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## G = C60.47D4order 480 = 25·3·5

### 47th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.47D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — D30⋊4C4 — C60.47D4
 Lower central C15 — C2×C30 — C60.47D4
 Upper central C1 — C22 — C2×C4

Generators and relations for C60.47D4
G = < a,b,c | a60=b4=c2=1, bab-1=a41, cac=a-1, cbc=a30b-1 >

Subgroups: 988 in 152 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], D5 [×2], C10, C10 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3 [×2], D15 [×2], C30, C30 [×2], C4.4D4, Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C4×Dic3, D6⋊C4 [×4], C2×D12, C6×Q8, C5×Dic3 [×2], C3×Dic5 [×2], C60 [×2], D30 [×6], C2×C30, D10⋊C4 [×4], C4×C20, C2×Dic10, C2×D20, C12.23D4, C3×Dic10 [×2], C6×Dic5 [×2], C10×Dic3 [×2], D60 [×2], C2×C60, C22×D15 [×2], C4.D20, D304C4 [×4], Dic3×C20, C6×Dic10, C2×D60, C60.47D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, C4.4D4, D20 [×2], C22×D5, Q83S3 [×2], C2×C3⋊D4, S3×D5, C2×D20, C4○D20 [×2], C12.23D4, C3⋊D20 [×2], C2×S3×D5, C4.D20, D60⋊C2 [×2], C2×C3⋊D20, C60.47D4

Smallest permutation representation of C60.47D4
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)
(1 173 119 235)(2 154 120 216)(3 135 61 197)(4 176 62 238)(5 157 63 219)(6 138 64 200)(7 179 65 181)(8 160 66 222)(9 141 67 203)(10 122 68 184)(11 163 69 225)(12 144 70 206)(13 125 71 187)(14 166 72 228)(15 147 73 209)(16 128 74 190)(17 169 75 231)(18 150 76 212)(19 131 77 193)(20 172 78 234)(21 153 79 215)(22 134 80 196)(23 175 81 237)(24 156 82 218)(25 137 83 199)(26 178 84 240)(27 159 85 221)(28 140 86 202)(29 121 87 183)(30 162 88 224)(31 143 89 205)(32 124 90 186)(33 165 91 227)(34 146 92 208)(35 127 93 189)(36 168 94 230)(37 149 95 211)(38 130 96 192)(39 171 97 233)(40 152 98 214)(41 133 99 195)(42 174 100 236)(43 155 101 217)(44 136 102 198)(45 177 103 239)(46 158 104 220)(47 139 105 201)(48 180 106 182)(49 161 107 223)(50 142 108 204)(51 123 109 185)(52 164 110 226)(53 145 111 207)(54 126 112 188)(55 167 113 229)(56 148 114 210)(57 129 115 191)(58 170 116 232)(59 151 117 213)(60 132 118 194)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 120)(74 119)(75 118)(76 117)(77 116)(78 115)(79 114)(80 113)(81 112)(82 111)(83 110)(84 109)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(121 212)(122 211)(123 210)(124 209)(125 208)(126 207)(127 206)(128 205)(129 204)(130 203)(131 202)(132 201)(133 200)(134 199)(135 198)(136 197)(137 196)(138 195)(139 194)(140 193)(141 192)(142 191)(143 190)(144 189)(145 188)(146 187)(147 186)(148 185)(149 184)(150 183)(151 182)(152 181)(153 240)(154 239)(155 238)(156 237)(157 236)(158 235)(159 234)(160 233)(161 232)(162 231)(163 230)(164 229)(165 228)(166 227)(167 226)(168 225)(169 224)(170 223)(171 222)(172 221)(173 220)(174 219)(175 218)(176 217)(177 216)(178 215)(179 214)(180 213)```

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D5 D6 D6 C4○D4 D10 D10 C3⋊D4 D20 C4○D20 Q8⋊3S3 S3×D5 C3⋊D20 C2×S3×D5 D60⋊C2 kernel C60.47D4 D30⋊4C4 Dic3×C20 C6×Dic10 C2×D60 C2×Dic10 C60 C4×Dic3 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C20 C12 C6 C10 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 1 2 2 2 1 4 4 2 4 8 16 2 2 4 2 8

Matrix representation of C60.47D4 in GL4(𝔽61) generated by

 29 59 0 0 2 2 0 0 0 0 2 49 0 0 46 60
,
 47 45 0 0 16 14 0 0 0 0 27 14 0 0 35 34
,
 29 59 0 0 54 32 0 0 0 0 60 12 0 0 0 1
`G:=sub<GL(4,GF(61))| [29,2,0,0,59,2,0,0,0,0,2,46,0,0,49,60],[47,16,0,0,45,14,0,0,0,0,27,35,0,0,14,34],[29,54,0,0,59,32,0,0,0,0,60,0,0,0,12,1] >;`

C60.47D4 in GAP, Magma, Sage, TeX

`C_{60}._{47}D_4`
`% in TeX`

`G:=Group("C60.47D4");`
`// GroupNames label`

`G:=SmallGroup(480,450);`
`// by ID`

`G=gap.SmallGroup(480,450);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,176,422,100,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^41,c*a*c=a^-1,c*b*c=a^30*b^-1>;`
`// generators/relations`

׿
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