Copied to
clipboard

## G = C6×C4○D20order 480 = 25·3·5

### Direct product of C6 and C4○D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C6×C4○D20
 Chief series C1 — C5 — C10 — C30 — C6×D5 — D5×C2×C6 — D5×C2×C12 — C6×C4○D20
 Lower central C5 — C10 — C6×C4○D20
 Upper central C1 — C2×C12 — C22×C12

Generators and relations for C6×C4○D20
G = < a,b,c,d | a6=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >

Subgroups: 912 in 328 conjugacy classes, 178 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], C5, C6, C6 [×2], C6 [×6], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], D5 [×4], C10, C10 [×2], C10 [×2], C12 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×10], C15, C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×4], D10 [×4], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×10], C3×D4 [×12], C3×Q8 [×4], C22×C6, C22×C6 [×2], C3×D5 [×4], C30, C30 [×2], C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, C22×C12, C22×C12 [×2], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C3×Dic5 [×4], C60 [×4], C6×D5 [×4], C6×D5 [×4], C2×C30, C2×C30 [×2], C2×C30 [×2], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, C6×C4○D4, C3×Dic10 [×4], D5×C12 [×8], C3×D20 [×4], C6×Dic5 [×2], C3×C5⋊D4 [×8], C2×C60 [×2], C2×C60 [×4], D5×C2×C6 [×2], C22×C30, C2×C4○D20, C6×Dic10, D5×C2×C12 [×2], C6×D20, C3×C4○D20 [×8], C6×C5⋊D4 [×2], C22×C60, C6×C4○D20
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], D5, C2×C6 [×35], C4○D4 [×2], C24, D10 [×7], C22×C6 [×15], C3×D5, C2×C4○D4, C22×D5 [×7], C3×C4○D4 [×2], C23×C6, C6×D5 [×7], C4○D20 [×2], C23×D5, C6×C4○D4, D5×C2×C6 [×7], C2×C4○D20, C3×C4○D20 [×2], D5×C22×C6, C6×C4○D20

Smallest permutation representation of C6×C4○D20
On 240 points
Generators in S240
(1 151 191 161 202 111)(2 152 192 162 203 112)(3 153 193 163 204 113)(4 154 194 164 205 114)(5 155 195 165 206 115)(6 156 196 166 207 116)(7 157 197 167 208 117)(8 158 198 168 209 118)(9 159 199 169 210 119)(10 160 200 170 211 120)(11 141 181 171 212 101)(12 142 182 172 213 102)(13 143 183 173 214 103)(14 144 184 174 215 104)(15 145 185 175 216 105)(16 146 186 176 217 106)(17 147 187 177 218 107)(18 148 188 178 219 108)(19 149 189 179 220 109)(20 150 190 180 201 110)(21 234 122 53 98 66)(22 235 123 54 99 67)(23 236 124 55 100 68)(24 237 125 56 81 69)(25 238 126 57 82 70)(26 239 127 58 83 71)(27 240 128 59 84 72)(28 221 129 60 85 73)(29 222 130 41 86 74)(30 223 131 42 87 75)(31 224 132 43 88 76)(32 225 133 44 89 77)(33 226 134 45 90 78)(34 227 135 46 91 79)(35 228 136 47 92 80)(36 229 137 48 93 61)(37 230 138 49 94 62)(38 231 139 50 95 63)(39 232 140 51 96 64)(40 233 121 52 97 65)
(1 228 11 238)(2 229 12 239)(3 230 13 240)(4 231 14 221)(5 232 15 222)(6 233 16 223)(7 234 17 224)(8 235 18 225)(9 236 19 226)(10 237 20 227)(21 107 31 117)(22 108 32 118)(23 109 33 119)(24 110 34 120)(25 111 35 101)(26 112 36 102)(27 113 37 103)(28 114 38 104)(29 115 39 105)(30 116 40 106)(41 195 51 185)(42 196 52 186)(43 197 53 187)(44 198 54 188)(45 199 55 189)(46 200 56 190)(47 181 57 191)(48 182 58 192)(49 183 59 193)(50 184 60 194)(61 213 71 203)(62 214 72 204)(63 215 73 205)(64 216 74 206)(65 217 75 207)(66 218 76 208)(67 219 77 209)(68 220 78 210)(69 201 79 211)(70 202 80 212)(81 180 91 170)(82 161 92 171)(83 162 93 172)(84 163 94 173)(85 164 95 174)(86 165 96 175)(87 166 97 176)(88 167 98 177)(89 168 99 178)(90 169 100 179)(121 146 131 156)(122 147 132 157)(123 148 133 158)(124 149 134 159)(125 150 135 160)(126 151 136 141)(127 152 137 142)(128 153 138 143)(129 154 139 144)(130 155 140 145)
(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 28)(22 27)(23 26)(24 25)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 80)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)(121 130)(122 129)(123 128)(124 127)(125 126)(131 140)(132 139)(133 138)(134 137)(135 136)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)(161 180)(162 179)(163 178)(164 177)(165 176)(166 175)(167 174)(168 173)(169 172)(170 171)(181 200)(182 199)(183 198)(184 197)(185 196)(186 195)(187 194)(188 193)(189 192)(190 191)(201 202)(203 220)(204 219)(205 218)(206 217)(207 216)(208 215)(209 214)(210 213)(211 212)(221 234)(222 233)(223 232)(224 231)(225 230)(226 229)(227 228)(235 240)(236 239)(237 238)

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

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

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

156 conjugacy classes

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

156 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 C6 D5 C4○D4 D10 D10 C3×D5 C3×C4○D4 C6×D5 C6×D5 C4○D20 C3×C4○D20 kernel C6×C4○D20 C6×Dic10 D5×C2×C12 C6×D20 C3×C4○D20 C6×C5⋊D4 C22×C60 C2×C4○D20 C2×Dic10 C2×C4×D5 C2×D20 C4○D20 C2×C5⋊D4 C22×C20 C22×C12 C30 C2×C12 C22×C6 C22×C4 C10 C2×C4 C23 C6 C2 # reps 1 1 2 1 8 2 1 2 2 4 2 16 4 2 2 4 12 2 4 8 24 4 16 32

Matrix representation of C6×C4○D20 in GL3(𝔽61) generated by

 48 0 0 0 1 0 0 0 1
,
 60 0 0 0 11 0 0 0 11
,
 60 0 0 0 2 32 0 29 7
,
 1 0 0 0 7 29 0 32 54
G:=sub<GL(3,GF(61))| [48,0,0,0,1,0,0,0,1],[60,0,0,0,11,0,0,0,11],[60,0,0,0,2,29,0,32,7],[1,0,0,0,7,32,0,29,54] >;

C6×C4○D20 in GAP, Magma, Sage, TeX

C_6\times C_4\circ D_{20}
% in TeX

G:=Group("C6xC4oD20");
// GroupNames label

G:=SmallGroup(480,1138);
// by ID

G=gap.SmallGroup(480,1138);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,268,1571,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^9>;
// generators/relations

׿
×
𝔽