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## G = C40.34D6order 480 = 25·3·5

### 34th non-split extension by C40 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C40.34D6
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D6.D10 — C40.34D6
 Lower central C15 — C30 — C40.34D6
 Upper central C1 — C4 — C8

Generators and relations for C40.34D6
G = < a,b,c | a40=b6=1, c2=a20, bab-1=a9, cac-1=a29, cbc-1=a20b-1 >

Subgroups: 540 in 124 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, D5 [×2], C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8 [×3], M4(2) [×3], C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4 [×2], C2×C12, C5×S3, C3×D5, D15, C30, C8○D4, C52C8, C52C8, C40, C40, Dic10, C4×D5, C4×D5, D20, C5⋊D4 [×2], C2×C20, S3×C8 [×2], C8⋊S3, C8⋊S3, C4.Dic3, C2×C24, C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C8×D5, C8×D5, C8⋊D5 [×2], C2×C52C8, C5×M4(2), C4○D20, C8○D12, C5×C3⋊C8, C3×C52C8, C153C8, C120, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, S3×C20, C4×D15, D20.2C4, S3×C52C8, D152C8, C20.32D6, D5×C24, C5×C8⋊S3, C40⋊S3, D6.D10, C40.34D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C8○D4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, C8○D12, C2×S3×D5, D20.2C4, C4×S3×D5, C40.34D6

Smallest permutation representation of C40.34D6
On 240 points
Generators in S240
```(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)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 205 200)(2 214 161 10 206 169)(3 223 162 19 207 178)(4 232 163 28 208 187)(5 201 164 37 209 196)(6 210 165)(7 219 166 15 211 174)(8 228 167 24 212 183)(9 237 168 33 213 192)(11 215 170)(12 224 171 20 216 179)(13 233 172 29 217 188)(14 202 173 38 218 197)(16 220 175)(17 229 176 25 221 184)(18 238 177 34 222 193)(21 225 180)(22 234 181 30 226 189)(23 203 182 39 227 198)(26 230 185)(27 239 186 35 231 194)(31 235 190)(32 204 191 40 236 199)(36 240 195)(41 119 157 77 83 153)(42 88 158 46 84 122)(43 97 159 55 85 131)(44 106 160 64 86 140)(45 115 121 73 87 149)(47 93 123 51 89 127)(48 102 124 60 90 136)(49 111 125 69 91 145)(50 120 126 78 92 154)(52 98 128 56 94 132)(53 107 129 65 95 141)(54 116 130 74 96 150)(57 103 133 61 99 137)(58 112 134 70 100 146)(59 81 135 79 101 155)(62 108 138 66 104 142)(63 117 139 75 105 151)(67 113 143 71 109 147)(68 82 144 80 110 156)(72 118 148 76 114 152)
(1 59 21 79)(2 48 22 68)(3 77 23 57)(4 66 24 46)(5 55 25 75)(6 44 26 64)(7 73 27 53)(8 62 28 42)(9 51 29 71)(10 80 30 60)(11 69 31 49)(12 58 32 78)(13 47 33 67)(14 76 34 56)(15 65 35 45)(16 54 36 74)(17 43 37 63)(18 72 38 52)(19 61 39 41)(20 50 40 70)(81 200 101 180)(82 189 102 169)(83 178 103 198)(84 167 104 187)(85 196 105 176)(86 185 106 165)(87 174 107 194)(88 163 108 183)(89 192 109 172)(90 181 110 161)(91 170 111 190)(92 199 112 179)(93 188 113 168)(94 177 114 197)(95 166 115 186)(96 195 116 175)(97 184 117 164)(98 173 118 193)(99 162 119 182)(100 191 120 171)(121 219 141 239)(122 208 142 228)(123 237 143 217)(124 226 144 206)(125 215 145 235)(126 204 146 224)(127 233 147 213)(128 222 148 202)(129 211 149 231)(130 240 150 220)(131 229 151 209)(132 218 152 238)(133 207 153 227)(134 236 154 216)(135 225 155 205)(136 214 156 234)(137 203 157 223)(138 232 158 212)(139 221 159 201)(140 210 160 230)```

