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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 — C30 — C2×D12⋊D5
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — S3×Dic5 — C2×S3×Dic5 — C2×D12⋊D5
 Lower central C15 — C30 — 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=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ece=b10c, ede=d-1 >

Subgroups: 1532 in 328 conjugacy classes, 116 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×6], C6, C6 [×2], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×2], C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], C12 [×4], D6 [×4], D6 [×8], C2×C6, C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×8], C4×S3 [×12], D12 [×4], D12 [×8], C2×Dic3, C2×C12, C2×C12 [×2], C3×Q8 [×4], C22×S3 [×2], C22×S3, C5×S3 [×4], D15 [×2], C30, C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C2×Dic5 [×2], C2×Dic5 [×9], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], S3×C2×C4 [×3], C2×D12, C2×D12 [×2], Q83S3 [×8], C6×Q8, C3×Dic5 [×4], Dic15 [×2], C60 [×2], S3×C10 [×4], S3×C10 [×4], D30 [×2], D30 [×2], C2×C30, C2×Dic10, C2×C4×D5, D42D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C2×Q83S3, S3×Dic5 [×8], C5⋊D12 [×8], C3×Dic10 [×4], C6×Dic5 [×2], C5×D12 [×4], C4×D15 [×4], C2×Dic15, C2×C60, S3×C2×C10 [×2], C22×D15, C2×D42D5, D12⋊D5 [×8], C2×S3×Dic5 [×2], C2×C5⋊D12 [×2], C6×Dic10, C10×D12, C2×C4×D15, C2×D12⋊D5
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], Q83S3 [×2], S3×C23, S3×D5, D42D5 [×2], C23×D5, C2×Q83S3, C2×S3×D5 [×3], C2×D42D5, D12⋊D5 [×2], C22×S3×D5, C2×D12⋊D5

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

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

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

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

66 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 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 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 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 6 6 30 30 2 2 2 10 10 10 10 15 15 15 15 2 2 2 2 2 2 ··· 2 12 ··· 12 4 4 20 20 20 20 4 4 4 4 4 4 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 D10 Q8⋊3S3 S3×D5 D4⋊2D5 C2×S3×D5 C2×S3×D5 D12⋊D5 kernel C2×D12⋊D5 D12⋊D5 C2×S3×Dic5 C2×C5⋊D12 C6×Dic10 C10×D12 C2×C4×D15 C2×Dic10 C2×D12 Dic10 C2×Dic5 C2×C20 C30 D12 C2×C12 C22×S3 C10 C2×C4 C6 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 2 4 2 1 4 8 2 4 2 2 4 4 2 8

Matrix representation of C2×D12⋊D5 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 60 60 0 0 0 0 0 0 16 52 0 0 0 0 15 45
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 60 0 0 0 0 0 0 0 45 17 0 0 0 0 46 16
,
 18 60 0 0 0 0 19 60 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 1
,
 28 53 0 0 0 0 14 33 0 0 0 0 0 0 60 0 0 0 0 0 1 1 0 0 0 0 0 0 54 23 0 0 0 0 43 7

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,1,60,0,0,0,0,0,0,16,15,0,0,0,0,52,45],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,45,46,0,0,0,0,17,16],[18,19,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,14,0,0,0,0,53,33,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,54,43,0,0,0,0,23,7] >;

C2×D12⋊D5 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,1079);
// by ID

G=gap.SmallGroup(480,1079);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,675,346,80,1356,18822]);
// 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,e*b*e=b^5,c*d=d*c,e*c*e=b^10*c,e*d*e=d^-1>;
// generators/relations

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