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## G = C2×C7⋊C9order 126 = 2·32·7

### Direct product of C2 and C7⋊C9

Aliases: C2×C7⋊C9, C14⋊C9, C72C18, C42.C3, C21.2C6, C6.(C7⋊C3), C3.(C2×C7⋊C3), SmallGroup(126,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×C7⋊C9
 Chief series C1 — C7 — C21 — C7⋊C9 — C2×C7⋊C9
 Lower central C7 — C2×C7⋊C9
 Upper central C1 — C6

Generators and relations for C2×C7⋊C9
G = < a,b,c | a2=b7=c9=1, ab=ba, ac=ca, cbc-1=b4 >

Character table of C2×C7⋊C9

 class 1 2 3A 3B 6A 6B 7A 7B 9A 9B 9C 9D 9E 9F 14A 14B 18A 18B 18C 18D 18E 18F 21A 21B 21C 21D 42A 42B 42C 42D size 1 1 1 1 1 1 3 3 7 7 7 7 7 7 3 3 7 7 7 7 7 7 3 3 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 1 1 -1 -1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 -1 -1 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ4 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 -1 1 1 -1 -1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 -1 -1 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ7 1 -1 ζ32 ζ3 ζ6 ζ65 1 1 ζ97 ζ92 ζ95 ζ9 ζ98 ζ94 -1 -1 -ζ95 -ζ94 -ζ97 -ζ9 -ζ98 -ζ92 ζ32 ζ3 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 linear of order 18 ρ8 1 -1 ζ3 ζ32 ζ65 ζ6 1 1 ζ92 ζ97 ζ94 ζ98 ζ9 ζ95 -1 -1 -ζ94 -ζ95 -ζ92 -ζ98 -ζ9 -ζ97 ζ3 ζ32 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 linear of order 18 ρ9 1 -1 ζ32 ζ3 ζ6 ζ65 1 1 ζ94 ζ95 ζ98 ζ97 ζ92 ζ9 -1 -1 -ζ98 -ζ9 -ζ94 -ζ97 -ζ92 -ζ95 ζ32 ζ3 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 linear of order 18 ρ10 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ94 ζ95 ζ98 ζ97 ζ92 ζ9 1 1 ζ98 ζ9 ζ94 ζ97 ζ92 ζ95 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 9 ρ11 1 -1 ζ3 ζ32 ζ65 ζ6 1 1 ζ95 ζ94 ζ9 ζ92 ζ97 ζ98 -1 -1 -ζ9 -ζ98 -ζ95 -ζ92 -ζ97 -ζ94 ζ3 ζ32 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 linear of order 18 ρ12 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ98 ζ9 ζ97 ζ95 ζ94 ζ92 1 1 ζ97 ζ92 ζ98 ζ95 ζ94 ζ9 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 9 ρ13 1 -1 ζ3 ζ32 ζ65 ζ6 1 1 ζ98 ζ9 ζ97 ζ95 ζ94 ζ92 -1 -1 -ζ97 -ζ92 -ζ98 -ζ95 -ζ94 -ζ9 ζ3 ζ32 ζ32 ζ3 ζ65 ζ65 ζ6 ζ6 linear of order 18 ρ14 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ97 ζ92 ζ95 ζ9 ζ98 ζ94 1 1 ζ95 ζ94 ζ97 ζ9 ζ98 ζ92 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 9 ρ15 1 -1 ζ32 ζ3 ζ6 ζ65 1 1 ζ9 ζ98 ζ92 ζ94 ζ95 ζ97 -1 -1 -ζ92 -ζ97 -ζ9 -ζ94 -ζ95 -ζ98 ζ32 ζ3 ζ3 ζ32 ζ6 ζ6 ζ65 ζ65 linear of order 18 ρ16 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ95 ζ94 ζ9 ζ92 ζ97 ζ98 1 1 ζ9 ζ98 ζ95 ζ92 ζ97 ζ94 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 9 ρ17 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ92 ζ97 ζ94 ζ98 ζ9 ζ95 1 1 ζ94 ζ95 ζ92 ζ98 ζ9 ζ97 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 9 ρ18 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ9 ζ98 ζ92 ζ94 ζ95 ζ97 1 1 ζ92 ζ97 ζ9 ζ94 ζ95 ζ98 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 9 ρ19 3 -3 3 3 -3 -3 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 1+√-7/2 1-√-7/2 0 0 0 0 0 0 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 