direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C20.17D4, C60.127D4, (C4×Dic5)⋊5C6, (D4×C10).5C6, (C6×D4).12D5, C10.52(C6×D4), C20.17(C3×D4), C23.D5⋊9C6, C23.7(C6×D5), (D4×C30).10C2, C30.406(C2×D4), (C22×C6).8D10, (C2×Dic10)⋊10C6, (C12×Dic5)⋊17C2, (C6×Dic10)⋊26C2, (C2×C12).362D10, C12.75(C5⋊D4), C15⋊23(C4.4D4), C30.239(C4○D4), (C2×C60).293C22, (C2×C30).368C23, C6.121(D4⋊2D5), (C22×C30).105C22, (C6×Dic5).251C22, C4.7(C3×C5⋊D4), C5⋊3(C3×C4.4D4), (C2×D4).6(C3×D5), (C2×C4).49(C6×D5), C2.12(C6×C5⋊D4), C22.58(D5×C2×C6), (C2×C20).30(C2×C6), C10.29(C3×C4○D4), C6.133(C2×C5⋊D4), C2.16(C3×D4⋊2D5), (C3×C23.D5)⋊25C2, (C22×C10).24(C2×C6), (C2×C10).51(C22×C6), (C2×Dic5).40(C2×C6), (C2×C6).364(C22×D5), SmallGroup(480,729)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C20.17D4
G = < a,b,c,d | a3=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd-1=b-1, dcd-1=b10c-1 >
Subgroups: 416 in 152 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C30, C30, C30, C4.4D4, Dic10, C2×Dic5, C2×C20, C5×D4, C22×C10, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C3×Dic5, C60, C2×C30, C2×C30, C4×Dic5, C23.D5, C2×Dic10, D4×C10, C3×C4.4D4, C3×Dic10, C6×Dic5, C2×C60, D4×C15, C22×C30, C20.17D4, C12×Dic5, C3×C23.D5, C6×Dic10, D4×C30, C3×C20.17D4
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, C2×D4, C4○D4, D10, C3×D4, C22×C6, C3×D5, C4.4D4, C5⋊D4, C22×D5, C6×D4, C3×C4○D4, C6×D5, D4⋊2D5, C2×C5⋊D4, C3×C4.4D4, C3×C5⋊D4, D5×C2×C6, C20.17D4, C3×D4⋊2D5, C6×C5⋊D4, C3×C20.17D4
►On 240 points
Generators in S240
(1 102 82)(2 103 83)(3 104 84)(4 105 85)(5 106 86)(6 107 87)(7 108 88)(8 109 89)(9 110 90)(10 111 91)(11 112 92)(12 113 93)(13 114 94)(14 115 95)(15 116 96)(16 117 97)(17 118 98)(18 119 99)(19 120 100)(20 101 81)(21 215 181)(22 216 182)(23 217 183)(24 218 184)(25 219 185)(26 220 186)(27 201 187)(28 202 188)(29 203 189)(30 204 190)(31 205 191)(32 206 192)(33 207 193)(34 208 194)(35 209 195)(36 210 196)(37 211 197)(38 212 198)(39 213 199)(40 214 200)(41 176 63)(42 177 64)(43 178 65)(44 179 66)(45 180 67)(46 161 68)(47 162 69)(48 163 70)(49 164 71)(50 165 72)(51 166 73)(52 167 74)(53 168 75)(54 169 76)(55 170 77)(56 171 78)(57 172 79)(58 173 80)(59 174 61)(60 175 62)(121 147 226)(122 148 227)(123 149 228)(124 150 229)(125 151 230)(126 152 231)(127 153 232)(128 154 233)(129 155 234)(130 156 235)(131 157 236)(132 158 237)(133 159 238)(134 160 239)(135 141 240)(136 142 221)(137 143 222)(138 144 223)(139 145 224)(140 146 225)
(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 74 186 153)(2 63 187 142)(3 72 188 151)(4 61 189 160)(5 70 190 149)(6 79 191 158)(7 68 192 147)(8 77 193 156)(9 66 194 145)(10 75 195 154)(11 64 196 143)(12 73 197 152)(13 62 198 141)(14 71 199 150)(15 80 200 159)(16 69 181 148)(17 78 182 157)(18 67 183 146)(19 76 184 155)(20 65 185 144)(21 227 117 47)(22 236 118 56)(23 225 119 45)(24 234 120 54)(25 223 101 43)(26 232 102 52)(27 221 103 41)(28 230 104 50)(29 239 105 59)(30 