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## G = C42⋊7D15order 480 = 25·3·5

### 6th semidirect product of C42 and D15 acting via D15/C15=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C42⋊7D15
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — D30⋊3C4 — C42⋊7D15
 Lower central C15 — C2×C30 — C42⋊7D15
 Upper central C1 — C22 — C42

Generators and relations for C427D15
G = < a,b,c,d | a4=b4=c15=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 1092 in 152 conjugacy classes, 55 normal (23 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 [×2], C2×C4 [×2], 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, Dic6 [×2], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×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], D6⋊C4 [×4], C4×C12, C2×Dic6, C2×D12, Dic15 [×2], C60 [×2], C60 [×2], D30 [×6], C2×C30, D10⋊C4 [×4], C4×C20, C2×Dic10, C2×D20, C427S3, Dic30 [×2], D60 [×2], C2×Dic15 [×2], C2×C60, C2×C60 [×2], C22×D15 [×2], C4.D20, D303C4 [×4], C4×C60, C2×Dic30, C2×D60, C427D15
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], D12 [×2], C22×S3, D15, C4.4D4, D20 [×2], C22×D5, C2×D12, C4○D12 [×2], D30 [×3], C2×D20, C4○D20 [×2], C427S3, D60 [×2], C22×D15, C4.D20, C2×D60, D6011C2 [×2], C427D15

Smallest permutation representation of C427D15
On 240 points
Generators in S240
```(1 159 39 121)(2 160 40 122)(3 161 41 123)(4 162 42 124)(5 163 43 125)(6 164 44 126)(7 165 45 127)(8 151 31 128)(9 152 32 129)(10 153 33 130)(11 154 34 131)(12 155 35 132)(13 156 36 133)(14 157 37 134)(15 158 38 135)(16 170 50 148)(17 171 51 149)(18 172 52 150)(19 173 53 136)(20 174 54 137)(21 175 55 138)(22 176 56 139)(23 177 57 140)(24 178 58 141)(25 179 59 142)(26 180 60 143)(27 166 46 144)(28 167 47 145)(29 168 48 146)(30 169 49 147)(61 223 99 185)(62 224 100 186)(63 225 101 187)(64 211 102 188)(65 212 103 189)(66 213 104 190)(67 214 105 191)(68 215 91 192)(69 216 92 193)(70 217 93 194)(71 218 94 195)(72 219 95 181)(73 220 96 182)(74 221 97 183)(75 222 98 184)(76 230 110 196)(77 231 111 197)(78 232 112 198)(79 233 113 199)(80 234 114 200)(81 235 115 201)(82 236 116 202)(83 237 117 203)(84 238 118 204)(85 239 119 205)(86 240 120 206)(87 226 106 207)(88 227 107 208)(89 228 108 209)(90 229 109 210)
(1 76 20 61)(2 77 21 62)(3 78 22 63)(4 79 23 64)(5 80 24 65)(6 81 25 66)(7 82 26 67)(8 83 27 68)(9 84 28 69)(10 85 29 70)(11 86 30 71)(12 87 16 72)(13 88 17 73)(14 89 18 74)(15 90 19 75)(31 117 46 91)(32 118 47 92)(33 119 48 93)(34 120 49 94)(35 106 50 95)(36 107 51 96)(37 108 52 97)(38 109 53 98)(39 110 54 99)(40 111 55 100)(41 112 56 101)(42 113 57 102)(43 114 58 103)(44 115 59 104)(45 116 60 105)(121 196 137 185)(122 197 138 186)(123 198 139 187)(124 199 140 188)(125 200 141 189)(126 201 142 190)(127 202 143 191)(128 203 144 192)(129 204 145 193)(130 205 146 194)(131 206 147 195)(132 207 148 181)(133 208 149 182)(134 209 150 183)(135 210 136 184)(151 237 166 215)(152 238 167 216)(153 239 168 217)(154 240 169 218)(155 226 170 219)(156 227 171 220)(157 228 172 221)(158 229 173 222)(159 230 174 223)(160 231 175 224)(161 232 176 225)(162 233 177 211)(163 234 178 212)(164 235 179 213)(165 236 180 214)
(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)
(1 184)(2 183)(3 182)(4 181)(5 195)(6 194)(7 193)(8 192)(9 191)(10 190)(11 189)(12 188)(13 187)(14 186)(15 185)(16 199)(17 198)(18 197)(19 196)(20 210)(21 209)(22 208)(23 207)(24 206)(25 205)(26 204)(27 203)(28 202)(29 201)(30 200)(31 215)(32 214)(33 213)(34 212)(35 211)(36 225)(37 224)(38 223)(39 222)(40 221)(41 220)(42 219)(43 218)(44 217)(45 216)(46 237)(47 236)(48 235)(49 234)(50 233)(51 232)(52 231)(53 230)(54 229)(55 228)(56 227)(57 226)(58 240)(59 239)(60 238)(61 173)(62 172)(63 171)(64 170)(65 169)(66 168)(67 167)(68 166)(69 180)(70 179)(71 178)(72 177)(73 176)(74 175)(75 174)(76 158)(77 157)(78 156)(79 155)(80 154)(81 153)(82 152)(83 151)(84 165)(85 164)(86 163)(87 162)(88 161)(89 160)(90 159)(91 144)(92 143)(93 142)(94 141)(95 140)(96 139)(97 138)(98 137)(99 136)(100 150)(101 149)(102 148)(103 147)(104 146)(105 145)(106 124)(107 123)(108 122)(109 121)(110 135)(111 134)(112 133)(113 132)(114 131)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)```

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

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

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

126 conjugacy classes

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

126 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D5 D6 C4○D4 D10 D12 D15 D20 C4○D12 D30 C4○D20 D60 D60⋊11C2 kernel C42⋊7D15 D30⋊3C4 C4×C60 C2×Dic30 C2×D60 C4×C20 C60 C4×C12 C2×C20 C30 C2×C12 C20 C42 C12 C10 C2×C4 C6 C4 C2 # reps 1 4 1 1 1 1 2 2 3 4 6 4 4 8 8 12 16 16 32

Matrix representation of C427D15 in GL4(𝔽61) generated by

 14 45 0 0 16 47 0 0 0 0 50 0 0 0 0 50
,
 50 0 0 0 0 50 0 0 0 0 14 45 0 0 16 47
,
 37 14 0 0 47 31 0 0 0 0 25 30 0 0 31 47
,
 29 2 0 0 7 32 0 0 0 0 20 53 0 0 27 41
`G:=sub<GL(4,GF(61))| [14,16,0,0,45,47,0,0,0,0,50,0,0,0,0,50],[50,0,0,0,0,50,0,0,0,0,14,16,0,0,45,47],[37,47,0,0,14,31,0,0,0,0,25,31,0,0,30,47],[29,7,0,0,2,32,0,0,0,0,20,27,0,0,53,41] >;`

C427D15 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_7D_{15}`
`% in TeX`

`G:=Group("C4^2:7D15");`
`// GroupNames label`

`G:=SmallGroup(480,840);`
`// by ID`

`G=gap.SmallGroup(480,840);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,254,100,2693,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^15=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

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