1+√-7/2 1-√-7/2 1-√-7/2 1+√-7/2 complex lifted from C2×C7⋊C3 ρ20 3 -3 3 3 -3 -3 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 1-√-7/2 1+√-7/2 0 0 0 0 0 0 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 1-√-7/2 1+√-7/2 1+√-7/2 1-√-7/2 complex lifted from C2×C7⋊C3 ρ21 3 3 3 3 3 3 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ22 3 3 3 3 3 3 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ23 3 -3 -3+3√-3/2 -3-3√-3/2 3-3√-3/2 3+3√-3/2 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 1-√-7/2 1+√-7/2 0 0 0 0 0 0 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 -ζ3ζ74-ζ3ζ72-ζ3ζ7 -ζ3ζ76-ζ3ζ75-ζ3ζ73 -ζ32ζ76-ζ32ζ75-ζ32ζ73 -ζ32ζ74-ζ32ζ72-ζ32ζ7 complex faithful, Schur index 3 ρ24 3 3 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 complex lifted from C7⋊C9, Schur index 3 ρ25 3 -3 -3-3√-3/2 -3+3√-3/2 3+3√-3/2 3-3√-3/2 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 1-√-7/2 1+√-7/2 0 0 0 0 0 0 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 -ζ32ζ74-ζ32ζ72-ζ32ζ7 -ζ32ζ76-ζ32ζ75-ζ32ζ73 -ζ3ζ76-ζ3ζ75-ζ3ζ73 -ζ3ζ74-ζ3ζ72-ζ3ζ7 complex faithful, Schur index 3 ρ26 3 3 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 complex lifted from C7⋊C9, Schur index 3 ρ27 3 -3 -3-3√-3/2 -3+3√-3/2 3+3√-3/2 3-3√-3/2 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 1+√-7/2 1-√-7/2 0 0 0 0 0 0 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 -ζ32ζ76-ζ32ζ75-ζ32ζ73 -ζ32ζ74-ζ32ζ72-ζ32ζ7 -ζ3ζ74-ζ3ζ72-ζ3ζ7 -ζ3ζ76-ζ3ζ75-ζ3ζ73 complex faithful, Schur index 3 ρ28 3 3 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 complex lifted from C7⋊C9, Schur index 3 ρ29 3 -3 -3+3√-3/2 -3-3√-3/2 3-3√-3/2 3+3√-3/2 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 1+√-7/2 1-√-7/2 0 0 0 0 0 0 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 -ζ3ζ76-ζ3ζ75-ζ3ζ73 -ζ3ζ74-ζ3ζ72-ζ3ζ7 -ζ32ζ74-ζ32ζ72-ζ32ζ7 -ζ32ζ76-ζ32ζ75-ζ32ζ73 complex faithful, Schur index 3 ρ30 3 3 -3-3√-3/2 -3+3√-3/2 -3-3√-3/2 -3+3√-3/2 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 complex lifted from C7⋊C9, Schur index 3

Smallest permutation representation of C2×C7⋊C9
Regular action on 126 points
Generators in S126
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 46)(9 47)(10 88)(11 89)(12 90)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 116)(20 117)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 73)(35 74)(36 75)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 100)(62 101)(63 102)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)
(1 93 86 67 73 111 104)(2 74 94 112 87 105 68)(3 88 75 106 95 69 113)(4 96 89 70 76 114 107)(5 77 97 115 90 108 71)(6 82 78 100 98 72 116)(7 99 83 64 79 117 101)(8 80 91 109 84 102 65)(9 85 81 103 92 66 110)(10 36 58 122 42 25 50)(11 43 28 26 59 51 123)(12 60 44 52 29 124 27)(13 30 61 125 45 19 53)(14 37 31 20 62 54 126)(15 63 38 46 32 118 21)(16 33 55 119 39 22 47)(17 40 34 23 56 48 120)(18 57 41 49 35 121 24)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)(82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126)