228 106 48)(31 237 107 57)(32 226 108 46)(33 235 109 55)(34 224 110 44)(35 233 111 53)(36 222 112 42)(37 231 113 51)(38 240 114 60)(39 229 115 49)(40 238 116 58)(81 178 219 138)(82 167 220 127)(83 176 201 136)(84 165 202 125)(85 174 203 134)(86 163 204 123)(87 172 205 132)(88 161 206 121)(89 170 207 130)(90 179 208 139)(91 168 209 128)(92 177 210 137)(93 166 211 126)(94 175 212 135)(95 164 213 124)(96 173 214 133)(97 162 215 122)(98 171 216 131)(99 180 217 140)(100 169 218 129)
(1 69 11 79)(2 68 12 78)(3 67 13 77)(4 66 14 76)(5 65 15 75)(6 64 16 74)(7 63 17 73)(8 62 18 72)(9 61 19 71)(10 80 20 70)(21 232 31 222)(22 231 32 221)(23 230 33 240)(24 229 34 239)(25 228 35 238)(26 227 36 237)(27 226 37 236)(28 225 38 235)(29 224 39 234)(30 223 40 233)(41 118 51 108)(42 117 52 107)(43 116 53 106)(44 115 54 105)(45 114 55 104)(46 113 56 103)(47 112 57 102)(48 111 58 101)(49 110 59 120)(50 109 60 119)(81 163 91 173)(82 162 92 172)(83 161 93 171)(84 180 94 170)(85 179 95 169)(86 178 96 168)(87 177 97 167)(88 176 98 166)(89 175 99 165)(90 174 100 164)(121 211 131 201)(122 210 132 220)(123 209 133 219)(124 208 134 218)(125 207 135 217)(126 206 136 216)(127 205 137 215)(128 204 138 214)(129 203 139 213)(130 202 140 212)(141 183 151 193)(142 182 152 192)(143 181 153 191)(144 200 154 190)(145 199 155 189)(146 198 156 188)(147 197 157 187)(148 196 158 186)(149 195 159 185)(150 194 160 184)
G:=sub<Sym(240)| (1,102,82)(2,103,83)(3,104,84)(4,105,85)(5,106,86)(6,107,87)(7,108,88)(8,109,89)(9,110,90)(10,111,91)(11,112,92)(12,113,93)(13,114,94)(14,115,95)(15,116,96)(16,117,97)(17,118,98)(18,119,99)(19,120,100)(20,101,81)(21,215,181)(22,216,182)(23,217,183)(24,218,184)(25,219,185)(26,220,186)(27,201,187)(28,202,188)(29,203,189)(30,204,190)(31,205,191)(32,206,192)(33,207,193)(34,208,194)(35,209,195)(36,210,196)(37,211,197)(38,212,198)(39,213,199)(40,214,200)(41,176,63)(42,177,64)(43,178,65)(44,179,66)(45,180,67)(46,161,68)(47,162,69)(48,163,70)(49,164,71)(50,165,72)(51,166,73)(52,167,74)(53,168,75)(54,169,76)(55,170,77)(56,171,78)(57,172,79)(58,173,80)(59,174,61)(60,175,62)(121,147,226)(122,148,227)(123,149,228)(124,150,229)(125,151,230)(126,152,231)(127,153,232)(128,154,233)(129,155,234)(130,156,235)(131,157,236)(132,158,237)(133,159,238)(134,160,239)(135,141,240)(136,142,221)(137,143,222)(138,144,223)(139,145,224)(140,146,225), (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), (1,74,186,153)(2,63,187,142)(3,72,188,151)(4,61,189,160)(5,70,190,149)(6,79,191,158)(7,68,192,147)(8,77,193,156)(9,66,194,145)(10,75,195,154)(11,64,196,143)(12,73,197,152)(13,62,198,141)(14,71,199,150)(15,80,200,159)(16,69,181,148)(17,78,182,157)(18,67,183,146)(19,76,184,155)(20,65,185,144)(21,227,117,47)(22,236,118,56)(23,225,119,45)(24,234,120,54)(25,223,101,43)(26,232,102,52)(27,221,103,41)(28,230,104,50)(29,239,105,59)(30,228,106,48)(31,237,107,57)(32,226,108,46)(33,235,109,55)(34,224,110,44)(35,233,111,53)(36,222,112,42