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## G = C2×Q8×D15order 480 = 25·3·5

### Direct product of C2, Q8 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×Q8×D15
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C2×C4×D15 — C2×Q8×D15
 Lower central C15 — C30 — C2×Q8×D15
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×Q8×D15
G = < a,b,c,d,e | a2=b4=d15=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1492 in 312 conjugacy classes, 135 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, Q8, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C22×C4, C2×Q8, C2×Q8, Dic5, C20, D10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C22×S3, D15, C30, C30, C22×Q8, Dic10, C4×D5, C2×Dic5, C2×C20, C5×Q8, C22×D5, C2×Dic6, S3×C2×C4, S3×Q8, C6×Q8, Dic15, C60, D30, C2×C30, C2×Dic10, C2×C4×D5, Q8×D5, Q8×C10, C2×S3×Q8, Dic30, C4×D15, C2×Dic15, C2×C60, Q8×C15, C22×D15, C2×Q8×D5, C2×Dic30, C2×C4×D15, Q8×D15, Q8×C30, C2×Q8×D15
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C24, D10, C22×S3, D15, C22×Q8, C22×D5, S3×Q8, S3×C23, D30, Q8×D5, C23×D5, C2×S3×Q8, C22×D15, C2×Q8×D5, Q8×D15, C23×D15, C2×Q8×D15

Smallest permutation representation of C2×Q8×D15
On 240 points
Generators in S240
(1 125)(2 126)(3 127)(4 128)(5 129)(6 130)(7 131)(8 132)(9 133)(10 134)(11 135)(12 121)(13 122)(14 123)(15 124)(16 148)(17 149)(18 150)(19 136)(20 137)(21 138)(22 139)(23 140)(24 141)(25 142)(26 143)(27 144)(28 145)(29 146)(30 147)(31 159)(32 160)(33 161)(34 162)(35 163)(36 164)(37 165)(38 151)(39 152)(40 153)(41 154)(42 155)(43 156)(44 157)(45 158)(46 169)(47 170)(48 171)(49 172)(50 173)(51 174)(52 175)(53 176)(54 177)(55 178)(56 179)(57 180)(58 166)(59 167)(60 168)(61 188)(62 189)(63 190)(64 191)(65 192)(66 193)(67 194)(68 195)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 225)(106 238)(107 239)(108 240)(109 226)(110 227)(111 228)(112 229)(113 230)(114 231)(115 232)(116 233)(117 234)(118 235)(119 236)(120 237)
(1 58 20 35)(2 59 21 36)(3 60 22 37)(4 46 23 38)(5 47 24 39)(6 48 25 40)(7 49 26 41)(8 50 27 42)(9 51 28 43)(10 52 29 44)(11 53 30 45)(12 54 16 31)(13 55 17 32)(14 56 18 33)(15 57 19 34)(61 98 83 113)(62 99 84 114)(63 100 85 115)(64 101 86 116)(65 102 87 117)(66 103 88 118)(67 104 89 119)(68 105 90 120)(69 91 76 106)(70 92 77 107)(71 93 78 108)(72 94 79 109)(73 95 80 110)(74 96 81 111)(75 97 82 112)(121 177 148 159)(122 178 149 160)(123 179 150 161)(124 180 136 162)(125 166 137 163)(126 167 138 164)(127 168 139 165)(128 169 140 151)(129 170 141 152)(130 171 142 153)(131 172 143 154)(132 173 144 155)(133 174 145 156)(134 175 146 157)(135 176 147 158)(181 211 196 238)(182 212 197 239)(183 213 198 240)(184 214 199 226)(185 215 200 227)(186 216 201 228)(187 217 202 229)(188 218 203 230)(189 219 204 231)(190 220 205 232)(191 221 206 233)(192 222 207 234)(193 223 208 235)(194 224 209 236)(195 225 210 237)
(1 200 20 185)(2 201 21 186)(3 202 22 187)(4 203 23 188)(5 204 24 189)(6 205 25 190)(7 206 26 191)(8 207 27 192)(9 208 28 193)(10 209 29 194)(11 210 30 195)(12 196 16 181)(13 197 17 182)(14 198 18 183)(15 199 19 184)(31 238 54 211)(32 239 55 212)(33 240 56 213)(34 226 57 214)(35 227 58 215)(36 228 59 216)(37 229 60 217)(38 230 46 218)(39 231 47 219)(40 232 48 220)(41 233 49 221)(42 234 50 222)(43 235 51 223)(44 236 52 224)(45 237 53 225)(61 128 83 140)(62 129 84 141)(63 130 85 142)(64 131 86 143)(65 132 87 144)(66 133 88 145)(67 134 89 146)(68 135 90 147)(69 121 76 148)(70 122 77 149)(71 123 78 150)(72 124 79 136)(73 125 80 137)(74 126 81 138)(75 127 82 139)(91 159 106 177)(92 160 107 178)(93 161 108 179)(94 162 109 180)(95 163 110 166)(96 164 111 167)(97 165 112 168)(98 151 113 169)(99 152 114 170)(100 153 115 171)(101 154 116 172)(102 155 117 173)(103 156 118 174)(104 157 119 175)(105 158 120 176)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 23)(17 22)(18 21)(19 20)(24 30)(25 29)(26 28)(31 38)(32 37)(33 36)(34 35)(39 45)(40 44)(41 43)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)(61 69)(62 68)(63 67)(64 66)(70 75)(71 74)(72 73)(76 83)(77 82)(78 81)(79 80)(84 90)(85 89)(86 88)(91 98)(92 97)(93 96)(94 95)(99 105)(100 104)(101 103)(106 113)(107 112)(108 111)(109 110)(114 120)(115 119)(116 118)(121 128)(122 127)(123 126)(124 125)(129 135)(130 134)(131 133)(136 137)(138 150)(139 149)(140 148)(141 147)(142 146)(143 145)(151 159)(152 158)(153 157)(154 156)(160 165)(161 164)(162 163)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(181 188)(182 187)(183 186)(184 185)(189 195)(190 194)(191 193)(196 203)(197 202)(198 201)(199 200)(204 210)(205 209)(206 208)(211 218)(212 217)(213 216)(214 215)(219 225)(220 224)(221 223)(226 227)(228 240)(229 239)(230 238)(231 237)(232 236)(233 235)

