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G = C2×C12.28D10order 480 = 25·3·5

Direct product of C2 and C12.28D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×C12.28D10
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C3⋊D20 — C2×C3⋊D20 — C2×C12.28D10
 Lower central C15 — C30 — C2×C12.28D10
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C12.28D10
G = < a,b,c,d | a2=b12=d2=1, c10=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b6c9 >

Subgroups: 1692 in 328 conjugacy classes, 116 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×4], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×6], C10, C10 [×2], Dic3 [×4], C12 [×2], C12 [×2], D6 [×8], C2×C6, C2×C6 [×4], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×10], C2×C10, Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C2×C12 [×5], C22×S3 [×2], C22×C6, C3×D5 [×2], D15 [×4], C30, C30 [×2], C2×C4○D4, C4×D5 [×4], C4×D5 [×8], D20 [×12], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5, C22×D5 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C5×Dic3 [×4], C3×Dic5 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], D30 [×4], D30 [×4], C2×C30, C2×C4×D5, C2×C4×D5 [×2], C2×D20 [×3], Q82D5 [×8], Q8×C10, C2×C4○D12, D30.C2 [×8], C3⋊D20 [×8], D5×C12 [×4], C6×Dic5, C5×Dic6 [×4], C10×Dic3 [×2], D60 [×4], C2×C60, D5×C2×C6, C22×D15 [×2], C2×Q82D5, C12.28D10 [×8], C2×D30.C2 [×2], C2×C3⋊D20 [×2], D5×C2×C12, C10×Dic6, C2×D60, C2×C12.28D10
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], C4○D12 [×2], S3×C23, S3×D5, Q82D5 [×2], C23×D5, C2×C4○D12, C2×S3×D5 [×3], C2×Q82D5, C12.28D10 [×2], C22×S3×D5, C2×C12.28D10

Smallest permutation representation of C2×C12.28D10
On 240 points
Generators in S240
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 41)(20 42)(21 226)(22 227)(23 228)(24 229)(25 230)(26 231)(27 232)(28 233)(29 234)(30 235)(31 236)(32 237)(33 238)(34 239)(35 240)(36 221)(37 222)(38 223)(39 224)(40 225)(61 202)(62 203)(63 204)(64 205)(65 206)(66 207)(67 208)(68 209)(69 210)(70 211)(71 212)(72 213)(73 214)(74 215)(75 216)(76 217)(77 218)(78 219)(79 220)(80 201)(81 177)(82 178)(83 179)(84 180)(85 161)(86 162)(87 163)(88 164)(89 165)(90 166)(91 167)(92 168)(93 169)(94 170)(95 171)(96 172)(97 173)(98 174)(99 175)(100 176)(101 195)(102 196)(103 197)(104 198)(105 199)(106 200)(107 181)(108 182)(109 183)(110 184)(111 185)(112 186)(113 187)(114 188)(115 189)(116 190)(117 191)(118 192)(119 193)(120 194)(121 154)(122 155)(123 156)(124 157)(125 158)(126 159)(127 160)(128 141)(129 142)(130 143)(131 144)(132 145)(133 146)(134 147)(135 148)(136 149)(137 150)(138 151)(139 152)(140 153)
(1 140 64 24 187 90 11 130 74 34 197 100)(2 81 198 35 75 131 12 91 188 25 65 121)(3 122 66 26 189 92 13 132 76 36 199 82)(4 83 200 37 77 133 14 93 190 27 67 123)(5 124 68 28 191 94 15 134 78 38 181 84)(6 85 182 39 79 135 16 95 192 29 69 125)(7 126 70 30 193 96 17 136 80 40 183 86)(8 87 184 21 61 137 18 97 194 31 71 127)(9 128 72 32 195 98 19 138 62 22 185 88)(10 89 186 23 63 139 20 99 196 33 73 129)(41 151 203 227 111 164 51 141 213 237 101 174)(42 175 102 238 214 142 52 165 112 228 204 152)(43 153 205 229 113 166 53 143 215 239 103 176)(44 177 104 240 216 144 54 167 114 230 206 154)(45 155 207 231 115 168 55 145 217 221 105 178)(46 179 106 222 218 146 56 169 116 232 208 156)(47 157 209 233 117 170 57 147 219 223 107 180)(48 161 108 224 220 148 58 171 118 234 210 158)(49 159 211 235 119 172 59 149 201 225 109 162)(50 163 110 226 202 150 60 173 120 236 212 160)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(81 88)(82 87)(83 86)(84 85)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)(121 128)(122 127)(123 126)(124 125)(129 140)(130 139)(131 138)(132 137)(133 136)(134 135)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)(155 160)(156 159)(157 158)(161 180)(162 179)(163 178)(164 177)(165 176)(166 175)(167 174)(168 173)(169 172)(170 171)(181 182)(183 200)(184 199)(185 198)(186 197)(187 196)(188 195)(189 194)(190 193)(191 192)(201 218)(202 217)(203 216)(204 215)(205 214)(206 213)(207 212)(208 211)(209 210)(219 220)(221 226)(222 225)(223 224)(227 240)(228 239)(229 238)(230 237)(231 236)(232 235)(233 234)

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

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

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

72 conjugacy classes

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

72 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 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 D6 C4○D4 D10 D10 D10 C4○D12 S3×D5 Q8⋊2D5 C2×S3×D5 C2×S3×D5 C12.28D10 kernel C2×C12.28D10 C12.28D10 C2×D30.C2 C2×C3⋊D20 D5×C2×C12 C10×Dic6 C2×D60 C2×C4×D5 C2×Dic6 C4×D5 C2×Dic5 C2×C20 C22×D5 C30 Dic6 C2×Dic3 C2×C12 C10 C2×C4 C6 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 2 4 1 1 1 4 8 4 2 8 2 4 4 2 8

Matrix representation of C2×C12.28D10 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 40 0 0 0 19 29
,
 1 44 0 0 17 17 0 0 0 0 32 20 0 0 25 29
,
 1 44 0 0 0 60 0 0 0 0 29 41 0 0 42 32
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,40,19,0,0,0,29],[1,17,0,0,44,17,0,0,0,0,32,25,0,0,20,29],[1,0,0,0,44,60,0,0,0,0,29,42,0,0,41,32] >;

C2×C12.28D10 in GAP, Magma, Sage, TeX

C_2\times C_{12}._{28}D_{10}
% in TeX

G:=Group("C2xC12.28D10");
// GroupNames label

G:=SmallGroup(480,1085);
// by ID

G=gap.SmallGroup(480,1085);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=d^2=1,c^10=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^6*c^9>;
// generators/relations

׿
×
𝔽