C₂×#14554864 #72244235

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

Aliases

direct product

C₂×#14554864

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

2308 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)(5,9,23)(6,17,40)(7,11,25)(10,24,20)(12,32,22)(13,26,47)(14,27,54)(15,29,57)(16,28,49)(18,41,67)(19,42,50)(21,33,62)(30,48,39)(31,60,45)(34,58,70)(35,52,71)(36,53,72)(37,55,77)(38,65,74)(43,51,75)(44,59,80)(46,61,68)(56,73,66)(63,76,83)(64,79,81)(69,78,82);(3,5,13)(4,9,26)(6,18,43)(7,16,36)(8,23,47)(10,30,58)(11,28,53)(12,29,52)(14,37,21)(15,35,22)(17,41,51)(19,38,63)(20,39,34)(24,48,70)(25,49,72)(27,55,33)(31,59,79)(32,57,71)(40,67,75)(42,65,76)(44,64,45)(46,66,69)(50,74,83)(54,77,62)(56,78,61)(60,80,81)(68,73,82);(3,6,19,7,21,46,22,20,45)(4,10,31,11,17,42,12,33,61)(5,14,38,16,39,66,15,18,44)(8,24,50,25,40,68,32,62,60)(9,27,56,28,30,59,29,41,65)(13,34,63,36,43,69,35,37,64)(23,48,73,49,67,80,57,54,74)(26,51,76,53,55,78,52,58,79)(47,70,81,72,75,83,71,77,82);(4,11,12)(5,15,16)(6,20,21)(9,28,29)(10,17,33)(13,35,36)(14,18,39)(19,45,46)(23,49,57)(24,40,62)(26,52,53)(27,41,30)(31,61,42)(51,55,58)(56,59,65)(63,64,69)(76,78,79)(81,83,82)

Conjugacy classes & character table

246 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#72221878
maximal subgroup of	C₂×#72221849, C₂×#72221878
maximal normal subgroup of	C₂×#72221849, C₂×#72221878

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

C2×#14554864 #72244235