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## G = C3×C20⋊2D4order 480 = 25·3·5

### Direct product of C3 and C20⋊2D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C20⋊2D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — D5×C2×C12 — C3×C20⋊2D4
 Lower central C5 — C2×C10 — C3×C20⋊2D4
 Upper central C1 — C2×C6 — C6×D4

Generators and relations for C3×C202D4
G = < a,b,c,d | a3=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, dcd=c-1 >

Subgroups: 608 in 188 conjugacy classes, 70 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], C5, C6 [×3], C6 [×4], C2×C4, C2×C4 [×5], D4 [×6], C23 [×2], C23, D5 [×2], C10 [×3], C10 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×10], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4, C2×D4 [×2], Dic5 [×3], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], C2×C12, C2×C12 [×5], C3×D4 [×6], C22×C6 [×2], C22×C6, C3×D5 [×2], C30 [×3], C30 [×2], C4⋊D4, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12, C6×D4, C6×D4 [×2], C3×Dic5 [×3], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C2×C30 [×6], C4⋊Dic5, C23.D5 [×2], C2×C4×D5, C2×C5⋊D4 [×2], D4×C10, C3×C4⋊D4, D5×C12 [×2], C6×Dic5, C6×Dic5 [×2], C3×C5⋊D4 [×4], C2×C60, D4×C15 [×2], D5×C2×C6, C22×C30 [×2], C202D4, C3×C4⋊Dic5, C3×C23.D5 [×2], D5×C2×C12, C6×C5⋊D4 [×2], D4×C30, C3×C202D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×4], C23, D5, C2×C6 [×7], C2×D4 [×2], C4○D4, D10 [×3], C3×D4 [×4], C22×C6, C3×D5, C4⋊D4, C5⋊D4 [×2], C22×D5, C6×D4 [×2], C3×C4○D4, C6×D5 [×3], D4×D5, D42D5, C2×C5⋊D4, C3×C4⋊D4, C3×C5⋊D4 [×2], D5×C2×C6, C202D4, C3×D4×D5, C3×D42D5, C6×C5⋊D4, C3×C202D4

