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## G = C3×C20.48D4order 480 = 25·3·5

### Direct product of C3 and C20.48D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C20.48D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C6×Dic10 — C3×C20.48D4
 Lower central C5 — C2×C10 — C3×C20.48D4
 Upper central C1 — C2×C6 — C22×C12

Generators and relations for C3×C20.48D4
G = < a,b,c,d | a3=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b10c-1 >

Subgroups: 384 in 148 conjugacy classes, 74 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C5, C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C10 [×3], C10 [×2], C12 [×2], C12 [×5], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C12 [×2], C2×C12 [×6], C3×Q8 [×2], C22×C6, C30 [×3], C30 [×2], C22⋊Q8, Dic10 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×2], C22×C10, C3×C22⋊C4 [×2], C3×C4⋊C4 [×3], C22×C12, C6×Q8, C3×Dic5 [×4], C60 [×2], C60, C2×C30, C2×C30 [×2], C2×C30 [×2], C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C2×Dic10, C22×C20, C3×C22⋊Q8, C3×Dic10 [×2], C6×Dic5 [×4], C2×C60 [×2], C2×C60 [×2], C22×C30, C20.48D4, C3×C10.D4 [×2], C3×C4⋊Dic5, C3×C23.D5 [×2], C6×Dic10, C22×C60, C3×C20.48D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], Q8 [×2], C23, D5, C2×C6 [×7], C2×D4, C2×Q8, C4○D4, D10 [×3], C3×D4 [×2], C3×Q8 [×2], C22×C6, C3×D5, C22⋊Q8, Dic10 [×2], C5⋊D4 [×2], C22×D5, C6×D4, C6×Q8, C3×C4○D4, C6×D5 [×3], C2×Dic10, C4○D20, C2×C5⋊D4, C3×C22⋊Q8, C3×Dic10 [×2], C3×C5⋊D4 [×2], D5×C2×C6, C20.48D4, C6×Dic10, C3×C4○D20, C6×C5⋊D4, C3×C20.48D4

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

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

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

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

138 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 ··· 10N 12A ··· 12H 12I ··· 12P 15A 15B 15C 15D 20A ··· 20P 30A ··· 30AB 60A ··· 60AF 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 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 1 1 2 2 2 2 20 20 20 20 2 2 1 ··· 1 2 2 2 2 2 ··· 2 2 ··· 2 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 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 Q8 D5 C4○D4 D10 D10 C3×D4 C3×Q8 C3×D5 C5⋊D4 Dic10 C3×C4○D4 C6×D5 C6×D5 C4○D20 C3×C5⋊D4 C3×Dic10 C3×C4○D20 kernel C3×C20.48D4 C3×C10.D4 C3×C4⋊Dic5 C3×C23.D5 C6×Dic10 C22×C60 C20.48D4 C10.D4 C4⋊Dic5 C23.D5 C2×Dic10 C22×C20 C60 C2×C30 C22×C12 C30 C2×C12 C22×C6 C20 C2×C10 C22×C4 C12 C2×C6 C10 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 1 1 2 4 2 4 2 2 2 2 2 2 4 2 4 4 4 8 8 4 8 4 8 16 16 16

Matrix representation of C3×C20.48D4 in GL4(𝔽61) generated by

 13 0 0 0 0 13 0 0 0 0 1 0 0 0 0 1
,
 37 58 0 0 0 33 0 0 0 0 20 0 0 0 0 58
,
 57 7 0 0 15 4 0 0 0 0 0 1 0 0 60 0
,
 57 7 0 0 15 4 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[37,0,0,0,58,33,0,0,0,0,20,0,0,0,0,58],[57,15,0,0,7,4,0,0,0,0,0,60,0,0,1,0],[57,15,0,0,7,4,0,0,0,0,0,1,0,0,1,0] >;

C3×C20.48D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}._{48}D_4
% in TeX

G:=Group("C3xC20.48D4");
// GroupNames label

G:=SmallGroup(480,717);
// by ID

G=gap.SmallGroup(480,717);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,701,344,590,18822]);
// Polycyclic

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

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