)(37,231,113,51)(38,240,114,60)(39,229,115,49)(40,238,116,58)(81,178,219,138)(82,167,220,127)(83,176,201,136)(84,165,202,125)(85,174,203,134)(86,163,204,123)(87,172,205,132)(88,161,206,121)(89,170,207,130)(90,179,208,139)(91,168,209,128)(92,177,210,137)(93,166,211,126)(94,175,212,135)(95,164,213,124)(96,173,214,133)(97,162,215,122)(98,171,216,131)(99,180,217,140)(100,169,218,129), (1,69,11,79)(2,68,12,78)(3,67,13,77)(4,66,14,76)(5,65,15,75)(6,64,16,74)(7,63,17,73)(8,62,18,72)(9,61,19,71)(10,80,20,70)(21,232,31,222)(22,231,32,221)(23,230,33,240)(24,229,34,239)(25,228,35,238)(26,227,36,237)(27,226,37,236)(28,225,38,235)(29,224,39,234)(30,223,40,233)(41,118,51,108)(42,117,52,107)(43,116,53,106)(44,115,54,105)(45,114,55,104)(46,113,56,103)(47,112,57,102)(48,111,58,101)(49,110,59,120)(50,109,60,119)(81,163,91,173)(82,162,92,172)(83,161,93,171)(84,180,94,170)(85,179,95,169)(86,178,96,168)(87,177,97,167)(88,176,98,166)(89,175,99,165)(90,174,100,164)(121,211,131,201)(122,210,132,220)(123,209,133,219)(124,208,134,218)(125,207,135,217)(126,206,136,216)(127,205,137,215)(128,204,138,214)(129,203,139,213)(130,202,140,212)(141,183,151,193)(142,182,152,192)(143,181,153,191)(144,200,154,190)(145,199,155,189)(146,198,156,188)(147,197,157,187)(148,196,158,186)(149,195,159,185)(150,194,160,184)>;
G:=Group( (1,102,82)(2,103,83)(3,104,84)(4,105,85)(5,106,86)(6,107,87)(7,108,88)(8,109,89)(9,110,90)(10,111,91)(11,112,92)(12,113,93)(13,114,94)(14,115,95)(15,116,96)(16,117,97)(17,118,98)(18,119,99)(19,120,100)(20,101,81)(21,215,181)(22,216,182)(23,217,183)(24,218,184)(25,219,185)(26,220,186)(27,201,187)(28,202,188)(29,203,189)(30,204,190)(31,205,191)(32,206,192)(33,207,193)(34,208,194)(35,209,195)(36,210,196)(37,211,197)(38,212,198)(39,213,199)(40,214,200)(41,176,63)(42,177,64)(43,178,65)(44,179,66)(45,180,67)(46,161,68)(47,162,69)(48,163,70)(49,164,71)(50,165,72)(51,166,73)(52,167,74)(53,168,75)(54,169,76)(55,170,77)(56,171,78)(57,172,79)(58,173,80)(59,174,61)(60,175,62)(121,147,226)(122,148,227)(123,149,228)(124,150,229)(125,151,230)(126,152,231)(127,153,232)(128,154,233)(129,155,234)(130,156,235)(131,157,236)(132,158,237)(133,159,238)(134,160,239)(135,141,240)(136,142,221)(137,143,222)(138,144,223)(139,145,224)(140,146,225), (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), (1,74,186,153)(2,63,187,142)(3,72,188,151)(4,61,189,160)(5,70,190,149)(6,79,191,158)(7,68,192,147)(8,77,193,156)(9,66,194,145)(10,75,195,154)(11,64,196,143)(12,73,197,152)(13,62,198,141)(14,71,199,150)(15,80,200,159)(16,69,181,148)(17,78,182,157)(18,67,183,146)(19,76,184,155)(20,65,185,144)(21,227,117,47)(22,236,118,56)(23,225,119,45)(24,234,120,54)(25,223,101,43)(26,232,102,52)(27,221,103,41)(28,230,104,50)(29,239,105,59)(30,228,106,48)(31,237,107,57)(32,226,108,46)(33,235,109,55)(34,224,110,44)(35,233,111,53)(36,222,112,42)(37,231,113,51)(38,240,114,60)(39,229,115,49)(40,238,116,58)(81,178,219,138)(82,167,220,127)(83,176,201,136