(8,21,40)(10,26,47)(11,25,48)(12,27,49)(16,37,20)(17,34,57)(18,36,55)(19,35,56)(22,43,60)(23,42,61)(28,41,62)(29,50,68)(30,52,70)(31,51,69)(38,58,73)(39,59,74)(44,64,75)(45,63,77)(46,65,76)(53,71,80)(54,72,81)(66,79,82)(67,78,83);(6,17,19)(11,30,29)(15,35,34)(23,44,45)(25,50,52)(36,59,58)(42,63,64)(51,71,72)(55,73,74)(65,78,79)(69,81,80)(76,82,83);(4,10,12)(5,14,16)(6,17,19)(8,22,28)(21,41,43)(23,45,44)(24,47,49)(25,50,52)(31,53,54)(40,60,62)(42,64,63)(46,66,67)(48,68,70)(51,71,72)(55,73,74)(61,75,77)(65,79,78)(76,83,82);(3,6,18)(4,11,31)(5,15,36)(7,17,38)(8,23,46)(9,25,51)(10,29,53)(12,30,54)(13,32,55)(14,34,58)(16,35,59)(19,39,20)(21,42,65)(22,44,66)(24,48,69)(26,50,71)(27,52,72)(28,45,67)(33,57,73)(37,56,74)(40,61,76)(41,63,78)(43,64,79)(47,68,80)(49,70,81)(60,75,82)(62,77,83)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72221898, C₂×#72221884
maximal subgroup of	C₂×#72221898, C₂×#72221884
maximal normal subgroup of	C₂×#72221898, C₂×#72221884

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2)(3,4,8)(5,9,21)(6,11,23)(7,12,22)(10,28,20)(13,24,40)(14,27,43)(15,25,42)(16,26,41)(17,30,44)(18,31,46)(19,29,45)(32,48,61)(33,49,60)(34,52,64)(35,50,63)(36,51,65)(37,47,62)(38,54,66)(39,53,67)(55,69,76)(56,68,77)(57,70,75)(58,72,79)(59,71,78)(73,81,82)(74,80,83), (3,5,13)(4,9,24)(6,15,32)(7,14,33)(8,21,40)(10,26,47)(11,25,48)(12,27,49)(16,37,20)(17,34,57)(18,36,55)(19,35,56)(22,43,60)(23,42,61)(28,41,62)(29,50,68)(30,52,70)(31,51,69)(38,58,73)(39,59,74)(44,64,75)(45,63,77)(46,65,76)(53,71,80)(54,72,81)(66,79,82)(67,78,83), (6,17,19)(11,30,29)(15,35,34)(23,44,45)(25,50,52)(36,59,58)(42,63,64)(51,71,72)(55,73,74)(65,78,79)(69,81,80)(76,82,83), (4,10,12)(5,14,16)(6,17,19)(8,22,28)(21,41,43)(23,45,44)(24,47,49)(25,50,52)(31,53,54)(40,60,62)(42,64,63)(46,66,67)(48,68,70)(51,71,72)(55,73,74)(61,75,77)(65,79,78)(76,83,82), (3,6,18)(4,11,31)(5,15,36)(7,17,38)(8,23,46)(9,25,51)(10,29,53)(12,30,54)(13,32,55)(14,34,58)(16,35,59)(19,39,20)(21,42,65)(22,44,66)(24,48,69)(26,50,71)(27,52,72)(28,45,67)(33,57,73)(37,56,74)(40,61,76)(41,63,78)(43,64,79)(47,68,80)(49,70,81)(60,75,82)(62,77,83));

C2×#72221849 #72244264