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## G = C3×C20⋊D4order 480 = 25·3·5

### Direct product of C3 and C20⋊D4

Series: Derived Chief Lower central Upper central

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

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

Subgroups: 800 in 216 conjugacy classes, 74 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×12], C5, C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×2], D4 [×12], C23 [×2], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C12 [×2], C12 [×4], C2×C6, C2×C6 [×12], C15, C42, C2×D4, C2×D4 [×5], Dic5 [×4], C20 [×2], D10 [×6], C2×C10, C2×C10 [×6], C2×C12, C2×C12 [×2], C3×D4 [×12], C22×C6 [×2], C22×C6 [×2], C3×D5 [×2], C30, C30 [×2], C30 [×2], C41D4, D20 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], C4×C12, C6×D4, C6×D4 [×5], C3×Dic5 [×4], C60 [×2], C6×D5 [×6], C2×C30, C2×C30 [×6], C4×Dic5, C2×D20, C2×C5⋊D4 [×4], D4×C10, C3×C41D4, C3×D20 [×2], C6×Dic5 [×2], C3×C5⋊D4 [×8], C2×C60, D4×C15 [×2], D5×C2×C6 [×2], C22×C30 [×2], C20⋊D4, C12×Dic5, C6×D20, C6×C5⋊D4 [×4], D4×C30, C3×C20⋊D4
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×6], C23, D5, C2×C6 [×7], C2×D4 [×3], D10 [×3], C3×D4 [×6], C22×C6, C3×D5, C41D4, C5⋊D4 [×2], C22×D5, C6×D4 [×3], C6×D5 [×3], D4×D5 [×2], C2×C5⋊D4, C3×C41D4, C3×C5⋊D4 [×2], D5×C2×C6, C20⋊D4, C3×D4×D5 [×2], C6×C5⋊D4, C3×C20⋊D4

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

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

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

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

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 ··· 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 15 15 15 15 20 20 20 20 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 4 4 20 20 1 1 2 2 10 10 10 10 2 2 1 ··· 1 4 4 4 4 20 20 20 20 2 ··· 2 4 ··· 4 2 2 2 2 10 ··· 10 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 D4 D5 D10 D10 C3×D4 C3×D4 C3×D5 C5⋊D4 C6×D5 C6×D5 C3×C5⋊D4 D4×D5 C3×D4×D5 kernel C3×C20⋊D4 C12×Dic5 C6×D20 C6×C5⋊D4 D4×C30 C20⋊D4 C4×Dic5 C2×D20 C2×C5⋊D4 D4×C10 C3×Dic5 C60 C6×D4 C2×C12 C22×C6 Dic5 C20 C2×D4 C12 C2×C4 C23 C4 C6 C2 # reps 1 1 1 4 1 2 2 2 8 2 4 2 2 2 4 8 4 4 8 4 8 16 4 8

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

 13 0 0 0 0 13 0 0 0 0 1 0 0 0 0 1
,
 0 60 0 0 1 44 0 0 0 0 1 2 0 0 60 60
,
 14 45 0 0 39 47 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 44 1 0 0 0 0 1 2 0 0 0 60
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,60,44,0,0,0,0,1,60,0,0,2,60],[14,39,0,0,45,47,0,0,0,0,60,0,0,0,0,60],[60,44,0,0,0,1,0,0,0,0,1,0,0,0,2,60] >;

C3×C20⋊D4 in GAP, Magma, Sage, TeX

C_3\times C_{20}\rtimes D_4
% in TeX

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

G:=SmallGroup(480,733);
// by ID

G=gap.SmallGroup(480,733);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,344,590,555,18822]);
// 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^9,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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