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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 — C3×D10⋊C4 — C3×C4.D20
 Lower central C5 — C2×C10 — C3×C4.D20
 Upper central C1 — C2×C6 — C4×C12

Generators and relations for C3×C4.D20
G = < a,b,c,d | a3=b4=c20=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 576 in 152 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], D5 [×2], C10, C10 [×2], C12 [×2], C12 [×4], C2×C6, C2×C6 [×6], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C2×C12, C2×C12 [×2], C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C22×C6 [×2], C3×D5 [×2], C30, C30 [×2], C4.4D4, Dic10 [×2], D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5 [×2], C4×C12, C3×C22⋊C4 [×4], C6×D4, C6×Q8, C3×Dic5 [×2], C60 [×2], C60 [×2], C6×D5 [×6], C2×C30, D10⋊C4 [×4], C4×C20, C2×Dic10, C2×D20, C3×C4.4D4, C3×Dic10 [×2], C3×D20 [×2], C6×Dic5 [×2], C2×C60, C2×C60 [×2], D5×C2×C6 [×2], C4.D20, C3×D10⋊C4 [×4], C4×C60, C6×Dic10, C6×D20, C3×C4.D20
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], C2×D4, C4○D4 [×2], D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C4.4D4, D20 [×2], C22×D5, C6×D4, C3×C4○D4 [×2], C6×D5 [×3], C2×D20, C4○D20 [×2], C3×C4.4D4, C3×D20 [×2], D5×C2×C6, C4.D20, C6×D20, C3×C4○D20 [×2], C3×C4.D20

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

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

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

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

138 conjugacy classes

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

138 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 type + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D5 C4○D4 D10 C3×D4 C3×D5 D20 C3×C4○D4 C6×D5 C4○D20 C3×D20 C3×C4○D20 kernel C3×C4.D20 C3×D10⋊C4 C4×C60 C6×Dic10 C6×D20 C4.D20 D10⋊C4 C4×C20 C2×Dic10 C2×D20 C60 C4×C12 C30 C2×C12 C20 C42 C12 C10 C2×C4 C6 C4 C2 # reps 1 4 1 1 1 2 8 2 2 2 2 2 4 6 4 4 8 8 12 16 16 32

Matrix representation of C3×C4.D20 in GL4(𝔽61) generated by

 47 0 0 0 0 47 0 0 0 0 13 0 0 0 0 13
,
 29 54 0 0 7 32 0 0 0 0 60 0 0 0 0 60
,
 4 50 0 0 11 0 0 0 0 0 59 32 0 0 29 54
,
 39 47 0 0 39 22 0 0 0 0 59 32 0 0 59 2
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,13,0,0,0,0,13],[29,7,0,0,54,32,0,0,0,0,60,0,0,0,0,60],[4,11,0,0,50,0,0,0,0,0,59,29,0,0,32,54],[39,39,0,0,47,22,0,0,0,0,59,59,0,0,32,2] >;

C3×C4.D20 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,668);
// by ID

G=gap.SmallGroup(480,668);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,176,590,268,18822]);
// Polycyclic

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

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