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G = C2×C6×D20order 480 = 25·3·5

Direct product of C2×C6 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C6×D20
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — D5×C22×C6 — C2×C6×D20
 Lower central C5 — C10 — C2×C6×D20
 Upper central C1 — C22×C6 — C22×C12

Generators and relations for C2×C6×D20
G = < a,b,c,d | a2=b6=c20=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1680 in 472 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C12, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, C20, D10, D10, C2×C10, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C22×D4, D20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C12, C6×D4, C23×C6, C60, C6×D5, C6×D5, C2×C30, C2×D20, C22×C20, C23×D5, D4×C2×C6, C3×D20, C2×C60, D5×C2×C6, D5×C2×C6, C22×C30, C22×D20, C6×D20, C22×C60, D5×C22×C6, C2×C6×D20
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C24, D10, C3×D4, C22×C6, C3×D5, C22×D4, D20, C22×D5, C6×D4, C23×C6, C6×D5, C2×D20, C23×D5, D4×C2×C6, C3×D20, D5×C2×C6, C22×D20, C6×D20, D5×C22×C6, C2×C6×D20

Smallest permutation representation of C2×C6×D20
On 240 points
Generators in S240
(1 133)(2 134)(3 135)(4 136)(5 137)(6 138)(7 139)(8 140)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 127)(16 128)(17 129)(18 130)(19 131)(20 132)(21 204)(22 205)(23 206)(24 207)(25 208)(26 209)(27 210)(28 211)(29 212)(30 213)(31 214)(32 215)(33 216)(34 217)(35 218)(36 219)(37 220)(38 201)(39 202)(40 203)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 61)(59 62)(60 63)(81 157)(82 158)(83 159)(84 160)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)(101 177)(102 178)(103 179)(104 180)(105 161)(106 162)(107 163)(108 164)(109 165)(110 166)(111 167)(112 168)(113 169)(114 170)(115 171)(116 172)(117 173)(118 174)(119 175)(120 176)(181 229)(182 230)(183 231)(184 232)(185 233)(186 234)(187 235)(188 236)(189 237)(190 238)(191 239)(192 240)(193 221)(194 222)(195 223)(196 224)(197 225)(198 226)(199 227)(200 228)
(1 194 96 177 75 205)(2 195 97 178 76 206)(3 196 98 179 77 207)(4 197 99 180 78 208)(5 198 100 161 79 209)(6 199 81 162 80 210)(7 200 82 163 61 211)(8 181 83 164 62 212)(9 182 84 165 63 213)(10 183 85 166 64 214)(11 184 86 167 65 215)(12 185 87 168 66 216)(13 186 88 169 67 217)(14 187 89 170 68 218)(15 188 90 171 69 219)(16 189 91 172 70 220)(17 190 92 173 71 201)(18 191 93 174 72 202)(19 192 94 175 73 203)(20 193 95 176 74 204)(21 132 221 151 120 51)(22 133 222 152 101 52)(23 134 223 153 102 53)(24 135 224 154 103 54)(25 136 225 155 104 55)(26 137 226 156 105 56)(27 138 227 157 106 57)(28 139 228 158 107 58)(29 140 229 159 108 59)(30 121 230 160 109 60)(31 122 231 141 110 41)(32 123 232 142 111 42)(33 124 233 143 112 43)(34 125 234 144 113 44)(35 126 235 145 114 45)(36 127 236 146 115 46)(37 128 237 147 116 47)(38 129 238 148 117 48)(39 130 239 149 118 49)(40 131 240 150 119 50)
(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)
(1 171)(2 170)(3 169)(4 168)(5 167)(6 166)(7 165)(8 164)(9 163)(10 162)(11 161)(12 180)(13 179)(14 178)(15 177)(16 176)(17 175)(18 174)(19 173)(20 172)(21 147)(22 146)(23 145)(24 144)(25 143)(26 142)(27 141)(28 160)(29 159)(30 158)(31 157)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 150)(39 149)(40 148)(41 227)(42 226)(43 225)(44 224)(45 223)(46 222)(47 221)(48 240)(49 239)(50 238)(51 237)(52 236)(53 235)(54 234)(55 233)(56 232)(57 231)(58 230)(59 229)(60 228)(61 182)(62 181)(63 200)(64 199)(65 198)(66 197)(67 196)(68 195)(69 194)(70 193)(71 192)(72 191)(73 190)(74 189)(75 188)(76 187)(77 186)(78 185)(79 184)(80 183)(81 214)(82 213)(83 212)(84 211)(85 210)(86 209)(87 208)(88 207)(89 206)(90 205)(91 204)(92 203)(93 202)(94 201)(95 220)(96 219)(97 218)(98 217)(99 216)(100 215)(101 127)(102 126)(103 125)(104 124)(105 123)(106 122)(107 121)(108 140)(109 139)(110 138)(111 137)(112 136)(113 135)(114 134)(115 133)(116 132)(117 131)(118 130)(119 129)(120 128)

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

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

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

156 conjugacy classes

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

156 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 C3 C6 C6 C6 D4 D5 D10 D10 C3×D4 C3×D5 D20 C6×D5 C6×D5 C3×D20 kernel C2×C6×D20 C6×D20 C22×C60 D5×C22×C6 C22×D20 C2×D20 C22×C20 C23×D5 C2×C30 C22×C12 C2×C12 C22×C6 C2×C10 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 1 2 2 24 2 4 4 2 12 2 8 4 16 24 4 32

Matrix representation of C2×C6×D20 in GL4(𝔽61) generated by

 60 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 48 0 0 0 0 47 0 0 0 0 47
,
 1 0 0 0 0 1 0 0 0 0 59 25 0 0 34 32
,
 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 60
G:=sub<GL(4,GF(61))| [60,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,48,0,0,0,0,47,0,0,0,0,47],[1,0,0,0,0,1,0,0,0,0,59,34,0,0,25,32],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,1,60] >;

C2×C6×D20 in GAP, Magma, Sage, TeX

C_2\times C_6\times D_{20}
% in TeX

G:=Group("C2xC6xD20");
// GroupNames label

G:=SmallGroup(480,1137);
// by ID

G=gap.SmallGroup(480,1137);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,1571,192,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^20=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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