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## G = C3×D20⋊6C4order 480 = 25·3·5

### Direct product of C3 and D20⋊6C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D20⋊6C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C6×D20 — C3×D20⋊6C4
 Lower central C5 — C10 — C20 — C3×D20⋊6C4
 Upper central C1 — C2×C6 — C2×C12 — C3×C4⋊C4

Generators and relations for C3×D206C4
G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b15c >

Subgroups: 432 in 100 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, C6 [×3], C6 [×2], C8, C2×C4, C2×C4, D4 [×3], C23, D5 [×2], C10 [×3], C12 [×2], C12, C2×C6, C2×C6 [×4], C15, C4⋊C4, C2×C8, C2×D4, C20 [×2], C20, D10 [×4], C2×C10, C24, C2×C12, C2×C12, C3×D4 [×3], C22×C6, C3×D5 [×2], C30 [×3], D4⋊C4, C52C8, D20 [×2], D20, C2×C20, C2×C20, C22×D5, C3×C4⋊C4, C2×C24, C6×D4, C60 [×2], C60, C6×D5 [×4], C2×C30, C2×C52C8, C5×C4⋊C4, C2×D20, C3×D4⋊C4, C3×C52C8, C3×D20 [×2], C3×D20, C2×C60, C2×C60, D5×C2×C6, D206C4, C6×C52C8, C15×C4⋊C4, C6×D20, C3×D206C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], D5, C12 [×2], C2×C6, C22⋊C4, D8, SD16, D10, C2×C12, C3×D4 [×2], C3×D5, D4⋊C4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C3×D8, C3×SD16, C6×D5, D10⋊C4, D4⋊D5, Q8⋊D5, C3×D4⋊C4, D5×C12, C3×D20, C3×C5⋊D4, D206C4, C3×D10⋊C4, C3×D4⋊D5, C3×Q8⋊D5, C3×D206C4

Smallest permutation representation of C3×D206C4
On 240 points
Generators in S240
(1 86 56)(2 87 57)(3 88 58)(4 89 59)(5 90 60)(6 91 41)(7 92 42)(8 93 43)(9 94 44)(10 95 45)(11 96 46)(12 97 47)(13 98 48)(14 99 49)(15 100 50)(16 81 51)(17 82 52)(18 83 53)(19 84 54)(20 85 55)(21 106 61)(22 107 62)(23 108 63)(24 109 64)(25 110 65)(26 111 66)(27 112 67)(28 113 68)(29 114 69)(30 115 70)(31 116 71)(32 117 72)(33 118 73)(34 119 74)(35 120 75)(36 101 76)(37 102 77)(38 103 78)(39 104 79)(40 105 80)(121 206 176)(122 207 177)(123 208 178)(124 209 179)(125 210 180)(126 211 161)(127 212 162)(128 213 163)(129 214 164)(130 215 165)(131 216 166)(132 217 167)(133 218 168)(134 219 169)(135 220 170)(136 201 171)(137 202 172)(138 203 173)(139 204 174)(140 205 175)(141 226 191)(142 227 192)(143 228 193)(144 229 194)(145 230 195)(146 231 196)(147 232 197)(148 233 198)(149 234 199)(150 235 200)(151 236 181)(152 237 182)(153 238 183)(154 239 184)(155 240 185)(156 221 186)(157 222 187)(158 223 188)(159 224 189)(160 225 190)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)(121 125)(122 124)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(141 145)(142 144)(146 160)(147 159)(148 158)(149 157)(150 156)(151 155)(152 154)(161 175)(162 174)(163 173)(164 172)(165 171)(166 170)(167 169)(176 180)(177 179)(181 185)(182 184)(186 200)(187 199)(188 198)(189 197)(190 196)(191 195)(192 194)(201 215)(202 214)(203 213)(204 212)(205 211)(206 210)(207 209)(216 220)(217 219)(221 235)(222 234)(223 233)(224 232)(225 231)(226 230)(227 229)(236 240)(237 239)
(1 151 21 131)(2 142 22 122)(3 153 23 133)(4 144 24 124)(5 155 25 135)(6 146 26 126)(7 157 27 137)(8 148 28 128)(9 159 29 139)(10 150 30 130)(11 141 31 121)(12 152 32 132)(13 143 33 123)(14 154 34 134)(15 145 35 125)(16 156 36 136)(17 147 37 127)(18 158 38 138)(19 149 39 129)(20 160 40 140)(41 196 66 161)(42 187 67 172)(43 198 68 163)(44 189 69 174)(45 200 70 165)(46 191 71 176)(47 182 72 167)(48 193 73 178)(49 184 74 169)(50 195 75 180)(51 186 76 171)(52 197 77 162)(53 188 78 173)(54 199 79 164)(55 190 80 175)(56 181 61 166)(57 192 62 177)(58 183 63 168)(59 194 64 179)(60 185 65 170)(81 221 101 201)(82 232 102 212)(83 223 103 203)(84 234 104 214)(85 225 105 205)(86 236 106 216)(87 227 107 207)(88 238 108 218)(89 229 109 209)(90 240 110 220)(91 231 111 211)(92 222 112 202)(93 233 113 213)(94 224 114 204)(95 235 115 215)(96 226 116 206)(97 237 117 217)(98 228 118 208)(99 239 119 219)(100 230 120 210)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 5A 5B 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 20A ··· 20L 24A ··· 24H 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 2 3 3 4 4 4 4 5 5 6 ··· 6 6 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 20 20 1 1 2 2 4 4 2 2 1 ··· 1 20 20 20 20 10 10 10 10 2 ··· 2 2 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4 10 ··· 10 2 ··· 2 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 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D4 D5 D8 SD16 D10 C3×D4 C3×D4 C3×D5 C4×D5 D20 C5⋊D4 C3×D8 C3×SD16 C6×D5 D5×C12 C3×D20 C3×C5⋊D4 D4⋊D5 Q8⋊D5 C3×D4⋊D5 C3×Q8⋊D5 kernel C3×D20⋊6C4 C6×C5⋊2C8 C15×C4⋊C4 C6×D20 D20⋊6C4 C3×D20 C2×C5⋊2C8 C5×C4⋊C4 C2×D20 D20 C60 C2×C30 C3×C4⋊C4 C30 C30 C2×C12 C20 C2×C10 C4⋊C4 C12 C12 C2×C6 C10 C10 C2×C4 C4 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 8 2 2 4 4

Matrix representation of C3×D206C4 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 1 0 0 0 0 1
,
 1 2 0 0 240 240 0 0 0 0 52 1 0 0 240 0
,
 1 0 0 0 240 240 0 0 0 0 52 1 0 0 189 189
,
 0 22 0 0 11 0 0 0 0 0 200 156 0 0 85 41
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,2,240,0,0,0,0,52,240,0,0,1,0],[1,240,0,0,0,240,0,0,0,0,52,189,0,0,1,189],[0,11,0,0,22,0,0,0,0,0,200,85,0,0,156,41] >;

C3×D206C4 in GAP, Magma, Sage, TeX

C_3\times D_{20}\rtimes_6C_4
% in TeX

G:=Group("C3xD20:6C4");
// GroupNames label

G:=SmallGroup(480,87);
// by ID

G=gap.SmallGroup(480,87);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,2524,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^15*c>;
// generators/relations

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