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## G = C62.12C32order 324 = 22·34

### 3rd non-split extension by C62 of C32 acting via C32/C3=C3

Aliases: C62.12C32, C9⋊(C3.A4), (C2×C18)⋊2C9, (C3×C9).4A4, C3.2(C9⋊A4), C222(C9⋊C9), (C6×C18).5C3, C32.17(C3×A4), (C2×C6).13- 1+2, (C2×C6).8(C3×C9), C3.4(C3×C3.A4), (C3×C3.A4).1C3, SmallGroup(324,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C62.12C32
 Chief series C1 — C22 — C2×C6 — C62 — C3×C3.A4 — C62.12C32
 Lower central C22 — C2×C6 — C62.12C32
 Upper central C1 — C32 — C3×C9

Generators and relations for C62.12C32
G = < a,b,c,d | a6=b6=1, c3=a2, d3=b2, ab=ba, cac-1=ab3, ad=da, cbc-1=a3b4, bd=db, dcd-1=b2c >

Smallest permutation representation of C62.12C32
On 162 points
Generators in S162
```(1 7 4)(2 45 5 39 8 42)(3 37 6 40 9 43)(10 24 13 27 16 21)(11 25 14 19 17 22)(12 18 15)(20 26 23)(28 52 31 46 34 49)(29 35 32)(30 54 33 48 36 51)(38 44 41)(47 53 50)(55 93 58 96 61 99)(56 94 59 97 62 91)(57 63 60)(64 74 67 77 70 80)(65 75 68 78 71 81)(66 72 69)(73 79 76)(82 106 85 100 88 103)(83 89 86)(84 108 87 102 90 105)(92 98 95)(101 107 104)(109 147 112 150 115 153)(110 148 113 151 116 145)(111 117 114)(118 137 121 140 124 143)(119 138 122 141 125 144)(120 126 123)(127 160 130 154 133 157)(128 134 131)(129 162 132 156 135 159)(136 142 139)(146 152 149)(155 161 158)
(1 131 95 38 158 63)(2 159 96)(3 133 97 40 160 56)(4 134 98 41 161 57)(5 162 99)(6 127 91 43 154 59)(7 128 92 44 155 60)(8 156 93)(9 130 94 37 157 62)(10 150 33)(11 110 34 19 151 52)(12 111 35 20 152 53)(13 153 36)(14 113 28 22 145 46)(15 114 29 23 146 47)(16 147 30)(17 116 31 25 148 49)(18 117 32 26 149 50)(21 112 54)(24 115 48)(27 109 51)(39 132 55)(42 135 58)(45 129 61)(64 143 87)(65 122 88 78 144 106)(66 123 89 79 136 107)(67 137 90)(68 125 82 81 138 100)(69 126 83 73 139 101)(70 140 84)(71 119 85 75 141 103)(72 120 86 76 142 104)(74 118 102)(77 121 105)(80 124 108)
(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)(127 128 129 130 131 132 133 134 135)(136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153)(154 155 156 157 158 159 160 161 162)
(1 142 149 95 72 18 158 86 32)(2 87 10 96 143 33 159 64 150)(3 65 34 97 88 151 160 144 11)(4 136 152 98 66 12 161 89 35)(5 90 13 99 137 36 162 67 153)(6 68 28 91 82 145 154 138 14)(7 139 146 92 69 15 155 83 29)(8 84 16 93 140 30 156 70 147)(9 71 31 94 85 148 157 141 17)(19 40 78 52 56 106 110 133 122)(20 134 107 53 41 123 111 57 79)(21 58 124 54 135 80 112 42 108)(22 43 81 46 59 100 113 127 125)(23 128 101 47 44 126 114 60 73)(24 61 118 48 129 74 115 45 102)(25 37 75 49 62 103 116 130 119)(26 131 104 50 38 120 117 63 76)(27 55 121 51 132 77 109 39 105)```

`G:=sub<Sym(162)| (1,7,4)(2,45,5,39,8,42)(3,37,6,40,9,43)(10,24,13,27,16,21)(11,25,14,19,17,22)(12,18,15)(20,26,23)(28,52,31,46,34,49)(29,35,32)(30,54,33,48,36,51)(38,44,41)(47,53,50)(55,93,58,96,61,99)(56,94,59,97,62,91)(57,63,60)(64,74,67,77,70,80)(65,75,68,78,71,81)(66,72,69)(73,79,76)(82,106,85,100,88,103)(83,89,86)(84,108,87,102,90,105)(92,98,95)(101,107,104)(109,147,112,150,115,153)(110,148,113,151,116,145)(111,117,114)(118,137,121,140,124,143)(119,138,122,141,125,144)(120,126,123)(127,160,130,154,133,157)(128,134,131)(129,162,132,156,135,159)(136,142,139)(146,152,149)(155,161,158), (1,131,95,38,158,63)(2,159,96)(3,133,97,40,160,56)(4,134,98,41,161,57)(5,