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## G = C3×C4⋊D20order 480 = 25·3·5

### Direct product of C3 and C4⋊D20

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C4⋊D20
G = < a,b,c,d | a3=b4=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 768 in 188 conjugacy classes, 70 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], C5, C6 [×3], C6 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×6], C23 [×3], D5 [×4], C10 [×3], C12 [×2], C12 [×3], C2×C6, C2×C6 [×10], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5, C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C2×C12, C2×C12 [×2], C2×C12 [×3], C3×D4 [×6], C22×C6 [×3], C3×D5 [×4], C30 [×3], C4⋊D4, C4×D5 [×2], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12, C6×D4 [×3], C3×Dic5, C60 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×8], C2×C30, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20 [×2], C3×C4⋊D4, D5×C12 [×2], C3×D20 [×6], C6×Dic5, C2×C60, C2×C60 [×2], D5×C2×C6, D5×C2×C6 [×2], C4⋊D20, C3×D10⋊C4 [×2], C15×C4⋊C4, D5×C2×C12, C6×D20, C6×D20 [×2], C3×C4⋊D20
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×4], C23, D5, C2×C6 [×7], C2×D4 [×2], C4○D4, D10 [×3], C3×D4 [×4], C22×C6, C3×D5, C4⋊D4, D20 [×2], C22×D5, C6×D4 [×2], C3×C4○D4, C6×D5 [×3], C2×D20, D4×D5, Q82D5, C3×C4⋊D4, C3×D20 [×2], D5×C2×C6, C4⋊D20, C6×D20, C3×D4×D5, C3×Q82D5, C3×C4⋊D20

Smallest permutation representation of C3×C4⋊D20
On 240 points
Generators in S240
(1 107 132)(2 108 133)(3 109 134)(4 110 135)(5 111 136)(6 112 137)(7 113 138)(8 114 139)(9 115 140)(10 116 121)(11 117 122)(12 118 123)(13 119 124)(14 120 125)(15 101 126)(16 102 127)(17 103 128)(18 104 129)(19 105 130)(20 106 131)(21 219 44)(22 220 45)(23 201 46)(24 202 47)(25 203 48)(26 204 49)(27 205 50)(28 206 51)(29 207 52)(30 208 53)(31 209 54)(32 210 55)(33 211 56)(34 212 57)(35 213 58)(36 214 59)(37 215 60)(38 216 41)(39 217 42)(40 218 43)(61 85 182)(62 86 183)(63 87 184)(64 88 185)(65 89 186)(66 90 187)(67 91 188)(68 92 189)(69 93 190)(70 94 191)(71 95 192)(72 96 193)(73 97 194)(74 98 195)(75 99 196)(76 100 197)(77 81 198)(78 82 199)(79 83 200)(80 84 181)(141 232 168)(142 233 169)(143 234 170)(144 235 171)(145 236 172)(146 237 173)(147 238 174)(148 239 175)(149 240 176)(150 221 177)(151 222 178)(152 223 179)(153 224 180)(154 225 161)(155 226 162)(156 227 163)(157 228 164)(158 229 165)(159 230 166)(160 231 167)
(1 196 231 206)(2 207 232 197)(3 198 233 208)(4 209 234 199)(5 200 235 210)(6 211 236 181)(7 182 237 212)(8 213 238 183)(9 184 239 214)(10 215 240 185)(11 186 221 216)(12 217 222 187)(13 188 223 218)(14 219 224 189)(15 190 225 220)(16 201 226 191)(17 192 227 202)(18 203 228 193)(19 194 229 204)(20 205 230 195)(21 153 92 125)(22 126 93 154)(23 155 94 127)(24 128 95 156)(25 157 96 129)(26 130 97 158)(27 159 98 131)(28 132 99 160)(29 141 100 133)(30 134 81 142)(31 143 82 135)(32 136 83 144)(33 145 84 137)(34 138 85 146)(35 147 86 139)(36 140 87 148)(37 149 88 121)(38 122 89 150)(39 151 90 123)(40 124 91 152)(41 117 65 177)(42 178 66 118)(43 119 67 179)(44 180 68 120)(45 101 69 161)(46 162 70 102)(47 103 71 163)(48 164 72 104)(49 105 73 165)(50 166 74 106)(51 107 75 167)(52 168 76 108)(53 109 77 169)(54 170 78 110)(55 111 79 171)(56 172 80 112)(57 113 61 173)(58 174 62 114)(59 115 63 175)(60 176 64 116)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 68)(62 67)(63 66)(64 65)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 100)(98 99)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)(121 122)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(131 132)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 150)(159 160)(161 172)(162 171)(163 170)(164 169)(165 168)(166 167)(173 180)(174 179)(175 178)(176 177)(181 190)(182 189)(183 188)(184 187)(185 186)(191 200)(192 199)(193 198)(194 197)(195 196)(201 210)(202 209)(203 208)(204 207)(205 206)(211 220)(212 219)(213 218)(214 217)(215 216)(221 240)(222 239)(223 238)(224 237)(225 236)(226 235)(227 234)(228 233)(229 232)(230 231)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 5 5 6 ··· 6 6 6 6 6 6 6 6 6 10 ··· 10 12 12 12 12 12 12 12 12 12 12 12 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 20 20 1 1 2 2 4 4 10 10 2 2 1 ··· 1 10 10 10 10 20 20 20 20 2 ··· 2 2 2 2 2 4 4 4 4 10 10 10 10 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D4 D5 C4○D4 D10 C3×D4 C3×D4 C3×D5 D20 C3×C4○D4 C6×D5 C3×D20 D4×D5 Q8⋊2D5 C3×D4×D5 C3×Q8⋊2D5 kernel C3×C4⋊D20 C3×D10⋊C4 C15×C4⋊C4 D5×C2×C12 C6×D20 C4⋊D20 D10⋊C4 C5×C4⋊C4 C2×C4×D5 C2×D20 C60 C6×D5 C3×C4⋊C4 C30 C2×C12 C20 D10 C4⋊C4 C12 C10 C2×C4 C4 C6 C6 C2 C2 # reps 1 2 1 1 3 2 4 2 2 6 2 2 2 2 6 4 4 4 8 4 12 16 2 2 4 4

Matrix representation of C3×C4⋊D20 in GL6(𝔽61)

 47 0 0 0 0 0 0 47 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 50 0 0 0 0 0 0 11
,
 21 20 0 0 0 0 45 40 0 0 0 0 0 0 43 60 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 60 0
,
 21 20 0 0 0 0 39 40 0 0 0 0 0 0 43 60 0 0 0 0 18 18 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(61))| [47,0,0,0,0,0,0,47,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,11],[21,45,0,0,0,0,20,40,0,0,0,0,0,0,43,1,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[21,39,0,0,0,0,20,40,0,0,0,0,0,0,43,18,0,0,0,0,60,18,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3×C4⋊D20 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes D_{20}
% in TeX

G:=Group("C3xC4:D20");
// GroupNames label

G:=SmallGroup(480,688);
// by ID

G=gap.SmallGroup(480,688);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,142,18822]);
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