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## G = C3⋊D80order 480 = 25·3·5

### The semidirect product of C3 and D80 acting via D80/D40=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C120 — C3⋊D80
 Chief series C1 — C5 — C15 — C30 — C60 — C120 — C3×D40 — C3⋊D80
 Lower central C15 — C30 — C60 — C120 — C3⋊D80
 Upper central C1 — C2 — C4 — C8

Generators and relations for C3⋊D80
G = < a,b,c | a3=b80=c2=1, bab-1=cac=a-1, cbc=b-1 >

40C2
120C2
20C22
60C22
40C6
40S3
8D5
24D5
10D4
30D4
20D6
20C2×C6
4D10
12D10
8D15
3C16
5D8
15D8
10C3×D4
10D12
2D20
6D20
4D30
15D16
5D24
3C80
3D40
2D60
3D80

Smallest permutation representation of C3⋊D80
On 240 points
Generators in S240
```(1 229 126)(2 127 230)(3 231 128)(4 129 232)(5 233 130)(6 131 234)(7 235 132)(8 133 236)(9 237 134)(10 135 238)(11 239 136)(12 137 240)(13 161 138)(14 139 162)(15 163 140)(16 141 164)(17 165 142)(18 143 166)(19 167 144)(20 145 168)(21 169 146)(22 147 170)(23 171 148)(24 149 172)(25 173 150)(26 151 174)(27 175 152)(28 153 176)(29 177 154)(30 155 178)(31 179 156)(32 157 180)(33 181 158)(34 159 182)(35 183 160)(36 81 184)(37 185 82)(38 83 186)(39 187 84)(40 85 188)(41 189 86)(42 87 190)(43 191 88)(44 89 192)(45 193 90)(46 91 194)(47 195 92)(48 93 196)(49 197 94)(50 95 198)(51 199 96)(52 97 200)(53 201 98)(54 99 202)(55 203 100)(56 101 204)(57 205 102)(58 103 206)(59 207 104)(60 105 208)(61 209 106)(62 107 210)(63 211 108)(64 109 212)(65 213 110)(66 111 214)(67 215 112)(68 113 216)(69 217 114)(70 115 218)(71 219 116)(72 117 220)(73 221 118)(74 119 222)(75 223 120)(76 121 224)(77 225 122)(78 123 226)(79 227 124)(80 125 228)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 80)(17 79)(18 78)(19 77)(20 76)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(81 208)(82 207)(83 206)(84 205)(85 204)(86 203)(87 202)(88 201)(89 200)(90 199)(91 198)(92 197)(93 196)(94 195)(95 194)(96 193)(97 192)(98 191)(99 190)(100 189)(101 188)(102 187)(103 186)(104 185)(105 184)(106 183)(107 182)(108 181)(109 180)(110 179)(111 178)(112 177)(113 176)(114 175)(115 174)(116 173)(117 172)(118 171)(119 170)(120 169)(121 168)(122 167)(123 166)(124 165)(125 164)(126 163)(127 162)(128 161)(129 240)(130 239)(131 238)(132 237)(133 236)(134 235)(135 234)(136 233)(137 232)(138 231)(139 230)(140 229)(141 228)(142 227)(143 226)(144 225)(145 224)(146 223)(147 222)(148 221)(149 220)(150 219)(151 218)(152 217)(153 216)(154 215)(155 214)(156 213)(157 212)(158 211)(159 210)(160 209)```

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 3 4 5A 5B 6A 6B 6C 8A 8B 10A 10B 12 15A 15B 16A 16B 16C 16D 20A 20B 20C 20D 24A 24B 30A 30B 40A ··· 40H 60A 60B 60C 60D 80A ··· 80P 120A ··· 120H order 1 2 2 2 3 4 5 5 6 6 6 8 8 10 10 12 15 15 16 16 16 16 20 20 20 20 24 24 30 30 40 ··· 40 60 60 60 60 80 ··· 80 120 ··· 120 size 1 1 40 120 2 2 2 2 2 40 40 2 2 2 2 4 4 4 6 6 6 6 2 2 2 2 4 4 4 4 2 ··· 2 4 4 4 4 6 ··· 6 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D5 D6 D8 D10 C3⋊D4 D16 D20 D40 D80 D4⋊S3 S3×D5 C3⋊D16 C3⋊D20 C3⋊D40 C3⋊D80 kernel C3⋊D80 C5×C3⋊C16 C3×D40 D120 D40 C60 C3⋊C16 C40 C30 C24 C20 C15 C12 C6 C3 C10 C8 C5 C4 C2 C1 # reps 1 1 1 1 1 1 2 1 2 2 2 4 4 8 16 1 2 2 2 4 8

Matrix representation of C3⋊D80 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 1 0 0 0 0 240 0
,
 227 42 0 0 0 0 199 212 0 0 0 0 0 0 237 136 0 0 0 0 156 58 0 0 0 0 0 0 101 70 0 0 0 0 171 140
,
 119 41 0 0 0 0 119 122 0 0 0 0 0 0 24 114 0 0 0 0 107 217 0 0 0 0 0 0 0 240 0 0 0 0 240 0

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[227,199,0,0,0,0,42,212,0,0,0,0,0,0,237,156,0,0,0,0,136,58,0,0,0,0,0,0,101,171,0,0,0,0,70,140],[119,119,0,0,0,0,41,122,0,0,0,0,0,0,24,107,0,0,0,0,114,217,0,0,0,0,0,0,0,240,0,0,0,0,240,0] >;`

C3⋊D80 in GAP, Magma, Sage, TeX

`C_3\rtimes D_{80}`
`% in TeX`

`G:=Group("C3:D80");`
`// GroupNames label`

`G:=SmallGroup(480,14);`
`// by ID`

`G=gap.SmallGroup(480,14);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,92,254,142,675,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=b^80=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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