Smallest permutation representation of C3×C202D4
On 240 points
Generators in S240
(1 182 41)(2 183 42)(3 184 43)(4 185 44)(5 186 45)(6 187 46)(7 188 47)(8 189 48)(9 190 49)(10 191 50)(11 192 51)(12 193 52)(13 194 53)(14 195 54)(15 196 55)(16 197 56)(17 198 57)(18 199 58)(19 200 59)(20 181 60)(21 205 126)(22 206 127)(23 207 128)(24 208 129)(25 209 130)(26 210 131)(27 211 132)(28 212 133)(29 213 134)(30 214 135)(31 215 136)(32 216 137)(33 217 138)(34 218 139)(35 219 140)(36 220 121)(37 201 122)(38 202 123)(39 203 124)(40 204 125)(61 84 105)(62 85 106)(63 86 107)(64 87 108)(65 88 109)(66 89 110)(67 90 111)(68 91 112)(69 92 113)(70 93 114)(71 94 115)(72 95 116)(73 96 117)(74 97 118)(75 98 119)(76 99 120)(77 100 101)(78 81 102)(79 82 103)(80 83 104)(141 228 175)(142 229 176)(143 230 177)(144 231 178)(145 232 179)(146 233 180)(147 234 161)(148 235 162)(149 236 163)(150 237 164)(151 238 165)(152 239 166)(153 240 167)(154 221 168)(155 222 169)(156 223 170)(157 224 171)(158 225 172)(159 226 173)(160 227 174)
(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 86 140 234)(2 85 121 233)(3 84 122 232)(4 83 123 231)(5 82 124 230)(6 81 125 229)(7 100 126 228)(8 99 127 227)(9 98 128 226)(10 97 129 225)(11 96 130 224)(12 95 131 223)(13 94 132 222)(14 93 133 221)(15 92 134 240)(16 91 135 239)(17 90 136 238)(18 89 137 237)(19 88 138 236)(20 87 139 235)(21 175 188 101)(22 174 189 120)(23 173 190 119)(24 172 191 118)(25 171 192 117)(26 170 193 116)(27 169 194 115)(28 168 195 114)(29 167 196 113)(30 166 197 112)(31 165 198 111)(32 164 199 110)(33 163 200 109)(34 162 181 108)(35 161 182 107)(36 180 183 106)(37 179 184 105)(38 178 185 104)(39 177 186 103)(40 176 187 102)(41 63 219 147)(42 62 220 146)(43 61 201 145)(44 80 202 144)(45 79 203 143)(46 78 204 142)(47 77 205 141)(48 76 206 160)(49 75 207 159)(50 74 208 158)(51 73 209 157)(52 72 210 156)(53 71 211 155)(54 70 212 154)(55 69 213 153)(56 68 214 152)(57 67 215 151)(58 66 216 150)(59 65 217 149)(60 64 218 148)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 29)(22 38)(23 27)(24 36)(26 34)(28 32)(31 39)(33 37)(42 50)(43 59)(44 48)(45 57)(47 55)(49 53)(52 60)(54 58)(61 149)(62 158)(63 147)(64 156)(65 145)(66 154)(67 143)(68 152)(69 141)(70 150)(71 159)(72 148)(73 157)(74 146)(75 155)(76 144)(77 153)(78 142)(79 151)(80 160)(81 229)(82 238)(83 227)(84 236)(85 225)(86 234)(87 223)(88 232)(89 221)(90 230)(91 239)(92 228)(93 237)(94 226)(95 235)(96 224)(97 233)(98 222)(99 231)(100 240)(101 167)(102 176)(103 165)(104 174)(105 163)(106 172)(107 161)(108 170)(109 179)(110 168)(111 177)(112 166)(113 175)(114 164)(115 173)(116 162)(117 171)(118 180)(119 169)(120 178)(121 129)(122 138)(123 127)(124 136)(126 134)(128 132)(131 139)(133 137)(181 193)(183 191)(184 200)(185 189)(186 198)(188 196)(190 194)(195 199)(201 217)(202 206)(203 215)(205 213)(207 211)(208 220)(210 218)(212 216)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 5 5 6 ··· 6 6 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 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 10 10 1 1 2 2 10 10 20 20 2 2 1 ··· 1 4 4 4 4 10 10 10 10 2 ··· 2 4 ··· 4 2 2 2 2 10 10 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 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4 D4 D5 C4○D4 D10 D10 C3×D4 C3×D4 C3×D5 C5⋊D4 C3×C4○D4 C6×D5 C6×D5 C3×C5⋊D4 D4×D5 D4⋊2D5 C3×D4×D5 C3×D4⋊2D5 kernel C3×C20⋊2D4 C3×C4⋊Dic5 C3×C23.D5 D5×C2×C12 C6×C5⋊D4 D4×C30 C20⋊2D4 C4⋊Dic5 C23.D5 C2×C4×D5 C2×C5⋊D4 D4×C10 C60 C6×D5 C6×D4 C30 C2×C12 C22×C6 C20 D10 C2×D4 C12 C10 C2×C4 C23 C4 C6 C6 C2 C2 # reps 1 1 2 1 2 1 2 2 4 2 4 2 2 2 2 2 2 4 4 4 4 8 4 4 8 16 2 2 4 4

Matrix representation of C3×C202D4 in GL6(𝔽61)

 13 0 0 0 0 0 0 13 0 0 0 0 0 0 47 0 0 0 0 0 0 47 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 0 1 0 0 0 0 60 17 0 0 0 0 0 0 50 0 0 0 0 0 34 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 44 1 0 0 0 0 0 0 1 15 0 0 0 0 8 60 0 0 0 0 0 0 0 60 0 0 0 0 1 0
,
 1 0 0 0 0 0 17 60 0 0 0 0 0 0 1 0 0 0 0 0 8 60 0 0 0 0 0 0 1 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[0,60,0,0,0,0,1,17,0,0,0,0,0,0,50,34,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,44,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,15,60,0,0,0,0,0,0,0,1,0,0,0,0,60,0],[1,17,0,0,0,0,0,60,0,0,0,0,0,0,1,8,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60] >;

C3×C202D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}\rtimes_2D_4
% in TeX

G:=Group("C3xC20:2D4");
// GroupNames label

G:=SmallGroup(480,731);
// by ID

G=gap.SmallGroup(480,731);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,590,555,18822]);
// 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^-1,d*b*d=b^9,d*c*d=c^-1>;
// generators/relations

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