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## G = C2×C10×D12order 480 = 25·3·5

### Direct product of C2×C10 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C10×D12
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — S3×C22×C10 — C2×C10×D12
 Lower central C3 — C6 — C2×C10×D12
 Upper central C1 — C22×C10 — C22×C20

Generators and relations for C2×C10×D12
G = < a,b,c,d | a2=b10=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1156 in 472 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, C10, C10, C10, C12, D6, D6, C2×C6, C15, C22×C4, C2×D4, C24, C20, C2×C10, C2×C10, D12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C30, C30, C22×D4, C2×C20, C5×D4, C22×C10, C22×C10, C2×D12, C22×C12, S3×C23, C60, S3×C10, S3×C10, C2×C30, C22×C20, D4×C10, C23×C10, C22×D12, C5×D12, C2×C60, S3×C2×C10, S3×C2×C10, C22×C30, D4×C2×C10, C10×D12, C22×C60, S3×C22×C10, C2×C10×D12
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C24, C2×C10, D12, C22×S3, C5×S3, C22×D4, C5×D4, C22×C10, C2×D12, S3×C23, S3×C10, D4×C10, C23×C10, C22×D12, C5×D12, S3×C2×C10, D4×C2×C10, C10×D12, S3×C22×C10, C2×C10×D12

Smallest permutation representation of C2×C10×D12
On 240 points
Generators in S240
(1 199)(2 200)(3 201)(4 202)(5 203)(6 204)(7 193)(8 194)(9 195)(10 196)(11 197)(12 198)(13 81)(14 82)(15 83)(16 84)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 141)(26 142)(27 143)(28 144)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 220)(38 221)(39 222)(40 223)(41 224)(42 225)(43 226)(44 227)(45 228)(46 217)(47 218)(48 219)(49 152)(50 153)(51 154)(52 155)(53 156)(54 145)(55 146)(56 147)(57 148)(58 149)(59 150)(60 151)(61 214)(62 215)(63 216)(64 205)(65 206)(66 207)(67 208)(68 209)(69 210)(70 211)(71 212)(72 213)(85 164)(86 165)(87 166)(88 167)(89 168)(90 157)(91 158)(92 159)(93 160)(94 161)(95 162)(96 163)(97 128)(98 129)(99 130)(100 131)(101 132)(102 121)(103 122)(104 123)(105 124)(106 125)(107 126)(108 127)(109 236)(110 237)(111 238)(112 239)(113 240)(114 229)(115 230)(116 231)(117 232)(118 233)(119 234)(120 235)(169 190)(170 191)(171 192)(172 181)(173 182)(174 183)(175 184)(176 185)(177 186)(178 187)(179 188)(180 189)
(1 169 106 33 87 208 52 219 110 23)(2 170 107 34 88 209 53 220 111 24)(3 171 108 35 89 210 54 221 112 13)(4 172 97 36 90 211 55 222 113 14)(5 173 98 25 91 212 56 223 114 15)(6 174 99 26 92 213 57 224 115 16)(7 175 100 27 93 214 58 225 116 17)(8 176 101 28 94 215 59 226 117 18)(9 177 102 29 95 216 60 227 118 19)(10 178 103 30 96 205 49 228 119 20)(11 179 104 31 85 206 50 217 120 21)(12 180 105 32 86 207 51 218 109 22)(37 238 80 200 191 126 138 167 68 156)(38 239 81 201 192 127 139 168 69 145)(39 240 82 202 181 128 140 157 70 146)(40 229 83 203 182 129 141 158 71 147)(41 230 84 204 183 130 142 159 72 148)(42 231 73 193 184 131 143 160 61 149)(43 232 74 194 185 132 144 161 62 150)(44 233 75 195 186 121 133 162 63 151)(45 234 76 196 187 122 134 163 64 152)(46 235 77 197 188 123 135 164 65 153)(47 236 78 198 189 124 136 165 66 154)(48 237 79 199 190 125 137 166 67 155)
(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)
(1 210)(2 209)(3 208)(4 207)(5 206)(6 205)(7 216)(8 215)(9 214)(10 213)(11 212)(12 211)(13 87)(14 86)(15 85)(16 96)(17 95)(18 94)(19 93)(20 92)(21 91)(22 90)(23 89)(24 88)(25 120)(26 119)(27 118)(28 117)(29 116)(30 115)(31 114)(32 113)(33 112)(34 111)(35 110)(36 109)(37 126)(38 125)(39 124)(40 123)(41 122)(42 121)(43 132)(44 131)(45 130)(46 129)(47 128)(48 127)(49 174)(50 173)(51 172)(52 171)(53 170)(54 169)(55 180)(56 179)(57 178)(58 177)(59 176)(60 175)(61 195)(62 194)(63 193)(64 204)(65 203)(66 202)(67 201)(68 200)(69 199)(70 198)(71 197)(72 196)(73 162)(74 161)(75 160)(76 159)(77 158)(78 157)(79 168)(80 167)(81 166)(82 165)(83 164)(84 163)(97 218)(98 217)(99 228)(100 227)(101 226)(102 225)(103 224)(104 223)(105 222)(106 221)(107 220)(108 219)(133 231)(134 230)(135 229)(136 240)(137 239)(138 238)(139 237)(140 236)(141 235)(142 234)(143 233)(144 232)(145 190)(146 189)(147 188)(148 187)(149 186)(150 185)(151 184)(152 183)(153 182)(154 181)(155 192)(156 191)

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

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

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

180 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A 4B 4C 4D 5A 5B 5C 5D 6A ··· 6G 10A ··· 10AB 10AC ··· 10BH 12A ··· 12H 15A 15B 15C 15D 20A ··· 20P 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 2 ··· 2 3 4 4 4 4 5 5 5 5 6 ··· 6 10 ··· 10 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 6 ··· 6 2 2 2 2 2 1 1 1 1 2 ··· 2 1 ··· 1 6 ··· 6 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 S3 D4 D6 D6 D12 C5×S3 C5×D4 S3×C10 S3×C10 C5×D12 kernel C2×C10×D12 C10×D12 C22×C60 S3×C22×C10 C22×D12 C2×D12 C22×C12 S3×C23 C22×C20 C2×C30 C2×C20 C22×C10 C2×C10 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 1 2 4 48 4 8 1 4 6 1 8 4 16 24 4 32

Matrix representation of C2×C10×D12 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 27 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 38 23 0 0 38 15
,
 1 0 0 0 0 60 0 0 0 0 1 1 0 0 0 60
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,38,38,0,0,23,15],[1,0,0,0,0,60,0,0,0,0,1,0,0,0,1,60] >;

C2×C10×D12 in GAP, Magma, Sage, TeX

C_2\times C_{10}\times D_{12}
% in TeX

G:=Group("C2xC10xD12");
// GroupNames label

G:=SmallGroup(480,1152);
// by ID

G=gap.SmallGroup(480,1152);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,2467,304,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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