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + + + + + + - - - image C1 C2 C2 C2 C2 S3 Q8 D5 D6 D6 D10 D10 D15 D30 D30 S3×Q8 Q8×D5 Q8×D15 kernel C2×Q8×D15 C2×Dic30 C2×C4×D15 Q8×D15 Q8×C30 Q8×C10 D30 C6×Q8 C2×C20 C5×Q8 C2×C12 C3×Q8 C2×Q8 C2×C4 Q8 C10 C6 C2 # reps 1 3 3 8 1 1 4 2 3 4 6 8 4 12 16 2 4 8

Matrix representation of C2×Q8×D15 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 60 59 0 0 1 1
,
 60 0 0 0 0 60 0 0 0 0 56 36 0 0 23 5
,
 56 30 0 0 40 16 0 0 0 0 1 0 0 0 0 1
,
 1 60 0 0 0 60 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,60,1,0,0,59,1],[60,0,0,0,0,60,0,0,0,0,56,23,0,0,36,5],[56,40,0,0,30,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,60,60,0,0,0,0,1,0,0,0,0,1] >;

C2×Q8×D15 in GAP, Magma, Sage, TeX

C_2\times Q_8\times D_{15}
% in TeX

G:=Group("C2xQ8xD15");
// GroupNames label

G:=SmallGroup(480,1172);
// by ID

G=gap.SmallGroup(480,1172);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,185,80,2693,18822]);
// Polycyclic

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

׿
×
𝔽