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## G = C40.2D6order 480 = 25·3·5

### 2nd non-split extension by C40 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C40.2D6
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — D20⋊5S3 — C40.2D6
 Lower central C15 — C30 — C60 — C40.2D6
 Upper central C1 — C2 — C4 — C8

Generators and relations for C40.2D6
G = < a,b,c | a40=b6=1, c2=a20, bab-1=a19, cac-1=a-1, cbc-1=b-1 >

Subgroups: 684 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C3×Q8, C5×S3, C3×D5, C30, C8.C22, C40, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C8⋊S3, Dic12, D4.S3, C3⋊Q16, C3×SD16, D42S3, S3×Q8, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, C40⋊C2, C40⋊C2, Dic20, C5×M4(2), C2×Dic10, C4○D20, D4.D6, C5×C3⋊C8, C120, D5×Dic3, S3×Dic5, C15⋊D4, C15⋊Q8, C3×Dic10, C3×D20, S3×C20, Dic30, C8.D10, C6.D20, C3⋊Dic20, C3×C40⋊C2, C5×C8⋊S3, Dic60, D205S3, S3×Dic10, C40.2D6
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8.C22, D20, C22×D5, S3×D4, S3×D5, C2×D20, D4.D6, C2×S3×D5, C8.D10, S3×D20, C40.2D6

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + - + + - + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 D20 D20 C8.C22 S3×D4 S3×D5 D4.D6 C2×S3×D5 C8.D10 S3×D20 C40.2D6 kernel C40.2D6 C6.D20 C3⋊Dic20 C3×C40⋊C2 C5×C8⋊S3 Dic60 D20⋊5S3 S3×Dic10 C40⋊C2 C5×Dic3 S3×C10 C8⋊S3 C40 Dic10 D20 C3⋊C8 C24 C4×S3 Dic3 D6 C15 C10 C8 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 4 4 1 1 2 2 2 4 4 8

Matrix representation of C40.2D6 in GL4(𝔽241) generated by

 107 181 214 121 133 240 25 239 27 120 134 60 216 2 108 1
,
 205 96 36 145 148 36 93 205 205 96 169 192 148 36 55 72
,
 151 54 90 187 30 90 211 151 180 133 90 187 181 61 211 151
`G:=sub<GL(4,GF(241))| [107,133,27,216,181,240,120,2,214,25,134,108,121,239,60,1],[205,148,205,148,96,36,96,36,36,93,169,55,145,205,192,72],[151,30,180,181,54,90,133,61,90,211,90,211,187,151,187,151] >;`

C40.2D6 in GAP, Magma, Sage, TeX

`C_{40}._2D_6`
`% in TeX`

`G:=Group("C40.2D6");`
`// GroupNames label`

`G:=SmallGroup(480,350);`
`// by ID`

`G=gap.SmallGroup(480,350);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,58,675,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;`
`// generators/relations`

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