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## G = C62.127D6order 432 = 24·33

### 11st non-split extension by C62 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — C62.127D6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C6×C18 — C2×C9⋊Dic3 — C62.127D6
 Lower central C3×C9 — C3×C18 — C62.127D6
 Upper central C1 — C22 — C23

Generators and relations for C62.127D6
G = < a,b,c,d | a6=b6=1, c6=b4, d2=a3b3, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b-1c5 >

Subgroups: 692 in 170 conjugacy classes, 83 normal (17 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C9, C32, Dic3, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C18, C3×C6, C3×C6, C3×C6, C2×Dic3, C22×C6, C22×C6, C3×C9, Dic9, C2×C18, C2×C18, C3⋊Dic3, C62, C62, C62, C6.D4, C3×C18, C3×C18, C3×C18, C2×Dic9, C22×C18, C2×C3⋊Dic3, C2×C62, C9⋊Dic3, C6×C18, C6×C18, C6×C18, C18.D4, C625C4, C2×C9⋊Dic3, C2×C6×C18, C62.127D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D9, C3⋊S3, C2×Dic3, C3⋊D4, Dic9, D18, C3⋊Dic3, C2×C3⋊S3, C6.D4, C9⋊S3, C2×Dic9, C9⋊D4, C2×C3⋊Dic3, C327D4, C9⋊Dic3, C2×C9⋊S3, C18.D4, C625C4, C2×C9⋊Dic3, C6.D18, C62.127D6

Smallest permutation representation of C62.127D6
On 216 points
Generators in S216
```(1 92 120 10 101 111)(2 93 121 11 102 112)(3 94 122 12 103 113)(4 95 123 13 104 114)(5 96 124 14 105 115)(6 97 125 15 106 116)(7 98 126 16 107 117)(8 99 109 17 108 118)(9 100 110 18 91 119)(19 194 130 163 53 68)(20 195 131 164 54 69)(21 196 132 165 37 70)(22 197 133 166 38 71)(23 198 134 167 39 72)(24 181 135 168 40 55)(25 182 136 169 41 56)(26 183 137 170 42 57)(27 184 138 171 43 58)(28 185 139 172 44 59)(29 186 140 173 45 60)(30 187 141 174 46 61)(31 188 142 175 47 62)(32 189 143 176 48 63)(33 190 144 177 49 64)(34 191 127 178 50 65)(35 192 128 179 51 66)(36 193 129 180 52 67)(73 154 206 82 145 215)(74 155 207 83 146 216)(75 156 208 84 147 199)(76 157 209 85 148 200)(77 158 210 86 149 201)(78 159 211 87 150 202)(79 160 212 88 151 203)(80 161 213 89 152 204)(81 162 214 90 153 205)
(1 154 13 148 7 160)(2 155 14 149 8 161)(3 156 15 150 9 162)(4 157 16 151 10 145)(5 158 17 152 11 146)(6 159 18 153 12 147)(19 178 31 172 25 166)(20 179 32 173 26 167)(21 180 33 174 27 168)(22 163 34 175 28 169)(23 164 35 176 29 170)(24 165 36 177 30 171)(37 193 49 187 43 181)(38 194 50 188 44 182)(39 195 51 189 45 183)(40 196 52 190 46 184)(41 197 53 191 47 185)(42 198 54 192 48 186)(55 132 67 144 61 138)(56 133 68 127 62 139)(57 134 69 128 63 140)(58 135 70 129 64 141)(59 136 71 130 65 142)(60 137 72 131 66 143)(73 123 85 117 79 111)(74 124 86 118 80 112)(75 125 87 119 81 113)(76 126 88 120 82 114)(77 109 89 121 83 115)(78 110 90 122 84 116)(91 205 103 199 97 211)(92 206 104 200 98 212)(93 207 105 201 99 213)(94 208 106 202 100 214)(95 209 107 203 101 215)(96 210 108 204 102 216)
(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)(163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)
