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## G = C2×C6.D20order 480 = 25·3·5

### Direct product of C2 and C6.D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×C6.D20
 Chief series C1 — C5 — C15 — C30 — C60 — C3×D20 — C6.D20 — C2×C6.D20
 Lower central C15 — C30 — C60 — C2×C6.D20
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C6.D20
G = < a,b,c,d | a2=b6=1, c20=d2=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=c19 >

Subgroups: 764 in 136 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, D5 [×2], C10, C10 [×2], Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C3⋊C8 [×2], Dic6 [×3], C2×Dic3, C2×C12, C3×D4 [×3], C22×C6, C3×D5 [×2], C30, C30 [×2], C2×SD16, C40 [×2], Dic10 [×3], D20 [×2], D20, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, D4.S3 [×4], C2×Dic6, C6×D4, Dic15 [×2], C60 [×2], C6×D5 [×4], C2×C30, C40⋊C2 [×4], C2×C40, C2×Dic10, C2×D20, C2×D4.S3, C5×C3⋊C8 [×2], C3×D20 [×2], C3×D20, Dic30 [×2], Dic30, C2×Dic15, C2×C60, D5×C2×C6, C2×C40⋊C2, C6.D20 [×4], C10×C3⋊C8, C6×D20, C2×Dic30, C2×C6.D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C2×SD16, D20 [×2], C22×D5, D4.S3 [×2], C2×C3⋊D4, S3×D5, C40⋊C2 [×2], C2×D20, C2×D4.S3, C3⋊D20 [×2], C2×S3×D5, C2×C40⋊C2, C6.D20 [×2], C2×C3⋊D20, C2×C6.D20

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 10A ··· 10F 12A 12B 15A 15B 20A ··· 20H 30A ··· 30F 40A ··· 40P 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 6 6 6 6 8 8 8 8 10 ··· 10 12 12 15 15 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 20 20 2 2 2 60 60 2 2 2 2 2 20 20 20 20 6 6 6 6 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4 6 ··· 6 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + + + - image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 SD16 D10 D10 C3⋊D4 C3⋊D4 D20 D20 C40⋊C2 D4.S3 S3×D5 C3⋊D20 C2×S3×D5 C3⋊D20 C6.D20 kernel C2×C6.D20 C6.D20 C10×C3⋊C8 C6×D20 C2×Dic30 C2×D20 C60 C2×C30 C2×C3⋊C8 D20 C2×C20 C30 C3⋊C8 C2×C12 C20 C2×C10 C12 C2×C6 C6 C10 C2×C4 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 2 2 1 4 4 2 2 2 4 4 16 2 2 2 2 2 8

Matrix representation of C2×C6.D20 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 240 0 0 0 0 240
,
 240 0 0 0 0 240 0 0 0 0 225 0 0 0 0 15
,
 94 166 0 0 106 200 0 0 0 0 0 1 0 0 240 0
,
 57 173 0 0 225 184 0 0 0 0 0 240 0 0 240 0
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[240,0,0,0,0,240,0,0,0,0,225,0,0,0,0,15],[94,106,0,0,166,200,0,0,0,0,0,240,0,0,1,0],[57,225,0,0,173,184,0,0,0,0,0,240,0,0,240,0] >;

C2×C6.D20 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC6.D20");
// GroupNames label

G:=SmallGroup(480,386);
// by ID

G=gap.SmallGroup(480,386);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=1,c^20=d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^19>;
// generators/relations

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