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## G = C2×D12.D5order 480 = 25·3·5

### Direct product of C2 and D12.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C2×D12.D5
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — D12.D5 — C2×D12.D5
 Lower central C15 — C30 — C60 — C2×D12.D5
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D12.D5
G = < a,b,c,d,e | a2=b12=c2=d5=1, e2=b9, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d-1 >

Subgroups: 668 in 136 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C24, Dic6, D12, D12, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C30, C2×SD16, C52C8, Dic10, C2×Dic5, C2×C20, C5×D4, C22×C10, C24⋊C2, C2×C24, C2×Dic6, C2×D12, Dic15, C60, S3×C10, C2×C30, C2×C52C8, D4.D5, C2×Dic10, D4×C10, C2×C24⋊C2, C3×C52C8, C5×D12, C5×D12, Dic30, Dic30, C2×Dic15, C2×C60, S3×C2×C10, C2×D4.D5, D12.D5, C6×C52C8, C10×D12, C2×Dic30, C2×D12.D5
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, D12, C22×S3, C2×SD16, C5⋊D4, C22×D5, C24⋊C2, C2×D12, S3×D5, D4.D5, C2×C5⋊D4, C2×C24⋊C2, C5⋊D12, C2×S3×D5, C2×D4.D5, D12.D5, C2×C5⋊D12, C2×D12.D5

Smallest permutation representation of C2×D12.D5
On 240 points
Generators in S240
(1 129)(2 130)(3 131)(4 132)(5 121)(6 122)(7 123)(8 124)(9 125)(10 126)(11 127)(12 128)(13 150)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 145)(21 146)(22 147)(23 148)(24 149)(25 226)(26 227)(27 228)(28 217)(29 218)(30 219)(31 220)(32 221)(33 222)(34 223)(35 224)(36 225)(37 163)(38 164)(39 165)(40 166)(41 167)(42 168)(43 157)(44 158)(45 159)(46 160)(47 161)(48 162)(49 235)(50 236)(51 237)(52 238)(53 239)(54 240)(55 229)(56 230)(57 231)(58 232)(59 233)(60 234)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)(85 191)(86 192)(87 181)(88 182)(89 183)(90 184)(91 185)(92 186)(93 187)(94 188)(95 189)(96 190)(97 113)(98 114)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(105 109)(106 110)(107 111)(108 112)(133 179)(134 180)(135 169)(136 170)(137 171)(138 172)(139 173)(140 174)(141 175)(142 176)(143 177)(144 178)(193 206)(194 207)(195 208)(196 209)(197 210)(198 211)(199 212)(200 213)(201 214)(202 215)(203 216)(204 205)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 23)(14 22)(15 21)(16 20)(17 19)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 42)(38 41)(39 40)(43 48)(44 47)(45 46)(49 53)(50 52)(54 60)(55 59)(56 58)(61 65)(62 64)(66 72)(67 71)(68 70)(73 77)(74 76)(78 84)(79 83)(80 82)(85 92)(86 91)(87 90)(88 89)(93 96)(94 95)(97 105)(98 104)(99 103)(100 102)(106 108)(109 113)(110 112)(114 120)(115 119)(116 118)(121 124)(122 123)(125 132)(126 131)(127 130)(128 129)(133 144)(134 143)(135 142)(136 141)(137 140)(138 139)(145 153)(146 152)(147 151)(148 150)(154 156)(157 162)(158 161)(159 160)(163 168)(164 167)(165 166)(169 176)(170 175)(171 174)(172 173)(177 180)(178 179)(181 184)(182 183)(185 192)(186 191)(187 190)(188 189)(193 199)(194 198)(195 197)(200 204)(201 203)(205 213)(206 212)(207 211)(208 210)(214 216)(217 228)(218 227)(219 226)(220 225)(221 224)(222 223)(229 233)(230 232)(234 240)(235 239)(236 238)
(1 34 173 160 189)(2 35 174 161 190)(3 36 175 162 191)(4 25 176 163 192)(5 26 177 164 181)(6 27 178 165 182)(7 28 179 166 183)(8 29 180 167 184)(9 30 169 168 185)(10 31 170 157 186)(11 32 171 158 187)(12 33 172 159 188)(13 112 82 216 52)(14 113 83 205 53)(15 114 84 206 54)(16 115 73 207 55)(17 116 74 208 56)(18 117 75 209 57)(19 118 76 210 58)(20 119 77 211 59)(21 120 78 212 60)(22 109 79 213 49)(23 110 80 214 50)(24 111 81 215 51)(37 86 132 226 142)(38 87 121 227 143)(39 88 122 228 144)(40 89 123 217 133)(41 90 124 218 134)(42 91 125 219 135)(43 92 126 220 136)(44 93 127 221 137)(45 94 128 222 138)(46 95 129 223 139)(47 96 130 224 140)(48 85 131 225 141)(61 194 229 153 99)(62 195 230 154 100)(63 196 231 155 101)(64 197 232 156 102)(65 198 233 145 103)(66 199 234 146 104)(67 200 235 147 105)(68 201 236 148 106)(69 202 237 149 107)(70 203 238 150 108)(71 204 239 151 97)(72 193 240 152 98)
(1 198 10 195 7 204 4 201)(2 199 11 196 8 193 5 202)(3 200 12 197 9 194 6 203)(13 48 22 45 19 42 16 39)(14 37 23 46 20 43 17 40)(15 38 24 47 21 44 18 41)(25 68 34 65 31 62 28 71)(26 69 35 66 32 63 29 72)(27 70 36 67 33 64 30 61)(49 94 58 91 55 88 52 85)(50 95 59 92 56 89 53 86)(51 96 60 93 57 90 54 87)(73 228 82 225 79 222 76 219)(74 217 83 226 80 223 77 220)(75 218 84 227 81 224 78 221)(97 176 106 173 103 170 100 179)(98 177 107 174 104 171 101 180)(99 178 108 175 105 172 102 169)(109 138 118 135 115 144 112 141)(110 139 119 136 116 133 113 142)(111 140 120 137 117 134 114 143)(121 215 130 212 127 209 124 206)(122 216 131 213 128 210 125 207)(123 205 132 214 129 211 126 208)(145 157 154 166 151 163 148 160)(146 158 155 167 152 164 149 161)(147 159 156 168 153 165 150 162)(181 237 190 234 187 231 184 240)(182 238 191 235 188 232 185 229)(183 239 192 236 189 233 186 230)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 24A ··· 24H 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 12 15 15 20 20 20 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 12 12 2 2 2 60 60 2 2 2 2 2 10 10 10 10 2 ··· 2 12 ··· 12 2 2 2 2 4 4 4 4 4 4 10 ··· 10 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 SD16 D10 D10 D12 D12 C5⋊D4 C5⋊D4 C24⋊C2 S3×D5 D4.D5 C5⋊D12 C2×S3×D5 C5⋊D12 D12.D5 kernel C2×D12.D5 D12.D5 C6×C5⋊2C8 C10×D12 C2×Dic30 C2×C5⋊2C8 C60 C2×C30 C2×D12 C5⋊2C8 C2×C20 C30 D12 C2×C12 C20 C2×C10 C12 C2×C6 C10 C2×C4 C6 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 2 2 1 4 4 2 2 2 4 4 8 2 4 2 2 2 8