162,99)(6,127,91,43,154,59)(7,128,92,44,155,60)(8,156,93)(9,130,94,37,157,62)(10,150,33)(11,110,34,19,151,52)(12,111,35,20,152,53)(13,153,36)(14,113,28,22,145,46)(15,114,29,23,146,47)(16,147,30)(17,116,31,25,148,49)(18,117,32,26,149,50)(21,112,54)(24,115,48)(27,109,51)(39,132,55)(42,135,58)(45,129,61)(64,143,87)(65,122,88,78,144,106)(66,123,89,79,136,107)(67,137,90)(68,125,82,81,138,100)(69,126,83,73,139,101)(70,140,84)(71,119,85,75,141,103)(72,120,86,76,142,104)(74,118,102)(77,121,105)(80,124,108), (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)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153)(154,155,156,157,158,159,160,161,162), (1,142,149,95,72,18,158,86,32)(2,87,10,96,143,33,159,64,150)(3,65,34,97,88,151,160,144,11)(4,136,152,98,66,12,161,89,35)(5,90,13,99,137,36,162,67,153)(6,68,28,91,82,145,154,138,14)(7,139,146,92,69,15,155,83,29)(8,84,16,93,140,30,156,70,147)(9,71,31,94,85,148,157,141,17)(19,40,78,52,56,106,110,133,122)(20,134,107,53,41,123,111,57,79)(21,58,124,54,135,80,112,42,108)(22,43,81,46,59,100,113,127,125)(23,128,101,47,44,126,114,60,73)(24,61,118,48,129,74,115,45,102)(25,37,75,49,62,103,116,130,119)(26,131,104,50,38,120,117,63,76)(27,55,121,51,132,77,109,39,105)>;`

`G:=Group( (1,7,4)(2,45,5,39,8,42)(3,37,6,40,9,43)(10,24,13,27,16,21)(11,25,14,19,17,22)(12,18,15)(20,26,23)(28,52,31,46,34,49)(29,35,32)(30,54,33,48,36,51)(38,44,41)(47,53,50)(55,93,58,96,61,99)(56,94,59,97,62,91)(57,63,60)(64,74,67,77,70,80)(65,75,68,78,71,81)(66,72,69)(73,79,76)(82,106,85,100,88,103)(83,89,86)(84,108,87,102,90,105)(92,98,95)(101,107,104)(109,147,112,150,115,153)(110,148,113,151,116,145)(111,117,114)(118,137,121,140,124,143)(119,138,122,141,125,144)(120,126,123)(127,160,130,154,133,157)(128,134,131)(129,162,132,156,135,159)(136,142,139)(146,152,149)(155,161,158), (1,131,95,38,158,63)(2,159,96)(3,133,97,40,160,56)(4,134,98,41,161,57)(5,162,99)(6,127,91,43,154,59)(7,128,92,44,155,60)(8,156,93)(9,130,94,37,157,62)(10,150,33)(11,110,34,19,151,52)(12,111,35,20,152,53)(13,153,36)(14,113,28,22,145,46)(15,114,29,23,146,47)(16,147,30)(17,116,31,25,148,49)(18,117,32,26,149,50)(21,112,54)(24,115,48)(27,109,51)(39,132,55)(42,135,58)(45,129,61)(64,143,87)(65,122,88,78,144,106)(66,123,89,79,136,107)(67,137,90)(68,125,82,81,138,100)(69,126,83,73,139,101)(70,140,84)(71,119,85,75,141,103)(72,120,86,76,142,104)(74,118,102)(77,121,105)(80,124,108), (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)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153)(154,155,156,157,158,159,160,161,162), (1,142,149,95,72,18,158,86,32)(2,87,10,96,143,33,159,64,150)(3,65,34,97,88,151,160,144,11)(4,136,152,98,66,12,161,89,35)(5,90,13,99,137,36,162,67,153)(6,68,28,91,82,145,154,138,14)(7,139,146,92,69,15,155,83,29)(8,84,16,93,140,30,156,70,147)(9,71,31,94,85,148,157,141,17)(19,40,78,52,56,106,110,133,122)(20,134,107,53,41,123,111,57,79)(21,58,124,54,135,80,112,42,108)(22,43,81,46,59,100,113,127,125)(23,128,101,47,44,126,114,60,73)(24,61,118,48,129,74,115,45,102)(25,37,75,49,62,103,116,130,119)(26,131,104,50,38,120,117,63,76)(27,55,121,51,132,77,109,39,105) );`

