C₂×#14554346 #72244228

order	4374
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁴ · Schur multiplier (H₂) = C₃⁸

Aliases

direct product

C₂×#14554346

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₃×C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₂×C₃⁴
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃⁴ ◁ G

4084 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)(6,17,19)(10,28,30)(15,38,41)(18,44,43)(22,49,51)(26,33,58)(29,60,59)(35,39,65)(40,45,66)(47,69,53)(50,73,72)(55,56,62)(57,61,63)(64,80,67)(68,70,75)(71,74,78)(76,79,77)(81,83,82);(3,4,8)(5,13,21)(6,10,22)(7,12,23)(9,24,16)(11,31,20)(14,34,46)(15,33,47)(17,28,49)(18,29,50)(19,30,51)(25,54,36)(26,53,41)(27,48,42)(32,52,37)(35,62,68)(38,58,69)(39,55,70)(40,63,71)(43,59,72)(44,60,73)(45,57,74)(56,75,65)(61,78,66)(64,79,81)(67,76,82)(77,83,80);(3,5,14,20,42,36,7,16,37)(4,9,25,11,13,32,12,27,34)(6,15,39,17,38,65,19,41,35)(8,21,46,31,48,54,23,24,52)(10,26,56,28,33,62,30,58,55)(18,45,64,43,40,67,44,66,80)(22,47,70,49,69,75,51,53,68)(29,61,76,59,57,77,60,63,79)(50,74,81,72,71,82,73,78,83);(3,6,18,7,19,44,20,17,43)(4,10,29,12,30,60,11,28,59)(5,15,40,16,41,45,42,38,66)(8,22,50,23,51,73,31,49,72)(9,26,57,27,58,61,13,33,63)(14,35,64,37,65,80,36,39,67)(21,47,71,24,53,74,48,69,78)(25,55,76,34,62,79,32,56,77)(46,68,81,52,75,83,54,70,82)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72221956
maximal subgroup of	C₂×#72221882, C₂×#72221956
maximal normal subgroup of	C₂×#72221882, C₂×#72221956

GAP

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

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

C2×#14554346 #72244228