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## G = C62.148D6order 432 = 24·33

### 32nd non-split extension by C62 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C62.148D6
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — C22×C33⋊C2 — C62.148D6
 Lower central C33 — C32×C6 — C62.148D6
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.148D6
G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 2984 in 476 conjugacy classes, 173 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C3⋊S3, C3×C6, C2×Dic3, C2×C12, C22×S3, C33, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, D6⋊C4, C33⋊C2, C32×C6, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C335C4, C32×C12, C2×C33⋊C2, C2×C33⋊C2, C3×C62, C6.11D12, C2×C335C4, C3×C6×C12, C22×C33⋊C2, C62.148D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C3⋊S3, C4×S3, D12, C3⋊D4, C2×C3⋊S3, D6⋊C4, C33⋊C2, C4×C3⋊S3, C12⋊S3, C327D4, C2×C33⋊C2, C6.11D12, C4×C33⋊C2, C3312D4, C3315D4, C62.148D6

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

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

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

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

114 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A ··· 3M 4A 4B 4C 4D 6A ··· 6AM 12A ··· 12AZ order 1 2 2 2 2 2 3 ··· 3 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 54 54 2 ··· 2 2 2 54 54 2 ··· 2 2 ··· 2

114 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D6 C4×S3 D12 C3⋊D4 kernel C62.148D6 C2×C33⋊5C4 C3×C6×C12 C22×C33⋊C2 C2×C33⋊C2 C6×C12 C32×C6 C62 C3×C6 C3×C6 C3×C6 # reps 1 1 1 1 4 13 2 13 26 26 26

Matrix representation of C62.148D6 in GL6(𝔽13)

 12 10 0 0 0 0 1 2 0 0 0 0 0 0 0 1 0 0 0 0 12 1 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 2 3 0 0 0 0 12 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 1 0
,
 4 4 0 0 0 0 3 0 0 0 0 0 0 0 3 3 0 0 0 0 10 6 0 0 0 0 0 0 2 2 0 0 0 0 11 4
,
 4 4 0 0 0 0 6 9 0 0 0 0 0 0 10 6 0 0 0 0 3 3 0 0 0 0 0 0 11 11 0 0 0 0 9 2

`G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,10,2,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[2,12,0,0,0,0,3,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[4,3,0,0,0,0,4,0,0,0,0,0,0,0,3,10,0,0,0,0,3,6,0,0,0,0,0,0,2,11,0,0,0,0,2,4],[4,6,0,0,0,0,4,9,0,0,0,0,0,0,10,3,0,0,0,0,6,3,0,0,0,0,0,0,11,9,0,0,0,0,11,2] >;`

C62.148D6 in GAP, Magma, Sage, TeX

`C_6^2._{148}D_6`
`% in TeX`

`G:=Group("C6^2.148D6");`
`// GroupNames label`

`G:=SmallGroup(432,506);`
`// by ID`

`G=gap.SmallGroup(432,506);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,1124,4037,14118]);`
`// Polycyclic`

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

׿
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