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## G = C2×D12⋊5D5order 480 = 25·3·5

### Direct product of C2 and D12⋊5D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D12⋊5D5
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C15⋊D4 — C2×C15⋊D4 — C2×D12⋊5D5
 Lower central C15 — C30 — C2×D12⋊5D5
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D125D5
G = < a,b,c,d,e | a2=b12=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b6c, ede=d-1 >

Subgroups: 1436 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 [×2], C10, C10 [×2], C10 [×4], Dic3 [×4], C12 [×2], C12 [×2], D6 [×4], D6 [×4], C2×C6, C2×C6 [×4], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×4], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×8], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12, C2×C12 [×5], C22×S3 [×2], C22×C6, C5×S3 [×4], C3×D5 [×2], C30, C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×10], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C3×Dic5 [×2], Dic15 [×4], C60 [×2], C6×D5 [×2], C6×D5 [×2], S3×C10 [×4], S3×C10 [×4], C2×C30, C2×Dic10, C2×C4×D5, D42D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C2×C4○D12, S3×Dic5 [×8], C15⋊D4 [×8], D5×C12 [×4], C6×Dic5, C5×D12 [×4], Dic30 [×4], C2×Dic15 [×2], C2×C60, D5×C2×C6, S3×C2×C10 [×2], C2×D42D5, D125D5 [×8], C2×S3×Dic5 [×2], C2×C15⋊D4 [×2], D5×C2×C12, C10×D12, C2×Dic30, C2×D125D5
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, D42D5 [×2], C23×D5, C2×C4○D12, C2×S3×D5 [×3], C2×D42D5, D125D5 [×2], C22×S3×D5, C2×D125D5

Smallest permutation representation of C2×D125D5
On 240 points
Generators in S240
(1 228)(2 217)(3 218)(4 219)(5 220)(6 221)(7 222)(8 223)(9 224)(10 225)(11 226)(12 227)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 133)(36 134)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 121)(61 158)(62 159)(63 160)(64 161)(65 162)(66 163)(67 164)(68 165)(69 166)(70 167)(71 168)(72 157)(73 117)(74 118)(75 119)(76 120)(77 109)(78 110)(79 111)(80 112)(81 113)(82 114)(83 115)(84 116)(85 154)(86 155)(87 156)(88 145)(89 146)(90 147)(91 148)(92 149)(93 150)(94 151)(95 152)(96 153)(97 239)(98 240)(99 229)(100 230)(101 231)(102 232)(103 233)(104 234)(105 235)(106 236)(107 237)(108 238)(169 190)(170 191)(171 192)(172 181)(173 182)(174 183)(175 184)(176 185)(177 186)(178 187)(179 188)(180 189)(193 213)(194 214)(195 215)(196 216)(197 205)(198 206)(199 207)(200 208)(201 209)(202 210)(203 211)(204 212)
(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)
(1 239)(2 238)(3 237)(4 236)(5 235)(6 234)(7 233)(8 232)(9 231)(10 230)(11 229)(12 240)(13 172)(14 171)(15 170)(16 169)(17 180)(18 179)(19 178)(20 177)(21 176)(22 175)(23 174)(24 173)(25 163)(26 162)(27 161)(28 160)(29 159)(30 158)(31 157)(32 168)(33 167)(34 166)(35 165)(36 164)(37 216)(38 215)(39 214)(40 213)(41 212)(42 211)(43 210)(44 209)(45 208)(46 207)(47 206)(48 205)(49 182)(50 181)(51 192)(52 191)(53 190)(54 189)(55 188)(56 187)(57 186)(58 185)(59 184)(60 183)(61 140)(62 139)(63 138)(64 137)(65 136)(66 135)(67 134)(68 133)(69 144)(70 143)(71 142)(72 141)(73 152)(74 151)(75 150)(76 149)(77 148)(78 147)(79 146)(80 145)(81 156)(82 155)(83 154)(84 153)(85 115)(86 114)(87 113)(88 112)(89 111)(90 110)(91 109)(92 120)(93 119)(94 118)(95 117)(96 116)(97 228)(98 227)(99 226)(100 225)(101 224)(102 223)(103 222)(104 221)(105 220)(106 219)(107 218)(108 217)(121 197)(122 196)(123 195)(124 194)(125 193)(126 204)(127 203)(128 202)(129 201)(130 200)(131 199)(132 198)
(1 183 113 203 143)(2 184 114 204 144)(3 185 115 193 133)(4 186 116 194 134)(5 187 117 195 135)(6 188 118 196 136)(7 189 119 197 137)(8 190 120 198 138)(9 191 109 199 139)(10 192 110 200 140)(11 181 111 201 141)(12 182 112 202 142)(13 146 44 157 99)(14 147 45 158 100)(15 148 46 159 101)(16 149 47 160 102)(17 150 48 161 103)(18 151 37 162 104)(19 152 38 163 105)(20 153 39 164 106)(21 154 40 165 107)(22 155 41 166 108)(23 156 42 167 97)(24 145 43 168 98)(25 220 178 73 215)(26 221 179 74 216)(27 222 180 75 205)(28 223 169 76 206)(29 224 170 77 207)(30 225 171 78 208)(31 226 172 79 209)(32 227 173 80 210)(33 228 174 81 211)(34 217 175 82 212)(35 218 176 83 213)(36 219 177 84 214)(49 88 128 71 240)(50 89 129 72 229)(51 90 130 61 230)(52 91 131 62 231)(53 92 132 63 232)(54 93 121 64 233)(55 94 122 65 234)(56 95 123 66 235)(57 96 124 67 236)(58 85 125 68 237)(59 86 126 69 238)(60 87 127 70 239)
(1 143)(2 144)(3 133)(4 134)(5 135)(6 136)(7 137)(8 138)(9 139)(10 140)(11 141)(12 142)(13 38)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 37)(25 220)(26 221)(27 222)(28 223)(29 224)(30 225)(31 226)(32 227)(33 228)(34 217)(35 218)(36 219)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 121)(61 236)(62 237)(63 238)(64 239)(65 240)(66 229)(67 230)(68 231)(69 232)(70 233)(71 234)(72 235)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 161)(98 162)(99 163)(100 164)(101 165)(102 166)(103 167)(104 168)(105 157)(106 158)(107 159)(108 160)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)(169 206)(170 207)(171 208)(172 209)(173 210)(174 211)(175 212)(176 213)(177 214)(178 215)(179 216)(180 205)(181 201)(182 202)(183 203)(184 204)(185 193)(186 194)(187 195)(188 196)(189 197)(190 198)(191 199)(192 200)

