Copied to
clipboard

## G = C2×Q8⋊3D15order 480 = 25·3·5

### Direct product of C2 and Q8⋊3D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×Q8⋊3D15
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C2×C4×D15 — C2×Q8⋊3D15
 Lower central C15 — C30 — C2×Q8⋊3D15
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×Q83D15
G = < a,b,c,d,e | a2=b4=d15=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1876 in 328 conjugacy classes, 127 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C3×Q8, C22×S3, D15, C30, C30, C2×C4○D4, C4×D5, D20, C2×Dic5, C2×C20, C5×Q8, C22×D5, S3×C2×C4, C2×D12, Q83S3, C6×Q8, Dic15, C60, D30, D30, C2×C30, C2×C4×D5, C2×D20, Q82D5, Q8×C10, C2×Q83S3, C4×D15, D60, C2×Dic15, C2×C60, Q8×C15, C22×D15, C2×Q82D5, C2×C4×D15, C2×D60, Q83D15, Q8×C30, C2×Q83D15
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, D15, C2×C4○D4, C22×D5, Q83S3, S3×C23, D30, Q82D5, C23×D5, C2×Q83S3, C22×D15, C2×Q82D5, Q83D15, C23×D15, C2×Q83D15

Smallest permutation representation of C2×Q83D15
On 240 points
Generators in S240
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 136)(9 137)(10 138)(11 139)(12 140)(13 141)(14 142)(15 143)(16 129)(17 130)(18 131)(19 132)(20 133)(21 134)(22 135)(23 121)(24 122)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 174)(32 175)(33 176)(34 177)(35 178)(36 179)(37 180)(38 166)(39 167)(40 168)(41 169)(42 170)(43 171)(44 172)(45 173)(46 165)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 161)(58 162)(59 163)(60 164)(61 206)(62 207)(63 208)(64 209)(65 210)(66 196)(67 197)(68 198)(69 199)(70 200)(71 201)(72 202)(73 203)(74 204)(75 205)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 181)(85 182)(86 183)(87 184)(88 185)(89 186)(90 187)(91 240)(92 226)(93 227)(94 228)(95 229)(96 230)(97 231)(98 232)(99 233)(100 234)(101 235)(102 236)(103 237)(104 238)(105 239)(106 217)(107 218)(108 219)(109 220)(110 221)(111 222)(112 223)(113 224)(114 225)(115 211)(116 212)(117 213)(118 214)(119 215)(120 216)
(1 76 21 61)(2 77 22 62)(3 78 23 63)(4 79 24 64)(5 80 25 65)(6 81 26 66)(7 82 27 67)(8 83 28 68)(9 84 29 69)(10 85 30 70)(11 86 16 71)(12 87 17 72)(13 88 18 73)(14 89 19 74)(15 90 20 75)(31 106 48 101)(32 107 49 102)(33 108 50 103)(34 109 51 104)(35 110 52 105)(36 111 53 91)(37 112 54 92)(38 113 55 93)(39 114 56 94)(40 115 57 95)(41 116 58 96)(42 117 59 97)(43 118 60 98)(44 119 46 99)(45 120 47 100)(121 208 146 190)(122 209 147 191)(123 210 148 192)(124 196 149 193)(125 197 150 194)(126 198 136 195)(127 199 137 181)(128 200 138 182)(129 201 139 183)(130 202 140 184)(131 203 141 185)(132 204 142 186)(133 205 143 187)(134 206 144 188)(135 207 145 189)(151 234 173 216)(152 235 174 217)(153 236 175 218)(154 237 176 219)(155 238 177 220)(156 239 178 221)(157 240 179 222)(158 226 180 223)(159 227 166 224)(160 228 167 225)(161 229 168 211)(162 230 169 212)(163 231 170 213)(164 232 171 214)(165 233 172 215)