)(84,165,202,125)(85,174,203,134)(86,163,204,123)(87,172,205,132)(88,161,206,121)(89,170,207,130)(90,179,208,139)(91,168,209,128)(92,177,210,137)(93,166,211,126)(94,175,212,135)(95,164,213,124)(96,173,214,133)(97,162,215,122)(98,171,216,131)(99,180,217,140)(100,169,218,129), (1,69,11,79)(2,68,12,78)(3,67,13,77)(4,66,14,76)(5,65,15,75)(6,64,16,74)(7,63,17,73)(8,62,18,72)(9,61,19,71)(10,80,20,70)(21,232,31,222)(22,231,32,221)(23,230,33,240)(24,229,34,239)(25,228,35,238)(26,227,36,237)(27,226,37,236)(28,225,38,235)(29,224,39,234)(30,223,40,233)(41,118,51,108)(42,117,52,107)(43,116,53,106)(44,115,54,105)(45,114,55,104)(46,113,56,103)(47,112,57,102)(48,111,58,101)(49,110,59,120)(50,109,60,119)(81,163,91,173)(82,162,92,172)(83,161,93,171)(84,180,94,170)(85,179,95,169)(86,178,96,168)(87,177,97,167)(88,176,98,166)(89,175,99,165)(90,174,100,164)(121,211,131,201)(122,210,132,220)(123,209,133,219)(124,208,134,218)(125,207,135,217)(126,206,136,216)(127,205,137,215)(128,204,138,214)(129,203,139,213)(130,202,140,212)(141,183,151,193)(142,182,152,192)(143,181,153,191)(144,200,154,190)(145,199,155,189)(146,198,156,188)(147,197,157,187)(148,196,158,186)(149,195,159,185)(150,194,160,184) );
G=PermutationGroup([[(1,102,82),(2,103,83),(3,104,84),(4,105,85),(5,106,86),(6,107,87),(7,108,88),(8,109,89),(9,110,90),(10,111,91),(11,112,92),(12,113,93),(13,114,94),(14,115,95),(15,116,96),(16,117,97),(17,118,98),(18,119,99),(19,120,100),(20,101,81),(21,215,181),(22,216,182),(23,217,183),(24,218,184),(25,219,185),(26,220,186),(27,201,187),(28,202,188),(29,203,189),(30,204,190),(31,205,191),(32,206,192),(33,207,193),(34,208,194),(35,209,195),(36,210,196),(37,211,197),(38,212,198),(39,213,199),(40,214,200),(41,176,63),(42,177,64),(43,178,65),(44,179,66),(45,180,67),(46,161,68),(47,162,69),(48,163,70),(49,164,71),(50,165,72),(51,166,73),(52,167,74),(53,168,75),(54,169,76),(55,170,77),(56,171,78),(57,172,79),(58,173,80),(59,174,61),(60,175,62),(121,147,226),(122,148,227),(123,149,228),(124,150,229),(125,151,230),(126,152,231),(127,153,232),(128,154,233),(129,155,234),(130,156,235),(131,157,236),(132,158,237),(133,159,238),(134,160,239),(135,141,240),(136,142,221),(137,143,222),(138,144,223),(139,145,224),(140,146,225)], [(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)], [(1,74,186,153),(2,63,187,142),(3,72,188,151),(4,61,189,160),(5,70,190,149),(6,79,191,158),(7,68,192,147),(8,77,193,156),(9,66,194,145),(10,75,195,154),(11,64,196,143),(12,73,197,152),(13,62,198,141),(14,71,199,150),(15,80,200,159),(16,69,181,148),(17,78,182,157),(18,67,183,146),(19,76,184,155),(20,65,185,144),(21,227,117,47),(22,236,118,56),(23,225,119,45),(24,234,120,54),(25,223,101,43),(26,232,102,52),(27,221,103,41),(28,230,104,50),(29,239,105,59),(30,228,106,48),(31,237,107,57),(32,226,108,46),(33,235,109,55),(34,224,110,44),(35,233,111,53),(36,222,112,42),(37,231,113,51),(38,240,114,60),(39,229,115,49),(40,238,116,58),(81,178,219,138