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

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

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

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 10G ··· 10N 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 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 10 ··· 10 12 12 12 12 12 12 12 12 15 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 6 6 6 6 10 10 2 2 2 5 5 5 5 30 30 30 30 2 2 2 2 2 10 10 10 10 2 ··· 2 12 ··· 12 2 2 2 2 10 10 10 10 4 4 4 4 4 4 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 D4⋊2D5 C2×S3×D5 C2×S3×D5 D12⋊5D5 kernel C2×D12⋊5D5 D12⋊5D5 C2×S3×Dic5 C2×C15⋊D4 D5×C2×C12 C10×D12 C2×Dic30 C2×C4×D5 C2×D12 C4×D5 C2×Dic5 C2×C20 C22×D5 C30 D12 C2×C12 C22×S3 C10 C2×C4 C6 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 2 4 1 1 1 4 8 2 4 8 2 4 4 2 8

Matrix representation of C2×D125D5 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 29 0 0 0 51 40 0 0 0 0 60 0 0 0 0 60
,
 46 47 0 0 16 15 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 1 0 0 0 0 0 60 0 0 1 17
,
 1 0 0 0 24 60 0 0 0 0 44 60 0 0 44 17
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[29,51,0,0,0,40,0,0,0,0,60,0,0,0,0,60],[46,16,0,0,47,15,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,60,17],[1,24,0,0,0,60,0,0,0,0,44,44,0,0,60,17] >;

C2×D125D5 in GAP, Magma, Sage, TeX

C_2\times D_{12}\rtimes_5D_5
% in TeX

G:=Group("C2xD12:5D5");
// GroupNames label

G:=SmallGroup(480,1084);
// by ID

G=gap.SmallGroup(480,1084);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^12=c^2=d^5=e^2=1,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=b^6*c,e*d*e=d^-1>;
// generators/relations

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