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## G = C2×C20.32D6order 480 = 25·3·5

### Direct product of C2 and C20.32D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×C20.32D6
 Chief series C1 — C5 — C15 — C30 — C60 — D5×C12 — C20.32D6 — C2×C20.32D6
 Lower central C15 — C30 — C2×C20.32D6
 Upper central C1 — C2×C4

Generators and relations for C2×C20.32D6
G = < a,b,c,d | a2=b20=c6=1, d2=b15, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b9, dcd-1=b10c-1 >

Subgroups: 476 in 136 conjugacy classes, 68 normal (34 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C2×M4(2), C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C22×C12, C3×Dic5, C60, C6×D5, C6×D5, C2×C30, C8⋊D5, C2×C52C8, C2×C40, C2×C4×D5, C2×C4.Dic3, C5×C3⋊C8, C153C8, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, C2×C8⋊D5, C20.32D6, C10×C3⋊C8, C2×C153C8, D5×C2×C12, C2×C20.32D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, M4(2), C22×C4, D10, C2×Dic3, C22×S3, C2×M4(2), C4×D5, C22×D5, C4.Dic3, C22×Dic3, S3×D5, C8⋊D5, C2×C4×D5, C2×C4.Dic3, D5×Dic3, C2×S3×D5, C2×C8⋊D5, C20.32D6, C2×D5×Dic3, C2×C20.32D6

Smallest permutation representation of C2×C20.32D6
On 240 points
Generators in S240
(1 147)(2 148)(3 149)(4 150)(5 151)(6 152)(7 153)(8 154)(9 155)(10 156)(11 157)(12 158)(13 159)(14 160)(15 141)(16 142)(17 143)(18 144)(19 145)(20 146)(21 185)(22 186)(23 187)(24 188)(25 189)(26 190)(27 191)(28 192)(29 193)(30 194)(31 195)(32 196)(33 197)(34 198)(35 199)(36 200)(37 181)(38 182)(39 183)(40 184)(41 234)(42 235)(43 236)(44 237)(45 238)(46 239)(47 240)(48 221)(49 222)(50 223)(51 224)(52 225)(53 226)(54 227)(55 228)(56 229)(57 230)(58 231)(59 232)(60 233)(61 168)(62 169)(63 170)(64 171)(65 172)(66 173)(67 174)(68 175)(69 176)(70 177)(71 178)(72 179)(73 180)(74 161)(75 162)(76 163)(77 164)(78 165)(79 166)(80 167)(81 206)(82 207)(83 208)(84 209)(85 210)(86 211)(87 212)(88 213)(89 214)(90 215)(91 216)(92 217)(93 218)(94 219)(95 220)(96 201)(97 202)(98 203)(99 204)(100 205)(101 125)(102 126)(103 127)(104 128)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 121)(118 122)(119 123)(120 124)
(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)
(1 185 201 147 21 96)(2 194 202 156 22 85)(3 183 203 145 23 94)(4 192 204 154 24 83)(5 181 205 143 25 92)(6 190 206 152 26 81)(7 199 207 141 27 90)(8 188 208 150 28 99)(9 197 209 159 29 88)(10 186 210 148 30 97)(11 195 211 157 31 86)(12 184 212 146 32 95)(13 193 213 155 33 84)(14 182 214 144 34 93)(15 191 215 153 35 82)(16 200 216 142 36 91)(17 189 217 151 37 100)(18 198 218 160 38 89)(19 187 219 149 39 98)(20 196 220 158 40 87)(41 69 102 236 174 128)(42 78 103 225 175 137)(43 67 104 234 176 126)(44 76 105 223 177 135)(45 65 106 232 178 124)(46 74 107 221 179 133)(47 63 108 230 180 122)(48 72 109 239 161 131)(49 61 110 228 162 140)(50 70 111 237 163 129)(51 79 112 226 164 138)(52 68 113 235 165 127)(53 77 114 224 166 136)(54 66 115 233 167 125)(55 75 116 222 168 134)(56 64 117 231 169 123)(57 73 118 240 170 132)(58 62 119 229 171 121)(59 71 120 238 172 130)(60 80 101 227 173 139)
(1 170 16 165 11 180 6 175)(2 179 17 174 12 169 7 164)(3 168 18 163 13 178 8 173)(4 177 19 172 14 167 9 162)(5 166 20 161 15 176 10 171)(21 57 36 52 31 47 26 42)(22 46 37 41 32 56 27 51)(23 55 38 50 33 45 28 60)(24 44 39 59 34 54 29 49)(25 53 40 48 35 43 30 58)(61 144 76 159 71 154 66 149)(62 153 77 148 72 143 67 158)(63 142 78 157 73 152 68 147)(64 151 79 146 74 141 69 156)(65 160 80 155 75 150 70 145)(81 127 96 122 91 137 86 132)(82 136 97 131 92 126 87 121)(83 125 98 140 93 135 88 130)(84 134 99 129 94 124 89 139)(85 123 100 138 95 133 90 128)(101 203 116 218 111 213 106 208)(102 212 117 207 112 202 107 217)(103 201 118 216 113 211 108 206)(104 210 119 205 114 220 109 215)(105 219 120 214 115 209 110 204)(181 234 196 229 191 224 186 239)(182 223 197 238 192 233 187 228)(183 232 198 227 193 222 188 237)(184 221 199 236 194 231 189 226)(185 230 200 225 195 240 190 235)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + - + - + + + - + - image C1 C2 C2 C2 C2 C4 C4 C4 S3 D5 Dic3 D6 Dic3 D6 Dic3 M4(2) D10 D10 C4×D5 C4×D5 C4.Dic3 C8⋊D5 S3×D5 D5×Dic3 C2×S3×D5 D5×Dic3 C20.32D6 kernel C2×C20.32D6 C20.32D6 C10×C3⋊C8 C2×C15⋊3C8 D5×C2×C12 D5×C12 C6×Dic5 D5×C2×C6 C2×C4×D5 C2×C3⋊C8 C4×D5 C4×D5 C2×Dic5 C2×C20 C22×D5 C30 C3⋊C8 C2×C12 C12 C2×C6 C10 C6 C2×C4 C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 1 2 2 2 1 1 1 4 4 2 4 4 8 16 2 2 2 2 8

Matrix representation of C2×C20.32D6 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 1 0 0 0 0 1
,
 177 0 0 0 0 177 0 0 0 0 177 64 0 0 223 195
,
 226 0 0 0 217 225 0 0 0 0 189 1 0 0 189 52
,
 10 141 0 0 10 231 0 0 0 0 196 115 0 0 95 45
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,1,0,0,0,0,1],[177,0,0,0,0,177,0,0,0,0,177,223,0,0,64,195],[226,217,0,0,0,225,0,0,0,0,189,189,0,0,1,52],[10,10,0,0,141,231,0,0,0,0,196,95,0,0,115,45] >;

C2×C20.32D6 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{32}D_6
% in TeX

G:=Group("C2xC20.32D6");
// GroupNames label

G:=SmallGroup(480,369);
// by ID

G=gap.SmallGroup(480,369);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^6=1,d^2=b^15,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^9,d*c*d^-1=b^10*c^-1>;
// generators/relations

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