(1 164 157 173)(2 28 158 19)(3 180 159 171)(4 26 160 35)(5 178 161 169)(6 24 162 33)(7 176 145 167)(8 22 146 31)(9 174 147 165)(10 20 148 29)(11 172 149 163)(12 36 150 27)(13 170 151 179)(14 34 152 25)(15 168 153 177)(16 32 154 23)(17 166 155 175)(18 30 156 21)(37 119 46 84)(38 74 47 109)(39 117 48 82)(40 90 49 125)(41 115 50 80)(42 88 51 123)(43 113 52 78)(44 86 53 121)(45 111 54 76)(55 214 64 97)(56 105 65 204)(57 212 66 95)(58 103 67 202)(59 210 68 93)(60 101 69 200)(61 208 70 91)(62 99 71 216)(63 206 72 107)(73 198 126 189)(75 196 110 187)(77 194 112 185)(79 192 114 183)(81 190 116 181)(83 188 118 197)(85 186 120 195)(87 184 122 193)(89 182 124 191)(92 131 209 140)(94 129 211 138)(96 127 213 136)(98 143 215 134)(100 141 199 132)(102 139 201 130)(104 137 203 128)(106 135 205 144)(108 133 207 142)```

`G:=sub<Sym(216)| (1,92,120,10,101,111)(2,93,121,11,102,112)(3,94,122,12,103,113)(4,95,123,13,104,114)(5,96,124,14,105,115)(6,97,125,15,106,116)(7,98,126,16,107,117)(8,99,109,17,108,118)(9,100,110,18,91,119)(19,194,130,163,53,68)(20,195,131,164,54,69)(21,196,132,165,37,70)(22,197,133,166,38,71)(23,198,134,167,39,72)(24,181,135,168,40,55)(25,182,136,169,41,56)(26,183,137,170,42,57)(27,184,138,171,43,58)(28,185,139,172,44,59)(29,186,140,173,45,60)(30,187,141,174,46,61)(31,188,142,175,47,62)(32,189,143,176,48,63)(33,190,144,177,49,64)(34,191,127,178,50,65)(35,192,128,179,51,66)(36,193,129,180,52,67)(73,154,206,82,145,215)(74,155,207,83,146,216)(75,156,208,84,147,199)(76,157,209,85,148,200)(77,158,210,86,149,201)(78,159,211,87,150,202)(79,160,212,88,151,203)(80,161,213,89,152,204)(81,162,214,90,153,205), (1,154,13,148,7,160)(2,155,14,149,8,161)(3,156,15,150,9,162)(4,157,16,151,10,145)(5,158,17,152,11,146)(6,159,18,153,12,147)(19,178,31,172,25,166)(20,179,32,173,26,167)(21,180,33,174,27,168)(22,163,34,175,28,169)(23,164,35,176,29,170)(24,165,36,177,30,171)(37,193,49,187,43,181)(38,194,50,188,44,182)(39,195,51,189,45,183)(40,196,52,190,46,184)(41,197,53,191,47,185)(42,198,54,192,48,186)(55,132,67,144,61,138)(56,133,68,127,62,139)(57,134,69,128,63,140)(58,135,70,129,64,141)(59,136,71,130,65,142)(60,137,72,131,66,143)(73,123,85,117,79,111)(74,124,86,118,80,112)(75,125,87,119,81,113)(76,126,88,120,82,114)(77,109,89,121,83,115)(78,110,90,122,84,116)(91,205,103,199,97,211)(92,206,104,200,98,212)(93,207,105,201,99,213)(94,208,106,202,100,214)(95,209,107,203,101,215)(96,210,108,204,102,216), (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)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216), (1,164,157,173)(2,28,158,19)(3,180,159,171)(4,26,160,35)(5,178,161,169)(6,24,162,33)(7,176,145,167)(8,22,146,31)(9,174,147,165)(10,20,148,29)(11,172,149,163)(12,36,150,27)(13,170,151,179)(14,34,152,25)(15,168,153,177)(16,32,154,23)(17,166,155,175)(18,30,156,21)(37,119,46,84)(38,74,47,109)(39,117,48,82)(40,90,49,125)(41,115,50,80)(42,88,51,123)(43,113,52,78)(44,86,53,121)(45,111,54,76)(55,214,64,97)(56,105,65,204)(57,212,66,95)(58,103,67,202)(59,210,68,93)(60,101,69,200)(61,208,70,91)(62,99,71,216)(63,206,72,107)(73,198,126,189)(75,196,110,187)(77,194,112,185)(79,192,114,183)(81,190,116,181)(83,188,118,197)(85,186,120,195)(87,184,122,193)(89,182,124,191)(92,131,209,140)(94,129,211,138)(96,127,213,136)(98,143,215,134)(100,141,199,132)(102,139,201,130)(104,137,203,128)(106,135,205,144)(108,133,207,142)>;`

