C₂×#14554867 #72244236

order	4374
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁷

Aliases

direct product

C₂×#14554867

Subgroups

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

2208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 83 points:

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

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#72221891
maximal subgroup of	C₂×#72221882, C₂×#72221891
maximal normal subgroup of	C₂×#72221882, C₂×#72221891

GAP

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

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

C2×#14554867 #72244236