G:=sub<Sym(126)| (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,88)(11,89)(12,90)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,116)(20,117)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,73)(35,74)(36,75)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,100)(62,101)(63,102)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126), (1,93,86,67,73,111,104)(2,74,94,112,87,105,68)(3,88,75,106,95,69,113)(4,96,89,70,76,114,107)(5,77,97,115,90,108,71)(6,82,78,100,98,72,116)(7,99,83,64,79,117,101)(8,80,91,109,84,102,65)(9,85,81,103,92,66,110)(10,36,58,122,42,25,50)(11,43,28,26,59,51,123)(12,60,44,52,29,124,27)(13,30,61,125,45,19,53)(14,37,31,20,62,54,126)(15,63,38,46,32,118,21)(16,33,55,119,39,22,47)(17,40,34,23,56,48,120)(18,57,41,49,35,121,24), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)>;

G:=Group( (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,46)(9,47)(10,88)(11,89)(12,90)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,116)(20,117)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,73)(35,74)(36,75)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,100)(62,101)(63,102)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126), (1,93,86,67,73,111,104)(2,74,94,112,87,105,68)(3,88,75,106,95,69,113)(4,96,89,70,76,114,107)(5,77,97,115,90,108,71)(6,82,78,100,98,72,116)(7,99,83,64,79,117,101)(8,80,91,109,84,102,65)(9,85,81,103,92,66,110)(10,36,58,122,42,25,50)(11,43,28,26,59,51,123)(12,60,44,52,29,124,27)(13,30,61,125,45,19,53)(14,37,31,20,62,54,126)(15,63,38,46,32,118,21)(16,33,55,119,39,22,47)(17,40,34,23,56,48,120)(18,57,41,49,35,121,24), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81)(82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126) );

G=PermutationGroup([(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,46),(9,47),(10,88),(11,89),(12,90),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,116),(20,117),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,73),(35,74),(36,75),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(55,103),(56,104),(57,105),(58,106),(59,107),(60,108),(61,100),(62,101),(63,102),(91,118),(92,119),(93,120),(94,121),(95,122),(96,123),(97,124),(98,125),(99,126)], [(1,93,86,67,73,111,104),(2,74,94,112,87,105,68),(3,88,75,106,95,69,113),(4,96,89,70,76,114,107),(5,77,97,115,90,108,71),(6,82,78,100,98,72,116),(7,99,83,64,79,117,101),(8,80,91,109,84,102,65),(9,85,81,103,92,66,110),(10,36,58,122,42,25,50),(11,43,28,26,59,51,123),(12,60,44,52,29,124,27),(13,30,61,125,45,19,53),(14,37,31,20,62,54,126),(15,63,38,46,32,118,21),(16,33,55,119,39,22,47),(17,40,34,23,56,48,120),(18,57,41,49,35,121,24)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81),(82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126)])

C2×C7⋊C9 is a maximal subgroup of   C7⋊C36  C18×C7⋊C3

Matrix representation of C2×C7⋊C9 in GL3(𝔽127) generated by

 126 0 0 0 126 0 0 0 126
,
 126 22 1 0 22 1 126 23 1
,
 81 31 5 42 75 50 40 34 98
G:=sub<GL(3,GF(127))| [126,0,0,0,126,0,0,0,126],[126,0,126,22,22,23,1,1,1],[81,42,40,31,75,34,5,50,98] >;

C2×C7⋊C9 in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes C_9
% in TeX

G:=Group("C2xC7:C9");
// GroupNames label

G:=SmallGroup(126,2);
// by ID

G=gap.SmallGroup(126,2);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,29,295]);
// Polycyclic

G:=Group<a,b,c|a^2=b^7=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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