C₂×#14555440 #72244227

order	4374
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₃⁵ · Schur multiplier (H₂) = C₃¹⁰

Aliases

direct product

C₂×#14555440

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₃³×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂×C₃⁵
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃²×3₋¹⁺² ◁ G

12040 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)(6,17,19)(10,28,30)(15,38,40)(18,43,42)(22,47,49)(26,33,55)(29,57,56)(35,61,63)(39,65,64)(45,67,51)(48,70,69)(53,58,60)(54,75,59)(62,79,78)(66,73,71)(68,72,81)(74,77,76)(80,82,83);(3,4,8)(5,13,21)(6,10,22)(7,12,23)(9,24,16)(11,31,20)(14,34,44)(15,33,45)(17,28,47)(18,29,48)(19,30,49)(25,52,36)(26,51,40)(27,46,41)(32,50,37)(35,60,66)(38,55,67)(39,59,68)(42,56,69)(43,57,70)(53,73,61)(54,72,65)(58,71,63)(62,77,80)(64,75,81)(74,83,78)(76,82,79);(3,5,14,20,41,36,7,16,37)(4,9,25,11,13,32,12,27,34)(6,15,35,17,38,61,19,40,63)(8,21,44,31,46,52,23,24,50)(10,26,53,28,33,58,30,55,60)(18,39,62,42,64,78,43,65,79)(22,45,66,47,67,73,49,51,71)(29,54,74,56,59,76,57,75,77)(48,68,80,69,81,83,70,72,82);(3,6,18,7,19,43,20,17,42)(4,10,29,12,30,57,11,28,56)(5,15,39,16,40,65,41,38,64)(8,22,48,23,49,70,31,47,69)(9,26,54,27,55,75,13,33,59)(14,35,62,37,63,79,36,61,78)(21,45,68,24,51,72,46,67,81)(25,53,74,34,60,77,32,58,76)(44,66,80,50,71,82,52,73,83);(4,11,12)(5,16,41)(6,19,17)(8,23,31)(14,36,37)(15,38,40)(18,42,43)(21,46,24)(22,47,49)(25,34,32)(26,55,33)(29,57,56)(53,58,60)(54,59,75)(62,79,78)(66,71,73)(68,72,81)(80,83,82)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72221921
maximal subgroup of	C₂×#72221882, C₂×#72221849, C₂×#72221921
maximal normal subgroup of	C₂×#72221882, C₂×#72221849, C₂×#72221921

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

C2×#14555440 #72244227