),(82,167,220,127),(83,176,201,136),(84,165,202,125),(85,174,203,134),(86,163,204,123),(87,172,205,132),(88,161,206,121),(89,170,207,130),(90,179,208,139),(91,168,209,128),(92,177,210,137),(93,166,211,126),(94,175,212,135),(95,164,213,124),(96,173,214,133),(97,162,215,122),(98,171,216,131),(99,180,217,140),(100,169,218,129)], [(1,69,11,79),(2,68,12,78),(3,67,13,77),(4,66,14,76),(5,65,15,75),(6,64,16,74),(7,63,17,73),(8,62,18,72),(9,61,19,71),(10,80,20,70),(21,232,31,222),(22,231,32,221),(23,230,33,240),(24,229,34,239),(25,228,35,238),(26,227,36,237),(27,226,37,236),(28,225,38,235),(29,224,39,234),(30,223,40,233),(41,118,51,108),(42,117,52,107),(43,116,53,106),(44,115,54,105),(45,114,55,104),(46,113,56,103),(47,112,57,102),(48,111,58,101),(49,110,59,120),(50,109,60,119),(81,163,91,173),(82,162,92,172),(83,161,93,171),(84,180,94,170),(85,179,95,169),(86,178,96,168),(87,177,97,167),(88,176,98,166),(89,175,99,165),(90,174,100,164),(121,211,131,201),(122,210,132,220),(123,209,133,219),(124,208,134,218),(125,207,135,217),(126,206,136,216),(127,205,137,215),(128,204,138,214),(129,203,139,213),(130,202,140,212),(141,183,151,193),(142,182,152,192),(143,181,153,191),(144,200,154,190),(145,199,155,189),(146,198,156,188),(147,197,157,187),(148,196,158,186),(149,195,159,185),(150,194,160,184)]])
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 | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | 12N | 12O | 12P | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| 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 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 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 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D5 | C4○D4 | D10 | D10 | C3×D4 | C3×D5 | C5⋊D4 | C3×C4○D4 | C6×D5 | C6×D5 | C3×C5⋊D4 | D4⋊2D5 | C3×D4⋊2D5 |
| kernel | C3×C20.17D4 | C12×Dic5 | C3×C23.D5 | C6×Dic10 | D4×C30 | C20.17D4 | C4×Dic5 | C23.D5 | C2×Dic10 | D4×C10 | C60 | C6×D4 | C30 | C2×C12 | C22×C6 | C20 | C2×D4 | C12 | C10 | C2×C4 | C23 | C4 | C6 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 8 | 16 | 4 | 8 |
Matrix representation of C3×C20.17D4 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 |
| 15 | 41 | 0 | 0 |
| 0 | 0 | 60 | 59 |
| 0 | 0 | 1 | 1 |
| 53 | 57 | 0 | 0 |
| 1 | 8 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 53 | 57 | 0 | 0 |
| 31 | 8 | 0 | 0 |
| 0 | 0 | 50 | 39 |
| 0 | 0 | 0 | 11 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[3,15,0,0,0,41,0,0,0,0,60,1,0,0,59,1],[53,1,0,0,57,8,0,0,0,0,11,0,0,0,0,11],[53,31,0,0,57,8,0,0,0,0,50,0,0,0,39,11] >;
C3×C20.17D4 in GAP, Magma, Sage, TeX
C_3\times C_{20}._{17}D_4 % in TeX
G:=Group("C3xC20.17D4"); // GroupNames label
G:=SmallGroup(480,729);
// by ID
G=gap.SmallGroup(480,729);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,176,1598,303,18822]);
// 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=b^9,d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations