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## G = (S3×C20)⋊5C4order 480 = 25·3·5

### 1st semidirect product of S3×C20 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — (S3×C20)⋊5C4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — (S3×C20)⋊5C4
 Lower central C15 — C30 — (S3×C20)⋊5C4
 Upper central C1 — C22 — C2×C4

Generators and relations for (S3×C20)⋊5C4
G = < a,b,c,d | a20=b3=c2=d4=1, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, bd=db, dcd-1=a10c >

Subgroups: 556 in 152 conjugacy classes, 68 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×9], C23, C10 [×3], C10 [×2], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×2], C30 [×3], C42⋊C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×5], C22×C10, C4×Dic3 [×2], C4⋊Dic3, D6⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C4×Dic5 [×2], C4⋊Dic5, C4⋊Dic5, C23.D5 [×2], C22×C20, C4⋊C47S3, C6×Dic5 [×2], S3×C20 [×4], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, C23.21D10, Dic3×Dic5 [×2], D6⋊Dic5 [×2], C3×C4⋊Dic5, C605C4, S3×C2×C20, (S3×C20)⋊5C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, C4○D4 [×2], Dic5 [×4], D10 [×3], C4×S3 [×2], C22×S3, C42⋊C2, C2×Dic5 [×6], C22×D5, S3×C2×C4, D42S3, Q83S3, S3×D5, C4○D20 [×2], C22×Dic5, C4⋊C47S3, S3×Dic5 [×2], C2×S3×D5, C23.21D10, D205S3, D60⋊C2, C2×S3×Dic5, (S3×C20)⋊5C4

Smallest permutation representation of (S3×C20)⋊5C4
On 240 points
Generators in S240
(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 89 134)(2 90 135)(3 91 136)(4 92 137)(5 93 138)(6 94 139)(7 95 140)(8 96 121)(9 97 122)(10 98 123)(11 99 124)(12 100 125)(13 81 126)(14 82 127)(15 83 128)(16 84 129)(17 85 130)(18 86 131)(19 87 132)(20 88 133)(21 177 57)(22 178 58)(23 179 59)(24 180 60)(25 161 41)(26 162 42)(27 163 43)(28 164 44)(29 165 45)(30 166 46)(31 167 47)(32 168 48)(33 169 49)(34 170 50)(35 171 51)(36 172 52)(37 173 53)(38 174 54)(39 175 55)(40 176 56)(61 145 232)(62 146 233)(63 147 234)(64 148 235)(65 149 236)(66 150 237)(67 151 238)(68 152 239)(69 153 240)(70 154 221)(71 155 222)(72 156 223)(73 157 224)(74 158 225)(75 159 226)(76 160 227)(77 141 228)(78 142 229)(79 143 230)(80 144 231)(101 187 219)(102 188 220)(103 189 201)(104 190 202)(105 191 203)(106 192 204)(107 193 205)(108 194 206)(109 195 207)(110 196 208)(111 197 209)(112 198 210)(113 199 211)(114 200 212)(115 181 213)(116 182 214)(117 183 215)(118 184 216)(119 185 217)(120 186 218)
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 161)(13 162)(14 163)(15 164)(16 165)(17 166)(18 167)(19 168)(20 169)(21 96)(22 97)(23 98)(24 99)(25 100)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 121)(58 122)(59 123)(60 124)(61 116)(62 117)(63 118)(64 119)(65 120)(66 101)(67 102)(68 103)(69 104)(70 105)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 113)(79 114)(80 115)(141 210)(142 211)(143 212)(144 213)(145 214)(146 215)(147 216)(148 217)(149 218)(150 219)(151 220)(152 201)(153 202)(154 203)(155 204)(156 205)(157 206)(158 207)(159 208)(160 209)(181 231)(182 232)(183 233)(184 234)(185 235)(186 236)(187 237)(188 238)(189 239)(190 240)(191 221)(192 222)(193 223)(194 224)(195 225)(196 226)(197 227)(198 228)(199 229)(200 230)
(1 209 180 160)(2 208 161 159)(3 207 162 158)(4 206 163 157)(5 205 164 156)(6 204 165 155)(7 203 166 154)(8 202 167 153)(9 201 168 152)(10 220 169 151)(11 219 170 150)(12 218 171 149)(13 217 172 148)(14 216 173 147)(15 215 174 146)(16 214 175 145)(17 213 176 144)(18 212 177 143)(19 211 178 142)(20 210 179 141)(21 79 131 200)(22 78 132 199)(23 77 133 198)(24 76 134 197)(25 75 135 196)(26 74 136 195)(27 73 137 194)(28 72 138 193)(29 71 139 192)(30 70 140 191)(31 69 121 190)(32 68 122 189)(33 67 123 188)(34 66 124 187)(35 65 125 186)(36 64 126 185)(37 63 127 184)(38 62 128 183)(39 61 129 182)(40 80 130 181)(41 226 90 110)(42 225 91 109)(43 224 92 108)(44 223 93 107)(45 222 94 106)(46 221 95 105)(47 240 96 104)(48 239 97 103)(49 238 98 102)(50 237 99 101)(51 236 100 120)(52 235 81 119)(53 234 82 118)(54 233 83 117)(55 232 84 116)(56 231 85 115)(57 230 86 114)(58 229 87 113)(59 228 88 112)(60 227 89 111)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5A 5B 6A 6B 6C 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 15A 15B 20A ··· 20H 20I ··· 20P 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 12 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 6 6 2 2 2 3 3 3 3 10 10 10 10 30 30 30 30 2 2 2 2 2 2 ··· 2 6 ··· 6 4 4 20 20 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

78 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + - + + + - + + - + - + image C1 C2 C2 C2 C2 C2 C4 S3 D5 D6 D6 C4○D4 Dic5 D10 D10 D10 C4×S3 C4○D20 D4⋊2S3 Q8⋊3S3 S3×D5 S3×Dic5 C2×S3×D5 D20⋊5S3 D60⋊C2 kernel (S3×C20)⋊5C4 Dic3×Dic5 D6⋊Dic5 C3×C4⋊Dic5 C60⋊5C4 S3×C2×C20 S3×C20 C4⋊Dic5 S3×C2×C4 C2×Dic5 C2×C20 C30 C4×S3 C2×Dic3 C2×C12 C22×S3 C20 C6 C10 C10 C2×C4 C4 C22 C2 C2 # reps 1 2 2 1 1 1 8 1 2 2 1 4 8 2 2 2 4 16 1 1 2 4 2 4 4

Matrix representation of (S3×C20)⋊5C4 in GL6(𝔽61)

 38 0 0 0 0 0 36 53 0 0 0 0 0 0 34 0 0 0 0 0 6 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 1 0 0 0 0 60 0
,
 60 0 0 0 0 0 17 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 7 8 0 0 0 0 55 54 0 0 0 0 0 0 51 1 0 0 0 0 21 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(61))| [38,36,0,0,0,0,0,53,0,0,0,0,0,0,34,6,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[60,17,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[7,55,0,0,0,0,8,54,0,0,0,0,0,0,51,21,0,0,0,0,1,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

(S3×C20)⋊5C4 in GAP, Magma, Sage, TeX

(S_3\times C_{20})\rtimes_5C_4
% in TeX

G:=Group("(S3xC20):5C4");
// GroupNames label

G:=SmallGroup(480,414);
// by ID

G=gap.SmallGroup(480,414);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^3=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^10*c>;
// generators/relations

׿
×
𝔽