(1 230 21 212)(2 231 22 213)(3 232 23 214)(4 233 24 215)(5 234 25 216)(6 235 26 217)(7 236 27 218)(8 237 28 219)(9 238 29 220)(10 239 30 221)(11 240 16 222)(12 226 17 223)(13 227 18 224)(14 228 19 225)(15 229 20 211)(31 193 48 196)(32 194 49 197)(33 195 50 198)(34 181 51 199)(35 182 52 200)(36 183 53 201)(37 184 54 202)(38 185 55 203)(39 186 56 204)(40 187 57 205)(41 188 58 206)(42 189 59 207)(43 190 60 208)(44 191 46 209)(45 192 47 210)(61 169 76 162)(62 170 77 163)(63 171 78 164)(64 172 79 165)(65 173 80 151)(66 174 81 152)(67 175 82 153)(68 176 83 154)(69 177 84 155)(70 178 85 156)(71 179 86 157)(72 180 87 158)(73 166 88 159)(74 167 89 160)(75 168 90 161)(91 129 111 139)(92 130 112 140)(93 131 113 141)(94 132 114 142)(95 133 115 143)(96 134 116 144)(97 135 117 145)(98 121 118 146)(99 122 119 147)(100 123 120 148)(101 124 106 149)(102 125 107 150)(103 126 108 136)(104 127 109 137)(105 128 110 138)
(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)
(1 187)(2 186)(3 185)(4 184)(5 183)(6 182)(7 181)(8 195)(9 194)(10 193)(11 192)(12 191)(13 190)(14 189)(15 188)(16 210)(17 209)(18 208)(19 207)(20 206)(21 205)(22 204)(23 203)(24 202)(25 201)(26 200)(27 199)(28 198)(29 197)(30 196)(31 221)(32 220)(33 219)(34 218)(35 217)(36 216)(37 215)(38 214)(39 213)(40 212)(41 211)(42 225)(43 224)(44 223)(45 222)(46 226)(47 240)(48 239)(49 238)(50 237)(51 236)(52 235)(53 234)(54 233)(55 232)(56 231)(57 230)(58 229)(59 228)(60 227)(61 133)(62 132)(63 131)(64 130)(65 129)(66 128)(67 127)(68 126)(69 125)(70 124)(71 123)(72 122)(73 121)(74 135)(75 134)(76 143)(77 142)(78 141)(79 140)(80 139)(81 138)(82 137)(83 136)(84 150)(85 149)(86 148)(87 147)(88 146)(89 145)(90 144)(91 151)(92 165)(93 164)(94 163)(95 162)(96 161)(97 160)(98 159)(99 158)(100 157)(101 156)(102 155)(103 154)(104 153)(105 152)(106 178)(107 177)(108 176)(109 175)(110 174)(111 173)(112 172)(113 171)(114 170)(115 169)(116 168)(117 167)(118 166)(119 180)(120 179)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3 4A ··· 4F 4G 4H 4I 4J 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 3 4 ··· 4 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 30 ··· 30 2 2 ··· 2 15 15 15 15 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 D5 D6 D6 C4○D4 D10 D10 D15 D30 D30 Q8⋊3S3 Q8⋊2D5 Q8⋊3D15 kernel C2×Q8⋊3D15 C2×C4×D15 C2×D60 Q8⋊3D15 Q8×C30 Q8×C10 C6×Q8 C2×C20 C5×Q8 C30 C2×C12 C3×Q8 C2×Q8 C2×C4 Q8 C10 C6 C2 # reps 1 3 3 8 1 1 2 3 4 4 6 8 4 12 16 2 4 8

Matrix representation of C2×Q83D15 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 20 30 0 0 0 0 9 41
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 37 25 0 0 0 0 55 24
,
 18 18 0 0 0 0 43 60 0 0 0 0 0 0 5 38 0 0 0 0 23 53 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 43 43 0 0 0 0 1 18 0 0 0 0 0 0 23 56 0 0 0 0 8 38 0 0 0 0 0 0 20 30 0 0 0 0 5 41

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,20,9,0,0,0,0,30,41],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,55,0,0,0,0,25,24],[18,43,0,0,0,0,18,60,0,0,0,0,0,0,5,23,0,0,0,0,38,53,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[43,1,0,0,0,0,43,18,0,0,0,0,0,0,23,8,0,0,0,0,56,38,0,0,0,0,0,0,20,5,0,0,0,0,30,41] >;

C2×Q83D15 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_3D_{15}
% in TeX

G:=Group("C2xQ8:3D15");
// GroupNames label

G:=SmallGroup(480,1173);
// by ID

G=gap.SmallGroup(480,1173);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,100,675,185,80,2693,18822]);
// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