Matrix representation of C2×D12.D5 in GL5(𝔽241)

 240 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 240 0 0 0 0 0 240 0 0 0 0 0 0 197 0 0 0 126 185
,
 240 0 0 0 0 0 240 0 0 0 0 75 1 0 0 0 0 0 56 153 0 0 0 126 185
,
 1 0 0 0 0 0 98 0 0 0 0 99 91 0 0 0 0 0 1 0 0 0 0 0 1
,
 240 0 0 0 0 0 91 237 0 0 0 22 150 0 0 0 0 0 119 15 0 0 0 209 160

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,0,126,0,0,0,197,185],[240,0,0,0,0,0,240,75,0,0,0,0,1,0,0,0,0,0,56,126,0,0,0,153,185],[1,0,0,0,0,0,98,99,0,0,0,0,91,0,0,0,0,0,1,0,0,0,0,0,1],[240,0,0,0,0,0,91,22,0,0,0,237,150,0,0,0,0,0,119,209,0,0,0,15,160] >;

C2×D12.D5 in GAP, Magma, Sage, TeX

C_2\times D_{12}.D_5
% in TeX

G:=Group("C2xD12.D5");
// GroupNames label

G:=SmallGroup(480,392);
// by ID

G=gap.SmallGroup(480,392);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^5=1,e^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=d^-1>;
// generators/relations

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