`G:=sub<Sym(240)| (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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,205,200)(2,214,161,10,206,169)(3,223,162,19,207,178)(4,232,163,28,208,187)(5,201,164,37,209,196)(6,210,165)(7,219,166,15,211,174)(8,228,167,24,212,183)(9,237,168,33,213,192)(11,215,170)(12,224,171,20,216,179)(13,233,172,29,217,188)(14,202,173,38,218,197)(16,220,175)(17,229,176,25,221,184)(18,238,177,34,222,193)(21,225,180)(22,234,181,30,226,189)(23,203,182,39,227,198)(26,230,185)(27,239,186,35,231,194)(31,235,190)(32,204,191,40,236,199)(36,240,195)(41,119,157,77,83,153)(42,88,158,46,84,122)(43,97,159,55,85,131)(44,106,160,64,86,140)(45,115,121,73,87,149)(47,93,123,51,89,127)(48,102,124,60,90,136)(49,111,125,69,91,145)(50,120,126,78,92,154)(52,98,128,56,94,132)(53,107,129,65,95,141)(54,116,130,74,96,150)(57,103,133,61,99,137)(58,112,134,70,100,146)(59,81,135,79,101,155)(62,108,138,66,104,142)(63,117,139,75,105,151)(67,113,143,71,109,147)(68,82,144,80,110,156)(72,118,148,76,114,152), (1,59,21,79)(2,48,22,68)(3,77,23,57)(4,66,24,46)(5,55,25,75)(6,44,26,64)(7,73,27,53)(8,62,28,42)(9,51,29,71)(10,80,30,60)(11,69,31,49)(12,58,32,78)(13,47,33,67)(14,76,34,56)(15,65,35,45)(16,54,36,74)(17,43,37,63)(18,72,38,52)(19,61,39,41)(20,50,40,70)(81,200,101,180)(82,189,102,169)(83,178,103,198)(84,167,104,187)(85,196,105,176)(86,185,106,165)(87,174,107,194)(88,163,108,183)(89,192,109,172)(90,181,110,161)(91,170,111,190)(92,199,112,179)(93,188,113,168)(94,177,114,197)(95,166,115,186)(96,195,116,175)(97,184,117,164)(98,173,118,193)(99,162,119,182)(100,191,120,171)(121,219,141,239)(122,208,142,228)(123,237,143,217)(124,226,144,206)(125,215,145,235)(126,204,146,224)(127,233,147,213)(128,222,148,202)(129,211,149,231)(130,240,150,220)(131,229,151,209)(132,218,152,238)(133,207,153,227)(134,236,154,216)(135,225,155,205)(136,214,156,234)(137,203,157,223)(138,232,158,212)(139,221,159,201)(140,210,160,230)>;`

`G:=Group( (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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,205,200)(2,214,161,10,206,169)(3,223,162,19,207,178)(4,232,163,28,208,187)(5,201,164,37,209,196)(6,210,165)(7,219,166,15,211,174)(8,228,167,24,212,183)(9,237,168,33,213,192)(11,215,170)(12,224,171,20,216,179)(13,233,172,29,217,188)(14,202,173,38,218,197)(16,220,175)(17,229,176,25,221,184)(18,238,177,34,222,193)(21,225,180)(22,234,181,30,226,189)(23,203,182,39,227,198)(26,230,185)(27,239,186,35,231,194)(31,235,190)(32,204,191,40,236,199)(36,240,195)(41,119,157,77,83,153)(42,88,158,46,84,122)(43,97,159,55,85,131)(44,106,160,64,86,140)(45,115,121,73,87,149)(47,93,123,51,89,127)(48,102,124,60,90,136)(49,111,125,69,91,145)(50,120,126,78,92,154)(52,98,128,56,94,132)(53,107,129,65,95,141)(54,116,130,74,96,150)(57,103,133,61,99,137)(58,112,134,70,100,146)(59,81,135,79,101,155)(62,108,138,66,104,142)(63,117,139,75,105,151)(67,113,143,71,109,147)(68,82,144,80,110,156)(72,118,148,76,114,152), (1,59,21,79)(2,48,22,68)(3,77,23,57)(4,66,24,46)(5,55,25,75)(6,44,26,64)(7,73,27,53)(8,62,28,42)(9,51,29,71)(10,80,30,60)(11,69,31,49)(12,58,32,78)(13,47,33,67)(14,76,34,56)(15,65,35,45)(16,54,36,74)(17,43,37,63)(18,72,38,52)(19,61,39,41)(20,50,40,70)(81,200,101,180)(82,189,102,169)(83,178,103,198)(84,167,104,187)(85,196,105,176)(86,185,106,165)(87,174,107,194)(88,163,108,183)(89,192,109,172)(90,181,110,161)(91,170,111,190)(92,199,112,179)(93,188,113,168)(94,177,114,197)(95,166,115,186)(96,195,116,175)(97,184,117,164)(98,173,118,193)(99,162,119,182)(100,191,120,171)(121,219,141,239)(122,208,142,228)(123,237,143,217)(124,226,144,206)(125,215,145,235)(126,204,146,224)(127,233,147,213)(128,222,148,202)(129,211,149,231)(130,240,150,220)(131,229,151,209)(132,218,152,238)(133,207,153,227)(134,236,154,216)(135,225,155,205)(136,214,156,234)(137,203,157,223)(138,232,158,212)(139,221,159,201)(140,210,160,230) );`

