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## G = C3×C20⋊4D4order 480 = 25·3·5

### Direct product of C3 and C20⋊4D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C20⋊4D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — C6×D20 — C3×C20⋊4D4
 Lower central C5 — C2×C10 — C3×C20⋊4D4
 Upper central C1 — C2×C6 — C4×C12

Generators and relations for C3×C204D4
G = < a,b,c,d | a3=b4=c20=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 960 in 216 conjugacy classes, 82 normal (14 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22, C22 [×12], C5, C6 [×3], C6 [×4], C2×C4 [×3], D4 [×12], C23 [×4], D5 [×4], C10 [×3], C12 [×6], C2×C6, C2×C6 [×12], C15, C42, C2×D4 [×6], C20 [×6], D10 [×12], C2×C10, C2×C12 [×3], C3×D4 [×12], C22×C6 [×4], C3×D5 [×4], C30 [×3], C41D4, D20 [×12], C2×C20 [×3], C22×D5 [×4], C4×C12, C6×D4 [×6], C60 [×6], C6×D5 [×12], C2×C30, C4×C20, C2×D20 [×6], C3×C41D4, C3×D20 [×12], C2×C60 [×3], D5×C2×C6 [×4], C204D4, C4×C60, C6×D20 [×6], C3×C204D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×6], C23, D5, C2×C6 [×7], C2×D4 [×3], D10 [×3], C3×D4 [×6], C22×C6, C3×D5, C41D4, D20 [×6], C22×D5, C6×D4 [×3], C6×D5 [×3], C2×D20 [×3], C3×C41D4, C3×D20 [×6], D5×C2×C6, C204D4, C6×D20 [×3], C3×C204D4

Smallest permutation representation of C3×C204D4
On 240 points
Generators in S240
(1 90 75)(2 91 76)(3 92 77)(4 93 78)(5 94 79)(6 95 80)(7 96 61)(8 97 62)(9 98 63)(10 99 64)(11 100 65)(12 81 66)(13 82 67)(14 83 68)(15 84 69)(16 85 70)(17 86 71)(18 87 72)(19 88 73)(20 89 74)(21 190 214)(22 191 215)(23 192 216)(24 193 217)(25 194 218)(26 195 219)(27 196 220)(28 197 201)(29 198 202)(30 199 203)(31 200 204)(32 181 205)(33 182 206)(34 183 207)(35 184 208)(36 185 209)(37 186 210)(38 187 211)(39 188 212)(40 189 213)(41 226 159)(42 227 160)(43 228 141)(44 229 142)(45 230 143)(46 231 144)(47 232 145)(48 233 146)(49 234 147)(50 235 148)(51 236 149)(52 237 150)(53 238 151)(54 239 152)(55 240 153)(56 221 154)(57 222 155)(58 223 156)(59 224 157)(60 225 158)(101 170 128)(102 171 129)(103 172 130)(104 173 131)(105 174 132)(106 175 133)(107 176 134)(108 177 135)(109 178 136)(110 179 137)(111 180 138)(112 161 139)(113 162 140)(114 163 121)(115 164 122)(116 165 123)(117 166 124)(118 167 125)(119 168 126)(120 169 127)
(1 141 34 116)(2 142 35 117)(3 143 36 118)(4 144 37 119)(5 145 38 120)(6 146 39 101)(7 147 40 102)(8 148 21 103)(9 149 22 104)(10 150 23 105)(11 151 24 106)(12 152 25 107)(13 153 26 108)(14 154 27 109)(15 155 28 110)(16 156 29 111)(17 157 30 112)(18 158 31 113)(19 159 32 114)(20 160 33 115)(41 181 163 88)(42 182 164 89)(43 183 165 90)(44 184 166 91)(45 185 167 92)(46 186 168 93)(47 187 169 94)(48 188 170 95)(49 189 171 96)(50 190 172 97)(51 191 173 98)(52 192 174 99)(53 193 175 100)(54 194 176 81)(55 195 177 82)(56 196 178 83)(57 197 179 84)(58 198 180 85)(59 199 161 86)(60 200 162 87)(61 234 213 129)(62 235 214 130)(63 236 215 131)(64 237 216 132)(65 238 217 133)(66 239 218 134)(67 240 219 135)(68 221 220 136)(69 222 201 137)(70 223 202 138)(71 224 203 139)(72 225 204 140)(73 226 205 121)(74 227 206 122)(75 228 207 123)(76 229 208 124)(77 230 209 125)(78 231 210 126)(79 232 211 127)(80 233 212 128)
(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)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 120)(12 119)(13 118)(14 117)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 148)(22 147)(23 146)(24 145)(25 144)(26 143)(27 142)(28 141)(29 160)(30 159)(31 158)(32 157)(33 156)(34 155)(35 154)(36 153)(37 152)(38 151)(39 150)(40 149)(41 199)(42 198)(43 197)(44 196)(45 195)(46 194)(47 193)(48 192)(49 191)(50 190)(51 189)(52 188)(53 187)(54 186)(55 185)(56 184)(57 183)(58 182)(59 181)(60 200)(61 131)(62 130)(63 129)(64 128)(65 127)(66 126)(67 125)(68 124)(69 123)(70 122)(71 121)(72 140)(73 139)(74 138)(75 137)(76 136)(77 135)(78 134)(79 133)(80 132)(81 168)(82 167)(83 166)(84 165)(85 164)(86 163)(87 162)(88 161)(89 180)(90 179)(91 178)(92 177)(93 176)(94 175)(95 174)(96 173)(97 172)(98 171)(99 170)(100 169)(201 228)(202 227)(203 226)(204 225)(205 224)(206 223)(207 222)(208 221)(209 240)(210 239)(211 238)(212 237)(213 236)(214 235)(215 234)(216 233)(217 232)(218 231)(219 230)(220 229)

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

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

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

138 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A ··· 4F 5A 5B 6A ··· 6F 6G ··· 6N 10A ··· 10F 12A ··· 12L 15A 15B 15C 15D 20A ··· 20X 30A ··· 30L 60A ··· 60AV order 1 2 2 2 2 2 2 2 3 3 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 1 20 20 20 20 1 1 2 ··· 2 2 2 1 ··· 1 20 ··· 20 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

138 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C3 C6 C6 D4 D5 D10 C3×D4 C3×D5 D20 C6×D5 C3×D20 kernel C3×C20⋊4D4 C4×C60 C6×D20 C20⋊4D4 C4×C20 C2×D20 C60 C4×C12 C2×C12 C20 C42 C12 C2×C4 C4 # reps 1 1 6 2 2 12 6 2 6 12 4 24 12 48

Matrix representation of C3×C204D4 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 36 4 0 0 57 25
,
 6 5 0 0 17 55 0 0 0 0 34 36 0 0 25 57
,
 60 0 0 0 39 1 0 0 0 0 34 36 0 0 34 27
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,36,57,0,0,4,25],[6,17,0,0,5,55,0,0,0,0,34,25,0,0,36,57],[60,39,0,0,0,1,0,0,0,0,34,34,0,0,36,27] >;

C3×C204D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}\rtimes_4D_4
% in TeX

G:=Group("C3xC20:4D4");
// GroupNames label

G:=SmallGroup(480,667);
// by ID

G=gap.SmallGroup(480,667);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,344,590,142,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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