`G=PermutationGroup([[(1,7,4),(2,45,5,39,8,42),(3,37,6,40,9,43),(10,24,13,27,16,21),(11,25,14,19,17,22),(12,18,15),(20,26,23),(28,52,31,46,34,49),(29,35,32),(30,54,33,48,36,51),(38,44,41),(47,53,50),(55,93,58,96,61,99),(56,94,59,97,62,91),(57,63,60),(64,74,67,77,70,80),(65,75,68,78,71,81),(66,72,69),(73,79,76),(82,106,85,100,88,103),(83,89,86),(84,108,87,102,90,105),(92,98,95),(101,107,104),(109,147,112,150,115,153),(110,148,113,151,116,145),(111,117,114),(118,137,121,140,124,143),(119,138,122,141,125,144),(120,126,123),(127,160,130,154,133,157),(128,134,131),(129,162,132,156,135,159),(136,142,139),(146,152,149),(155,161,158)], [(1,131,95,38,158,63),(2,159,96),(3,133,97,40,160,56),(4,134,98,41,161,57),(5,162,99),(6,127,91,43,154,59),(7,128,92,44,155,60),(8,156,93),(9,130,94,37,157,62),(10,150,33),(11,110,34,19,151,52),(12,111,35,20,152,53),(13,153,36),(14,113,28,22,145,46),(15,114,29,23,146,47),(16,147,30),(17,116,31,25,148,49),(18,117,32,26,149,50),(21,112,54),(24,115,48),(27,109,51),(39,132,55),(42,135,58),(45,129,61),(64,143,87),(65,122,88,78,144,106),(66,123,89,79,136,107),(67,137,90),(68,125,82,81,138,100),(69,126,83,73,139,101),(70,140,84),(71,119,85,75,141,103),(72,120,86,76,142,104),(74,118,102),(77,121,105),(80,124,108)], [(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),(127,128,129,130,131,132,133,134,135),(136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153),(154,155,156,157,158,159,160,161,162)], [(1,142,149,95,72,18,158,86,32),(2,87,10,96,143,33,159,64,150),(3,65,34,97,88,151,160,144,11),(4,136,152,98,66,12,161,89,35),(5,90,13,99,137,36,162,67,153),(6,68,28,91,82,145,154,138,14),(7,139,146,92,69,15,155,83,29),(8,84,16,93,140,30,156,70,147),(9,71,31,94,85,148,157,141,17),(19,40,78,52,56,106,110,133,122),(20,134,107,53,41,123,111,57,79),(21,58,124,54,135,80,112,42,108),(22,43,81,46,59,100,113,127,125),(23,128,101,47,44,126,114,60,73),(24,61,118,48,129,74,115,45,102),(25,37,75,49,62,103,116,130,119),(26,131,104,50,38,120,117,63,76),(27,55,121,51,132,77,109,39,105)]])`

60 conjugacy classes

 class 1 2 3A ··· 3H 6A ··· 6H 9A ··· 9F 9G ··· 9X 18A ··· 18R order 1 2 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 ··· 1 3 ··· 3 3 ··· 3 12 ··· 12 3 ··· 3

60 irreducible representations

 dim 1 1 1 1 3 3 3 3 3 type + + image C1 C3 C3 C9 A4 3- 1+2 C3.A4 C3×A4 C9⋊A4 kernel C62.12C32 C3×C3.A4 C6×C18 C2×C18 C3×C9 C2×C6 C9 C32 C3 # reps 1 6 2 18 1 6 6 2 18

Matrix representation of C62.12C32 in GL6(𝔽19)

 11 0 0 0 0 0 0 11 0 0 0 0 0 0 11 0 0 0 0 0 0 7 0 0 0 0 0 0 12 0 0 0 0 17 0 12
,
 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 2 1 7
,
 12 5 0 0 0 0 11 4 13 0 0 0 8 9 3 0 0 0 0 0 0 0 1 0 0 0 0 16 8 17 0 0 0 11 2 11
,
 1 10 0 0 0 0 0 18 1 0 0 0 1 18 0 0 0 0 0 0 0 17 0 0 0 0 0 0 16 0 0 0 0 1 6 5

`G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,17,0,0,0,0,12,0,0,0,0,0,0,12],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,12,0,2,0,0,0,0,12,1,0,0,0,0,0,7],[12,11,8,0,0,0,5,4,9,0,0,0,0,13,3,0,0,0,0,0,0,0,16,11,0,0,0,1,8,2,0,0,0,0,17,11],[1,0,1,0,0,0,10,18,18,0,0,0,0,1,0,0,0,0,0,0,0,17,0,1,0,0,0,0,16,6,0,0,0,0,0,5] >;`

C62.12C32 in GAP, Magma, Sage, TeX

`C_6^2._{12}C_3^2`
`% in TeX`

`G:=Group("C6^2.12C3^2");`
`// GroupNames label`

`G:=SmallGroup(324,48);`
`// by ID`

`G=gap.SmallGroup(324,48);`
`# by ID`

`G:=PCGroup([6,-3,-3,-3,-3,-2,2,54,361,115,4864,8753]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^3=a^2,d^3=b^2,a*b=b*a,c*a*c^-1=a*b^3,a*d=d*a,c*b*c^-1=a^3*b^4,b*d=d*b,d*c*d^-1=b^2*c>;`
`// generators/relations`

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