`G:=Group( (1,92,120,10,101,111)(2,93,121,11,102,112)(3,94,122,12,103,113)(4,95,123,13,104,114)(5,96,124,14,105,115)(6,97,125,15,106,116)(7,98,126,16,107,117)(8,99,109,17,108,118)(9,100,110,18,91,119)(19,194,130,163,53,68)(20,195,131,164,54,69)(21,196,132,165,37,70)(22,197,133,166,38,71)(23,198,134,167,39,72)(24,181,135,168,40,55)(25,182,136,169,41,56)(26,183,137,170,42,57)(27,184,138,171,43,58)(28,185,139,172,44,59)(29,186,140,173,45,60)(30,187,141,174,46,61)(31,188,142,175,47,62)(32,189,143,176,48,63)(33,190,144,177,49,64)(34,191,127,178,50,65)(35,192,128,179,51,66)(36,193,129,180,52,67)(73,154,206,82,145,215)(74,155,207,83,146,216)(75,156,208,84,147,199)(76,157,209,85,148,200)(77,158,210,86,149,201)(78,159,211,87,150,202)(79,160,212,88,151,203)(80,161,213,89,152,204)(81,162,214,90,153,205), (1,154,13,148,7,160)(2,155,14,149,8,161)(3,156,15,150,9,162)(4,157,16,151,10,145)(5,158,17,152,11,146)(6,159,18,153,12,147)(19,178,31,172,25,166)(20,179,32,173,26,167)(21,180,33,174,27,168)(22,163,34,175,28,169)(23,164,35,176,29,170)(24,165,36,177,30,171)(37,193,49,187,43,181)(38,194,50,188,44,182)(39,195,51,189,45,183)(40,196,52,190,46,184)(41,197,53,191,47,185)(42,198,54,192,48,186)(55,132,67,144,61,138)(56,133,68,127,62,139)(57,134,69,128,63,140)(58,135,70,129,64,141)(59,136,71,130,65,142)(60,137,72,131,66,143)(73,123,85,117,79,111)(74,124,86,118,80,112)(75,125,87,119,81,113)(76,126,88,120,82,114)(77,109,89,121,83,115)(78,110,90,122,84,116)(91,205,103,199,97,211)(92,206,104,200,98,212)(93,207,105,201,99,213)(94,208,106,202,100,214)(95,209,107,203,101,215)(96,210,108,204,102,216), (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)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216), (1,164,157,173)(2,28,158,19)(3,180,159,171)(4,26,160,35)(5,178,161,169)(6,24,162,33)(7,176,145,167)(8,22,146,31)(9,174,147,165)(10,20,148,29)(11,172,149,163)(12,36,150,27)(13,170,151,179)(14,34,152,25)(15,168,153,177)(16,32,154,23)(17,166,155,175)(18,30,156,21)(37,119,46,84)(38,74,47,109)(39,117,48,82)(40,90,49,125)(41,115,50,80)(42,88,51,123)(43,113,52,78)(44,86,53,121)(45,111,54,76)(55,214,64,97)(56,105,65,204)(57,212,66,95)(58,103,67,202)(59,210,68,93)(60,101,69,200)(61,208,70,91)(62,99,71,216)(63,206,72,107)(73,198,126,189)(75,196,110,187)(77,194,112,185)(79,192,114,183)(81,190,116,181)(83,188,118,197)(85,186,120,195)(87,184,122,193)(89,182,124,191)(92,131,209,140)(94,129,211,138)(96,127,213,136)(98,143,215,134)(100,141,199,132)(102,139,201,130)(104,137,203,128)(106,135,205,144)(108,133,207,142) );`

`G=PermutationGroup([[(1,92,120,10,101,111),(2,93,121,11,102,112),(3,94,122,12,103,113),(4,95,123,13,104,114),(5,96,124,14,105,115),(6,97,125,15,106,116),(7,98,126,16,107,117),(8,99,109,17,108,118),(9,100,110,18,91,119),(19,194,130,163,53,68),(20,195,131,164,54,69),(21,196,132,165,37,70),(22,197,133,166,38,71),(23,198,134,167,39,72),(24,181,135,168,40,55),(25,182,136,169,41,56),(26,183,137,170,42,57),(27,184,138,171,43,58),(28,185,139,172,44,59),(29,186,140,173,45,60),(30,187,141,174,46,61),(31,188,142,175,47,62),(32,189,143,176,48,63),(33,190,144,177,49,64),(34,191,127,178,50,65),(35,192,128,179,51,66),(36,193,129,180,52,67),(73,154,206,82,145,215),(74,155,207,83,146,216),(75,156,208,84,147,199),(76,157,209,85,148,200),(77,158,210,86,149,201),(78,159,211,87,150,202),(79,160,212,88,151,203),(80,161,213,89,152,204),(81,162,214,90,153,205)], [(1,154,13,148,7,160),(2,155,14,149,8,161),(3,156,15,150,9,162),(4,157,16,151,10,145),(5,158,17,152,11,146),(6,159