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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, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C5×S3, D15, C30, C30, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, S3×C2×C4, C2×D12, C2×D12, Q83S3, C6×Q8, C3×Dic5, Dic15, C60, S3×C10, S3×C10, D30, D30, C2×C30, C2×Dic10, C2×C4×D5, D42D5, C22×Dic5, C2×C5⋊D4, D4×C10, C2×Q83S3, S3×Dic5, C5⋊D12, C3×Dic10, C6×Dic5, C5×D12, C4×D15, C2×Dic15, C2×C60, S3×C2×C10, C22×D15, C2×D42D5, D12⋊D5, C2×S3×Dic5, C2×C5⋊D12, C6×Dic10, C10×D12, C2×C4×D15, C2×D12⋊D5
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, Q83S3, S3×C23, S3×D5, D42D5, C23×D5, C2×Q83S3, C2×S3×D5, C2×D42D5, D12⋊D5, C22×S3×D5, C2×D12⋊D5

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

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

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

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

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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