C₂×#14554857 #72244168

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

Aliases

direct product

C₂×#14554857

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

2232 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)(3,4,8)(5,13,22)(6,10,23)(7,12,24)(9,25,16)(11,33,21)(14,35,46)(15,30,47)(17,32,50)(18,27,52)(19,31,48)(20,29,49)(26,54,64)(28,53,41)(34,51,37)(36,60,68)(38,56,74)(39,61,69)(40,58,70)(42,62,71)(43,55,75)(44,57,72)(45,59,73)(63,79,80)(65,77,81)(66,78,82)(67,76,83);(6,17,20)(10,29,32)(15,18,40)(27,30,58)(31,59,62)(36,38,43)(48,71,73)(55,56,60)(57,61,78)(69,72,82)(76,77,79)(80,81,83);(3,5,14,7,16,37,21,41,64)(4,9,26,12,28,35,11,13,34)(6,18,43,17,40,36,20,15,38)(8,22,46,24,25,51,33,53,54)(10,30,60,32,27,56,29,58,55)(19,39,63,42,44,65,45,66,67)(23,47,68,50,52,74,49,70,75)(31,61,79,62,57,77,59,78,76)(48,72,83,71,82,80,73,69,81);(3,6,19)(4,10,31)(5,15,39)(7,17,42)(8,23,48)(9,27,57)(11,29,59)(12,32,62)(13,30,61)(14,36,63)(16,18,44)(20,45,21)(22,47,69)(24,50,71)(25,52,72)(26,55,76)(28,58,78)(33,49,73)(34,56,77)(35,60,79)(37,38,65)(40,66,41)(43,67,64)(46,68,80)(51,74,81)(53,70,82)(54,75,83)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72221903, C₂×#72221886, C₂×#72221923
maximal subgroup of	C₂×#72221882, C₂×#72221903, C₂×#72221886, …
maximal normal subgroup of	C₂×#72221882, C₂×#72221903, C₂×#72221886, …

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

C2×#14554857 #72244168