`G=PermutationGroup([(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),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,205,200),(2,214,161,10,206,169),(3,223,162,19,207,178),(4,232,163,28,208,187),(5,201,164,37,209,196),(6,210,165),(7,219,166,15,211,174),(8,228,167,24,212,183),(9,237,168,33,213,192),(11,215,170),(12,224,171,20,216,179),(13,233,172,29,217,188),(14,202,173,38,218,197),(16,220,175),(17,229,176,25,221,184),(18,238,177,34,222,193),(21,225,180),(22,234,181,30,226,189),(23,203,182,39,227,198),(26,230,185),(27,239,186,35,231,194),(31,235,190),(32,204,191,40,236,199),(36,240,195),(41,119,157,77,83,153),(42,88,158,46,84,122),(43,97,159,55,85,131),(44,106,160,64,86,140),(45,115,121,73,87,149),(47,93,123,51,89,127),(48,102,124,60,90,136),(49,111,125,69,91,145),(50,120,126,78,92,154),(52,98,128,56,94,132),(53,107,129,65,95,141),(54,116,130,74,96,150),(57,103,133,61,99,137),(58,112,134,70,100,146),(59,81,135,79,101,155),(62,108,138,66,104,142),(63,117,139,75,105,151),(67,113,143,71,109,147),(68,82,144,80,110,156),(72,118,148,76,114,152)], [(1,59,21,79),(2,48,22,68),(3,77,23,57),(4,66,24,46),(5,55,25,75),(6,44,26,64),(7,73,27,53),(8,62,28,42),(9,51,29,71),(10,80,30,60),(11,69,31,49),(12,58,32,78),(13,47,33,67),(14,76,34,56),(15,65,35,45),(16,54,36,74),(17,43,37,63),(18,72,38,52),(19,61,39,41),(20,50,40,70),(81,200,101,180),(82,189,102,169),(83,178,103,198),(84,167,104,187),(85,196,105,176),(86,185,106,165),(87,174,107,194),(88,163,108,183),(89,192,109,172),(90,181,110,161),(91,170,111,190),(92,199,112,179),(93,188,113,168),(94,177,114,197),(95,166,115,186),(96,195,116,175),(97,184,117,164),(98,173,118,193),(99,162,119,182),(100,191,120,171),(121,219,141,239),(122,208,142,228),(123,237,143,217),(124,226,144,206),(125,215,145,235),(126,204,146,224),(127,233,147,213),(128,222,148,202),(129,211,149,231),(130,240,150,220),(131,229,151,209),(132,218,152,238),(133,207,153,227),(134,236,154,216),(135,225,155,205),(136,214,156,234),(137,203,157,223),(138,232,158,212),(139,221,159,201),(140,210,160,230)])`

72 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A 10B 10C 10D 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 24E 24F 24G 24H 30A 30B 40A 40B 40C 40D 40E 40F 40G 40H 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 2 3 4 4 4 4 4 5 5 6 6 6 8 8 8 8 8 8 8 8 8 8 10 10 10 10 12 12 12 12 15 15 20 20 20 20 20 20 24 24 24 24 24 24 24 24 30 30 40 40 40 40 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 6 10 30 2 1 1 6 10 30 2 2 2 10 10 2 2 5 5 5 5 6 6 30 30 2 2 12 12 2 2 10 10 4 4 2 2 2 2 12 12 2 2 2 2 10 10 10 10 4 4 4 4 4 4 12 12 12 12 4 4 4 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D5 D6 D6 D6 D10 D10 D10 C4×S3 C4×S3 C8○D4 C4×D5 C4×D5 C8○D12 S3×D5 C2×S3×D5 D20.2C4 C4×S3×D5 C40.34D6 kernel C40.34D6 S3×C5⋊2C8 D15⋊2C8 C20.32D6 D5×C24 C5×C8⋊S3 C40⋊S3 D6.D10 C15⋊D4 C3⋊D20 C5⋊D12 C15⋊Q8 C8×D5 C8⋊S3 C5⋊2C8 C40 C4×D5 C3⋊C8 C24 C4×S3 Dic5 D10 C15 Dic3 D6 C5 C8 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 1 2 1 1 1 2 2 2 2 2 4 4 4 8 2 2 4 4 8

Matrix representation of C40.34D6 in GL4(𝔽241) generated by

 195 46 0 0 195 177 0 0 0 0 233 0 0 0 0 8
,
 189 52 0 0 240 52 0 0 0 0 15 0 0 0 0 16
,
 52 189 0 0 1 189 0 0 0 0 0 225 0 0 226 0
`G:=sub<GL(4,GF(241))| [195,195,0,0,46,177,0,0,0,0,233,0,0,0,0,8],[189,240,0,0,52,52,0,0,0,0,15,0,0,0,0,16],[52,1,0,0,189,189,0,0,0,0,0,226,0,0,225,0] >;`

C40.34D6 in GAP, Magma, Sage, TeX

`C_{40}._{34}D_6`
`% in TeX`

`G:=Group("C40.34D6");`
`// GroupNames label`

`G:=SmallGroup(480,342);`
`// by ID`

`G=gap.SmallGroup(480,342);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,219,58,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=a^9,c*a*c^-1=a^29,c*b*c^-1=a^20*b^-1>;`
`// generators/relations`

׿
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