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## G = C60.8C23order 480 = 25·3·5

### 8th non-split extension by C60 of C23 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C60.8C23
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — D5×Dic6 — C60.8C23
 Lower central C15 — C30 — C60 — C60.8C23
 Upper central C1 — C2 — C4 — D4

Generators and relations for C60.8C23
G = < a,b,c,d | a60=b2=d2=1, c2=a30, bab=a49, cac-1=a11, dad=a31, bc=cb, dbd=a30b, dcd=a45c >

Subgroups: 572 in 120 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, C6, C6 [×2], C8 [×2], C2×C4 [×3], D4, D4, Q8 [×4], D5, C10, C10, Dic3 [×2], C12, C12 [×2], C2×C6 [×2], C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5, Dic5 [×2], C20, C20, D10, C2×C10, C3⋊C8, C3⋊C8, Dic6, Dic6 [×2], C2×Dic3, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C3×D5, C30, C30, C8.C22, C52C8, C40, Dic10, Dic10 [×2], C4×D5, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C4.Dic3, D4.S3, D4.S3, C3⋊Q16 [×2], C2×Dic6, C3×C4○D4, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C6×D5, C2×C30, C8⋊D5, Dic20, D4.D5, C5⋊Q16, C5×SD16, D42D5, Q8×D5, Q8.14D6, C5×C3⋊C8, C153C8, D5×Dic3, C15⋊Q8, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, C5×Dic6, Dic30, D4×C15, SD16⋊D5, C20.32D6, C15⋊Q16, C3⋊Dic20, C5×D4.S3, D4.D15, D5×Dic6, C3×D42D5, C60.8C23
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, Q8.14D6, C2×S3×D5, SD16⋊D5, D5×C3⋊D4, C60.8C23

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 5A 5B 6A 6B 6C 6D 8A 8B 10A 10B 10C 10D 12A 12B 12C 12D 12E 15A 15B 20A 20B 20C 20D 30A 30B 30C 30D 30E 30F 40A 40B 40C 40D 60A 60B order 1 2 2 2 3 4 4 4 4 4 5 5 6 6 6 6 8 8 10 10 10 10 12 12 12 12 12 15 15 20 20 20 20 30 30 30 30 30 30 40 40 40 40 60 60 size 1 1 4 10 2 2 10 12 20 60 2 2 2 4 4 20 12 60 2 2 8 8 4 10 10 20 20 4 4 4 4 24 24 4 4 8 8 8 8 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 Q8.14D6 C2×S3×D5 SD16⋊D5 D5×C3⋊D4 C60.8C23 kernel C60.8C23 C20.32D6 C15⋊Q16 C3⋊Dic20 C5×D4.S3 D4.D15 D5×Dic6 C3×D4⋊2D5 D4⋊2D5 C3×Dic5 C6×D5 D4.S3 Dic10 C4×D5 C5×D4 C3⋊C8 Dic6 C3×D4 Dic5 D10 C15 D4 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 C60.8C23 in GL6(𝔽241)

 52 52 0 0 0 0 189 240 0 0 0 0 0 0 99 99 0 0 0 0 142 198 0 0 0 0 118 234 142 142 0 0 94 79 99 43
,
 1 0 0 0 0 0 189 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 151 228 240 0 0 0 128 68 0 240
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 81 220 0 0 0 0 60 160 0 0 0 0 35 135 5 19 0 0 189 84 24 236
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 225 153 5 0 0 0 162 71 0 5 0 0 101 4 16 88 0 0 146 108 79 170

`G:=sub<GL(6,GF(241))| [52,189,0,0,0,0,52,240,0,0,0,0,0,0,99,142,118,94,0,0,99,198,234,79,0,0,0,0,142,99,0,0,0,0,142,43],[1,189,0,0,0,0,0,240,0,0,0,0,0,0,1,0,151,128,0,0,0,1,228,68,0,0,0,0,240,0,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,81,60,35,189,0,0,220,160,135,84,0,0,0,0,5,24,0,0,0,0,19,236],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,225,162,101,146,0,0,153,71,4,108,0,0,5,0,16,79,0,0,0,5,88,170] >;`

C60.8C23 in GAP, Magma, Sage, TeX

`C_{60}._8C_2^3`
`% in TeX`

`G:=Group("C60.8C2^3");`
`// GroupNames label`

`G:=SmallGroup(480,560);`
`// by ID`

`G=gap.SmallGroup(480,560);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,422,135,346,185,80,1356,18822]);`
`// Polycyclic`

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

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