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## G = C60.C23order 480 = 25·3·5

### 36th non-split extension by C60 of C23 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C60.C23
 Chief series C1 — C5 — C15 — C30 — C60 — C3×Dic10 — D15⋊Q8 — C60.C23
 Lower central C15 — C30 — C60 — C60.C23
 Upper central C1 — C2 — C4 — Q8

Generators and relations for C60.C23
G = < a,b,c,d | a60=1, b2=c2=d2=a30, bab-1=a19, cac-1=a11, dad-1=a31, bc=cb, dbd-1=a45b, dcd-1=a15c >

Subgroups: 748 in 120 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22 [×2], C5, S3 [×2], C6, C8 [×2], C2×C4 [×3], D4 [×2], Q8, Q8 [×3], D5 [×2], C10, Dic3 [×2], C12, C12 [×2], D6 [×2], C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×2], C20, C20 [×2], D10 [×2], C3⋊C8, C24, Dic6, Dic6, C4×S3 [×3], D12 [×2], C3×Q8, C3×Q8, D15 [×2], C30, C8.C22, C52C8, C40, Dic10, Dic10, C4×D5 [×3], D20 [×2], C5×Q8, C5×Q8, C8⋊S3, C24⋊C2, Q82S3, C3⋊Q16, C3×Q16, S3×Q8, Q83S3, C5×Dic3, C3×Dic5, Dic15, C60, C60, D30, D30, C8⋊D5, C40⋊C2, Q8⋊D5, C5⋊Q16, C5×Q16, Q8×D5, Q82D5, Q16⋊S3, C5×C3⋊C8, C3×C52C8, D30.C2, C15⋊Q8, C3×Dic10, C5×Dic6, C4×D15, C4×D15, D60, D60, Q8×C15, Q16⋊D5, D30.5C4, C15⋊SD16, Dic6⋊D5, C3×C5⋊Q16, C5×C3⋊Q16, D15⋊Q8, Q83D15, C60.C23
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8.C22, C22×D5, S3×D4, S3×D5, D4×D5, Q16⋊S3, C2×S3×D5, Q16⋊D5, D10⋊D6, C60.C23

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 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 C8.C22 S3×D4 S3×D5 D4×D5 Q16⋊S3 C2×S3×D5 Q16⋊D5 D10⋊D6 C60.C23 kernel C60.C23 D30.5C4 C15⋊SD16 Dic6⋊D5 C3×C5⋊Q16 C5×C3⋊Q16 D15⋊Q8 Q8⋊3D15 C5⋊Q16 Dic15 D30 C3⋊Q16 C5⋊2C8 Dic10 C5×Q8 C3⋊C8 Dic6 C3×Q8 C15 C10 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 1 1 2 2 2 2 4 4 2

Matrix representation of C60.C23 in GL6(𝔽241)

 0 240 0 0 0 0 1 240 0 0 0 0 0 0 52 52 137 137 0 0 189 240 104 2 0 0 52 52 189 189 0 0 189 240 52 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 120 0 25 0 0 0 26 121 146 216 0 0 12 0 121 0 0 0 99 229 215 120
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 121 0 216 0 0 0 0 121 0 216 0 0 229 0 120 0 0 0 0 229 0 120
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 195 128 148 154 0 0 113 46 87 93 0 0 28 205 46 113 0 0 36 213 128 195

`G:=sub<GL(6,GF(241))| [0,1,0,0,0,0,240,240,0,0,0,0,0,0,52,189,52,189,0,0,52,240,52,240,0,0,137,104,189,52,0,0,137,2,189,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,120,26,12,99,0,0,0,121,0,229,0,0,25,146,121,215,0,0,0,216,0,120],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,121,0,229,0,0,0,0,121,0,229,0,0,216,0,120,0,0,0,0,216,0,120],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,195,113,28,36,0,0,128,46,205,213,0,0,148,87,46,128,0,0,154,93,113,195] >;`

C60.C23 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(480,588);`
`// by ID`

`G=gap.SmallGroup(480,588);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,254,303,100,675,346,185,80,1356,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^60=1,b^2=c^2=d^2=a^30,b*a*b^-1=a^19,c*a*c^-1=a^11,d*a*d^-1=a^31,b*c=c*b,d*b*d^-1=a^45*b,d*c*d^-1=a^15*c>;`
`// generators/relations`

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