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## G = C60.39C23order 480 = 25·3·5

### 39th non-split extension by C60 of C23 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C60.39C23
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — C12.28D10 — C60.39C23
 Lower central C15 — C30 — C60 — C60.39C23
 Upper central C1 — C2 — C4 — Q8

Generators and relations for C60.39C23
G = < a,b,c,d | a60=b2=1, c2=d2=a30, bab=a49, cac-1=a11, dad-1=a31, cbc-1=a30b, bd=db, dcd-1=a45c >

Subgroups: 652 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3, C6, C6, C8 [×2], C2×C4 [×3], D4 [×2], Q8, Q8 [×3], D5 [×2], C10, Dic3, C12, C12 [×3], D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5, Dic5, C20, C20 [×2], D10, D10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12 [×2], C3×Q8, C3×Q8 [×2], C3×D5, D15, C30, C8.C22, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5 [×2], D20 [×2], C5×Q8, C5×Q8, C4.Dic3, Q82S3 [×2], C3⋊Q16, C3⋊Q16, C4○D12, C6×Q8, C5×Dic3, C3×Dic5, C3×Dic5, C60, C60, C6×D5, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q82D5, Q8.11D6, C5×C3⋊C8, C153C8, D30.C2, C3⋊D20, C3×Dic10, C3×Dic10, D5×C12, D5×C12, C5×Dic6, D60, Q8×C15, Q16⋊D5, C20.32D6, C15⋊SD16, C15⋊Q16, C5×C3⋊Q16, Q82D15, C12.28D10, C3×Q8×D5, C60.39C23
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8.C22, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, Q8.11D6, C2×S3×D5, Q16⋊D5, D5×C3⋊D4, C60.39C23

Smallest permutation representation of C60.39C23
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)
(2 50)(3 39)(4 28)(5 17)(7 55)(8 44)(9 33)(10 22)(12 60)(13 49)(14 38)(15 27)(18 54)(19 43)(20 32)(23 59)(24 48)(25 37)(29 53)(30 42)(34 58)(35 47)(40 52)(45 57)(61 97)(62 86)(63 75)(65 113)(66 102)(67 91)(68 80)(70 118)(71 107)(72 96)(73 85)(76 112)(77 101)(78 90)(81 117)(82 106)(83 95)(87 111)(88 100)(92 116)(93 105)(98 110)(103 115)(108 120)(121 163)(122 152)(123 141)(124 130)(125 179)(126 168)(127 157)(128 146)(129 135)(131 173)(132 162)(133 151)(134 140)(136 178)(137 167)(138 156)(139 145)(142 172)(143 161)(144 150)(147 177)(148 166)(149 155)(153 171)(154 160)(158 176)(159 165)(164 170)(169 175)(174 180)(181 223)(182 212)(183 201)(184 190)(185 239)(186 228)(187 217)(188 206)(189 195)(191 233)(192 222)(193 211)(194 200)(196 238)(197 227)(198 216)(199 205)(202 232)(203 221)(204 210)(207 237)(208 226)(209 215)(213 231)(214 220)(218 236)(219 225)(224 230)(229 235)(234 240)
(1 142 31 172)(2 153 32 123)(3 164 33 134)(4 175 34 145)(5 126 35 156)(6 137 36 167)(7 148 37 178)(8 159 38 129)(9 170 39 140)(10 121 40 151)(11 132 41 162)(12 143 42 173)(13 154 43 124)(14 165 44 135)(15 176 45 146)(16 127 46 157)(17 138 47 168)(18 149 48 179)(19 160 49 130)(20 171 50 141)(21 122 51 152)(22 133 52 163)(23 144 53 174)(24 155 54 125)(25 166 55 136)(26 177 56 147)(27 128 57 158)(28 139 58 169)(29 150 59 180)(30 161 60 131)(61 229 91 199)(62 240 92 210)(63 191 93 221)(64 202 94 232)(65 213 95 183)(66 224 96 194)(67 235 97 205)(68 186 98 216)(69 197 99 227)(70 208 100 238)(71 219 101 189)(72 230 102 200)(73 181 103 211)(74 192 104 222)(75 203 105 233)(76 214 106 184)(77 225 107 195)(78 236 108 206)(79 187 109 217)(80 198 110 228)(81 209 111 239)(82 220 112 190)(83 231 113 201)(84 182 114 212)(85 193 115 223)(86 204 116 234)(87 215 117 185)(88 226 118 196)(89 237 119 207)(90 188 120 218)
(1 74 31 104)(2 105 32 75)(3 76 33 106)(4 107 34 77)(5 78 35 108)(6 109 36 79)(7 80 37 110)(8 111 38 81)(9 82 39 112)(10 113 40 83)(11 84 41 114)(12 115 42 85)(13 86 43 116)(14 117 44 87)(15 88 45 118)(16 119 46 89)(17 90 47 120)(18 61 48 91)(19 92 49 62)(20 63 50 93)(21 94 51 64)(22 65 52 95)(23 96 53 66)(24 67 54 97)(25 98 55 68)(26 69 56 99)(27 100 57 70)(28 71 58 101)(29 102 59 72)(30 73 60 103)(121 216 151 186)(122 187 152 217)(123 218 153 188)(124 189 154 219)(125 220 155 190)(126 191 156 221)(127 222 157 192)(128 193 158 223)(129 224 159 194)(130 195 160 225)(131 226 161 196)(132 197 162 227)(133 228 163 198)(134 199 164 229)(135 230 165 200)(136 201 166 231)(137 232 167 202)(138 203 168 233)(139 234 169 204)(140 205 170 235)(141 236 171 206)(142 207 172 237)(143 238 173 208)(144 209 174 239)(145 240 175 210)(146 211 176 181)(147 182 177 212)(148 213 178 183)(149 184 179 214)(150 215 180 185)```

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

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

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

45 conjugacy classes

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

45 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 8 type + + + + + + + + + + + + + + + + + + - + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 D6 D10 D10 D10 C3⋊D4 C3⋊D4 C8.C22 S3×D5 D4×D5 Q8.11D6 C2×S3×D5 Q16⋊D5 D5×C3⋊D4 C60.39C23 kernel C60.39C23 C20.32D6 C15⋊SD16 C15⋊Q16 C5×C3⋊Q16 Q8⋊2D15 C12.28D10 C3×Q8×D5 Q8×D5 C3×Dic5 C6×D5 C3⋊Q16 Dic10 C4×D5 C5×Q8 C3⋊C8 Dic6 C3×Q8 Dic5 D10 C15 Q8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 1 2 2 2 2 4 4 2

Matrix representation of C60.39C23 in GL8(𝔽241)

 1 189 0 0 0 0 0 0 52 189 0 0 0 0 0 0 97 0 190 189 0 0 0 0 127 144 51 0 0 0 0 0 0 0 0 0 226 188 71 209 0 0 0 0 154 15 89 89 0 0 0 0 0 0 0 16 0 0 0 0 0 0 225 0
,
 189 52 0 0 0 0 0 0 240 52 0 0 0 0 0 0 144 0 51 52 0 0 0 0 114 97 191 190 0 0 0 0 0 0 0 0 1 0 142 142 0 0 0 0 0 1 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 240
,
 145 0 180 219 0 0 0 0 98 2 44 22 0 0 0 0 227 227 239 0 0 0 0 0 140 236 206 96 0 0 0 0 0 0 0 0 219 0 67 67 0 0 0 0 0 0 0 1 0 0 0 0 187 1 22 22 0 0 0 0 0 240 0 0
,
 1 0 104 239 0 0 0 0 228 229 4 2 0 0 0 0 50 50 12 0 0 0 0 0 190 190 194 240 0 0 0 0 0 0 0 0 141 87 227 19 0 0 0 0 223 100 168 168 0 0 0 0 0 0 80 180 0 0 0 0 0 0 180 161

`G:=sub<GL(8,GF(241))| [1,52,97,127,0,0,0,0,189,189,0,144,0,0,0,0,0,0,190,51,0,0,0,0,0,0,189,0,0,0,0,0,0,0,0,0,226,154,0,0,0,0,0,0,188,15,0,0,0,0,0,0,71,89,0,225,0,0,0,0,209,89,16,0],[189,240,144,114,0,0,0,0,52,52,0,97,0,0,0,0,0,0,51,191,0,0,0,0,0,0,52,190,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,142,0,240,0,0,0,0,0,142,0,0,240],[145,98,227,140,0,0,0,0,0,2,227,236,0,0,0,0,180,44,239,206,0,0,0,0,219,22,0,96,0,0,0,0,0,0,0,0,219,0,187,0,0,0,0,0,0,0,1,240,0,0,0,0,67,0,22,0,0,0,0,0,67,1,22,0],[1,228,50,190,0,0,0,0,0,229,50,190,0,0,0,0,104,4,12,194,0,0,0,0,239,2,0,240,0,0,0,0,0,0,0,0,141,223,0,0,0,0,0,0,87,100,0,0,0,0,0,0,227,168,80,180,0,0,0,0,19,168,180,161] >;`

C60.39C23 in GAP, Magma, Sage, TeX

`C_{60}._{39}C_2^3`
`% in TeX`

`G:=Group("C60.39C2^3");`
`// GroupNames label`

`G:=SmallGroup(480,591);`
`// by ID`

`G=gap.SmallGroup(480,591);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,135,100,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^60=b^2=1,c^2=d^2=a^30,b*a*b=a^49,c*a*c^-1=a^11,d*a*d^-1=a^31,c*b*c^-1=a^30*b,b*d=d*b,d*c*d^-1=a^45*c>;`
`// generators/relations`

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