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## G = C3×C20.17D4order 480 = 25·3·5

### Direct product of C3 and C20.17D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C20.17D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C12×Dic5 — C3×C20.17D4
 Lower central C5 — C2×C10 — C3×C20.17D4
 Upper central C1 — C2×C6 — C6×D4

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 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C10, C10 [×2], C10 [×2], C12 [×2], C12 [×4], C2×C6, C2×C6 [×6], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×6], C2×C12, C2×C12 [×4], C3×D4 [×2], C3×Q8 [×2], C22×C6 [×2], C30, C30 [×2], C30 [×2], C4.4D4, Dic10 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×2], C22×C10 [×2], C4×C12, C3×C22⋊C4 [×4], C6×D4, C6×Q8, C3×Dic5 [×4], C60 [×2], C2×C30, C2×C30 [×6], C4×Dic5, C23.D5 [×4], C2×Dic10, D4×C10, C3×C4.4D4, C3×Dic10 [×2], C6×Dic5 [×4], C2×C60, D4×C15 [×2], C22×C30 [×2], C20.17D4, C12×Dic5, C3×C23.D5 [×4], C6×Dic10, D4×C30, C3×C20.17D4
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], D42D5 [×2], C2×C5⋊D4, C3×C4.4D4, C3×C5⋊D4 [×2], D5×C2×C6, C20.17D4, C3×D42D5 [×2], C6×C5⋊D4, C3×C20.17D4

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

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

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

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

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

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