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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 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×4], C24 [×2], Dic6 [×3], D12 [×2], D12, C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C30, C30 [×2], C2×SD16, C52C8 [×2], Dic10 [×3], C2×Dic5, C2×C20, C5×D4 [×3], C22×C10, C24⋊C2 [×4], C2×C24, C2×Dic6, C2×D12, Dic15 [×2], C60 [×2], S3×C10 [×4], C2×C30, C2×C52C8, D4.D5 [×4], C2×Dic10, D4×C10, C2×C24⋊C2, C3×C52C8 [×2], C5×D12 [×2], C5×D12, Dic30 [×2], Dic30, C2×Dic15, C2×C60, S3×C2×C10, C2×D4.D5, D12.D5 [×4], C6×C52C8, C10×D12, C2×Dic30, C2×D12.D5
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], SD16 [×2], C2×D4, D10 [×3], D12 [×2], C22×S3, C2×SD16, C5⋊D4 [×2], C22×D5, C24⋊C2 [×2], C2×D12, S3×D5, D4.D5 [×2], C2×C5⋊D4, C2×C24⋊C2, C5⋊D12 [×2], C2×S3×D5, C2×D4.D5, D12.D5 [×2], C2×C5⋊D12, C2×D12.D5

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

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

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

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

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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