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## G = C3×C20.23D4order 480 = 25·3·5

### Direct product of C3 and C20.23D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C20.23D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — C6×D20 — C3×C20.23D4
 Lower central C5 — C2×C10 — C3×C20.23D4
 Upper central C1 — C2×C6 — C6×Q8

Generators and relations for C3×C20.23D4
G = < a,b,c,d | a3=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd=b-1, dcd=b10c-1 >

Subgroups: 544 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, D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×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, C4×Dic5, D10⋊C4 [×4], C2×D20, Q8×C10, C3×C4.4D4, C3×D20 [×2], C6×Dic5 [×2], C2×C60, C2×C60 [×2], Q8×C15 [×2], D5×C2×C6 [×2], C20.23D4, C12×Dic5, C3×D10⋊C4 [×4], C6×D20, Q8×C30, C3×C20.23D4
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, C5⋊D4 [×2], C22×D5, C6×D4, C3×C4○D4 [×2], C6×D5 [×3], Q82D5 [×2], C2×C5⋊D4, C3×C4.4D4, C3×C5⋊D4 [×2], D5×C2×C6, C20.23D4, C3×Q82D5 [×2], C6×C5⋊D4, C3×C20.23D4

Smallest permutation representation of C3×C20.23D4
On 240 points
Generators in S240
(1 87 35)(2 88 36)(3 89 37)(4 90 38)(5 91 39)(6 92 40)(7 93 21)(8 94 22)(9 95 23)(10 96 24)(11 97 25)(12 98 26)(13 99 27)(14 100 28)(15 81 29)(16 82 30)(17 83 31)(18 84 32)(19 85 33)(20 86 34)(41 214 103)(42 215 104)(43 216 105)(44 217 106)(45 218 107)(46 219 108)(47 220 109)(48 201 110)(49 202 111)(50 203 112)(51 204 113)(52 205 114)(53 206 115)(54 207 116)(55 208 117)(56 209 118)(57 210 119)(58 211 120)(59 212 101)(60 213 102)(61 156 138)(62 157 139)(63 158 140)(64 159 121)(65 160 122)(66 141 123)(67 142 124)(68 143 125)(69 144 126)(70 145 127)(71 146 128)(72 147 129)(73 148 130)(74 149 131)(75 150 132)(76 151 133)(77 152 134)(78 153 135)(79 154 136)(80 155 137)(161 224 189)(162 225 190)(163 226 191)(164 227 192)(165 228 193)(166 229 194)(167 230 195)(168 231 196)(169 232 197)(170 233 198)(171 234 199)(172 235 200)(173 236 181)(174 237 182)(175 238 183)(176 239 184)(177 240 185)(178 221 186)(179 222 187)(180 223 188)
(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)
(1 161 56 145)(2 170 57 154)(3 179 58 143)(4 168 59 152)(5 177 60 141)(6 166 41 150)(7 175 42 159)(8 164 43 148)(9 173 44 157)(10 162 45 146)(11 171 46 155)(12 180 47 144)(13 169 48 153)(14 178 49 142)(15 167 50 151)(16 176 51 160)(17 165 52 149)(18 174 53 158)(19 163 54 147)(20 172 55 156)(21 183 104 64)(22 192 105 73)(23 181 106 62)(24 190 107 71)(25 199 108 80)(26 188 109 69)(27 197 110 78)(28 186 111 67)(29 195 112 76)(30 184 113 65)(31 193 114 74)(32 182 115 63)(33 191 116 72)(34 200 117 61)(35 189 118 70)(36 198 119 79)(37 187 120 68)(38 196 101 77)(39 185 102 66)(40 194 103 75)(81 230 203 133)(82 239 204 122)(83 228 205 131)(84 237 206 140)(85 226 207 129)(86 235 208 138)(87 224 209 127)(88 233 210 136)(89 222 211 125)(90 231 212 134)(91 240 213 123)(92 229 214 132)(93 238 215 121)(94 227 216 130)(95 236 217 139)(96 225 218 128)(97 234 219 137)(98 223 220 126)(99 232 201 135)(100 221 202 124)
(1 41)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 117)(22 116)(23 115)(24 114)(25 113)(26 112)(27 111)(28 110)(29 109)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 120)(39 119)(40 118)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 80)(76 79)(77 78)(81 220)(82 219)(83 218)(84 217)(85 216)(86 215)(87 214)(88 213)(89 212)(90 211)(91 210)(92 209)(93 208)(94 207)(95 206)(96 205)(97 204)(98 203)(99 202)(100 201)(121 128)(122 127)(123 126)(124 125)(129 140)(130 139)(131 138)(132 137)(133 136)(134 135)(141 144)(142 143)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 154)(152 153)(161 176)(162 175)(163 174)(164 173)(165 172)(166 171)(167 170)(168 169)(177 180)(178 179)(181 192)(182 191)(183 190)(184 189)(185 188)(186 187)(193 200)(194 199)(195 198)(196 197)(221 222)(223 240)(224 239)(225 238)(226 237)(227 236)(228 235)(229 234)(230 233)(231 232)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A ··· 6F 6G 6H 6I 6J 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12P 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 5 5 6 ··· 6 6 6 6 6 10 ··· 10 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 20 20 1 1 2 2 4 4 10 10 10 10 2 2 1 ··· 1 20 20 20 20 2 ··· 2 2 2 2 2 4 4 4 4 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 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D5 C4○D4 D10 C3×D4 C3×D5 C5⋊D4 C3×C4○D4 C6×D5 C3×C5⋊D4 Q8⋊2D5 C3×Q8⋊2D5 kernel C3×C20.23D4 C12×Dic5 C3×D10⋊C4 C6×D20 Q8×C30 C20.23D4 C4×Dic5 D10⋊C4 C2×D20 Q8×C10 C60 C6×Q8 C30 C2×C12 C20 C2×Q8 C12 C10 C2×C4 C4 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 2 2 4 6 4 4 8 8 12 16 4 8

Matrix representation of C3×C20.23D4 in GL4(𝔽61) generated by

 47 0 0 0 0 47 0 0 0 0 13 0 0 0 0 13
,
 60 1 0 0 16 44 0 0 0 0 50 39 0 0 0 11
,
 22 31 0 0 6 39 0 0 0 0 50 0 0 0 0 50
,
 44 60 0 0 44 17 0 0 0 0 50 39 0 0 11 11
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,13,0,0,0,0,13],[60,16,0,0,1,44,0,0,0,0,50,0,0,0,39,11],[22,6,0,0,31,39,0,0,0,0,50,0,0,0,0,50],[44,44,0,0,60,17,0,0,0,0,50,11,0,0,39,11] >;

C3×C20.23D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}._{23}D_4
% in TeX

G:=Group("C3xC20.23D4");
// GroupNames label

G:=SmallGroup(480,740);
// by ID

G=gap.SmallGroup(480,740);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,344,590,555,268,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^9,d*b*d=b^-1,d*c*d=b^10*c^-1>;
// generators/relations

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