C₂×#14554332 #72244204

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

Aliases

direct product

C₂×#14554332

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

4048 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)(6,18,44)(10,30,19)(15,40,20)(21,48,45)(24,31,59)(28,32,37)(33,46,49)(38,41,69)(42,47,50)(52,60,55)(53,64,61)(56,62,67)(58,65,63)(68,71,70)(72,75,73)(74,76,78)(77,80,79)(81,82,83);(3,4,8)(5,13,35)(6,19,24)(7,12,25)(9,26,17)(10,31,18)(11,34,22)(14,27,51)(15,28,55)(16,29,23)(20,37,60)(21,49,53)(30,59,44)(32,52,40)(33,64,48)(36,54,43)(38,56,73)(39,57,66)(41,62,72)(42,63,74)(45,46,61)(47,58,76)(50,65,78)(67,75,69)(68,77,83)(70,79,82)(71,80,81);(3,5,14,7,17,39,22,16,43)(4,9,27,12,29,57,11,13,36)(6,20,38,18,15,41,44,40,69)(8,23,51,25,35,66,34,26,54)(10,32,62,30,37,67,19,28,56)(21,50,68,45,47,70,48,42,71)(24,52,73,31,60,72,59,55,75)(33,65,80,49,58,77,46,63,79)(53,74,83,61,78,82,64,76,81);(3,6,21,7,18,45,22,44,48)(4,10,33,12,30,49,11,19,46)(5,15,42,17,40,50,16,20,47)(8,24,53,25,31,61,34,59,64)(9,28,58,29,32,63,13,37,65)(14,38,68,39,41,70,43,69,71)(23,52,74,35,60,78,26,55,76)(27,56,77,57,62,79,36,67,80)(51,72,81,66,75,83,54,73,82)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₂×#72221946
maximal subgroup of	C₂×#72221882, C₂×#72221946
maximal normal subgroup of	C₂×#72221882, C₂×#72221946

GAP

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

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

C2×#14554332 #72244204