,18,153,12,147),(19,178,31,172,25,166),(20,179,32,173,26,167),(21,180,33,174,27,168),(22,163,34,175,28,169),(23,164,35,176,29,170),(24,165,36,177,30,171),(37,193,49,187,43,181),(38,194,50,188,44,182),(39,195,51,189,45,183),(40,196,52,190,46,184),(41,197,53,191,47,185),(42,198,54,192,48,186),(55,132,67,144,61,138),(56,133,68,127,62,139),(57,134,69,128,63,140),(58,135,70,129,64,141),(59,136,71,130,65,142),(60,137,72,131,66,143),(73,123,85,117,79,111),(74,124,86,118,80,112),(75,125,87,119,81,113),(76,126,88,120,82,114),(77,109,89,121,83,115),(78,110,90,122,84,116),(91,205,103,199,97,211),(92,206,104,200,98,212),(93,207,105,201,99,213),(94,208,106,202,100,214),(95,209,107,203,101,215),(96,210,108,204,102,216)], [(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),(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198),(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)], [(1,164,157,173),(2,28,158,19),(3,180,159,171),(4,26,160,35),(5,178,161,169),(6,24,162,33),(7,176,145,167),(8,22,146,31),(9,174,147,165),(10,20,148,29),(11,172,149,163),(12,36,150,27),(13,170,151,179),(14,34,152,25),(15,168,153,177),(16,32,154,23),(17,166,155,175),(18,30,156,21),(37,119,46,84),(38,74,47,109),(39,117,48,82),(40,90,49,125),(41,115,50,80),(42,88,51,123),(43,113,52,78),(44,86,53,121),(45,111,54,76),(55,214,64,97),(56,105,65,204),(57,212,66,95),(58,103,67,202),(59,210,68,93),(60,101,69,200),(61,208,70,91),(62,99,71,216),(63,206,72,107),(73,198,126,189),(75,196,110,187),(77,194,112,185),(79,192,114,183),(81,190,116,181),(83,188,118,197),(85,186,120,195),(87,184,122,193),(89,182,124,191),(92,131,209,140),(94,129,211,138),(96,127,213,136),(98,143,215,134),(100,141,199,132),(102,139,201,130),(104,137,203,128),(106,135,205,144),(108,133,207,142)]])`

114 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6AB 9A ··· 9I 18A ··· 18BK order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 2 2 2 2 2 2 54 54 54 54 2 ··· 2 2 ··· 2 2 ··· 2

114 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + + - + image C1 C2 C2 C4 S3 S3 D4 Dic3 D6 Dic3 D6 D9 C3⋊D4 C3⋊D4 Dic9 D18 C9⋊D4 kernel C62.127D6 C2×C9⋊Dic3 C2×C6×C18 C6×C18 C22×C18 C2×C62 C3×C18 C2×C18 C2×C18 C62 C62 C22×C6 C18 C3×C6 C2×C6 C2×C6 C6 # reps 1 2 1 4 3 1 2 6 3 2 1 9 12 4 18 9 36

Matrix representation of C62.127D6 in GL5(𝔽37)

 36 0 0 0 0 0 27 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 11 0 0 0 0 0 27 0 0 0 0 0 10 0 0 0 0 0 26
,
 1 0 0 0 0 0 3 0 0 0 0 0 12 0 0 0 0 0 16 0 0 0 0 0 7
,
 6 0 0 0 0 0 0 28 0 0 0 4 0 0 0 0 0 0 0 34 0 0 0 12 0

`G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,27,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,11,0,0,0,0,0,27,0,0,0,0,0,10,0,0,0,0,0,26],[1,0,0,0,0,0,3,0,0,0,0,0,12,0,0,0,0,0,16,0,0,0,0,0,7],[6,0,0,0,0,0,0,4,0,0,0,28,0,0,0,0,0,0,0,12,0,0,0,34,0] >;`

C62.127D6 in GAP, Magma, Sage, TeX

`C_6^2._{127}D_6`
`% in TeX`

`G:=Group("C6^2.127D6");`
`// GroupNames label`

`G:=SmallGroup(432,198);`
`// by ID`

`G=gap.SmallGroup(432,198);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,6164,662,4037,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^4,d^2=a^3*b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^5>;`
`// generators/relations`

׿
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