C₂×#72221882 #72244263

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

Aliases

direct product

C₂×#72221882

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

28452 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)(5,13,34)(9,25,14)(15,37,35)(16,18,42)(22,26,52)(27,36,38)(28,53,30)(39,44,40)(41,45,66)(47,56,54)(48,55,70)(57,58,61)(59,74,62)(64,67,65)(68,76,75)(69,72,82)(77,78,79)(80,83,81);(3,4,8)(5,14,22)(6,10,23)(7,12,24)(9,26,13)(11,33,21)(15,38,47)(16,30,48)(17,32,51)(18,28,55)(19,31,49)(20,29,50)(25,52,34)(27,56,37)(35,36,54)(39,61,68)(40,58,75)(41,62,69)(42,53,70)(43,63,71)(44,57,76)(45,59,72)(46,60,73)(64,79,80)(65,78,81)(66,74,82)(67,77,83);(6,17,20)(10,29,32)(16,18,42)(28,30,53)(31,60,63)(39,40,44)(49,71,73)(57,58,61)(59,62,74)(69,72,82)(77,78,79)(80,81,83);(3,5,15,7,13,35,21,34,37)(4,9,27,12,25,38,11,14,36)(6,18,44,17,42,39,20,16,40)(8,22,47,24,26,54,33,52,56)(10,30,61,32,28,58,29,53,57)(19,41,64,43,45,65,46,66,67)(23,48,68,51,55,75,50,70,76)(31,62,79,63,59,78,60,74,77)(49,72,83,71,82,80,73,69,81);(3,6,19)(4,10,31)(5,16,41)(7,17,43)(8,23,49)(9,28,59)(11,29,60)(12,32,63)(13,18,45)(14,30,62)(15,39,64)(20,46,21)(22,48,69)(24,51,71)(25,53,74)(26,55,72)(27,57,77)(33,50,73)(34,42,66)(35,40,65)(36,58,78)(37,44,67)(38,61,79)(47,68,80)(52,70,82)(54,75,81)(56,76,83)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72221920, C₂×#72221899
maximal subgroup of	C₂×#72221920, C₂×#72221899
maximal normal subgroup of	C₂×#72221920, C₂×#72221899

GAP

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

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

C2×#72221882 #72244263