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## G = C2×D60⋊C2order 480 = 25·3·5

### Direct product of C2 and D60⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D60⋊C2
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — D30.C2 — C2×D30.C2 — C2×D60⋊C2
 Lower central C15 — C30 — C2×D60⋊C2
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D60⋊C2
G = < a,b,c,d | a2=b60=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b41, dcd=b10c >

Subgroups: 1660 in 328 conjugacy classes, 116 normal (26 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 [×4], C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], C12 [×4], D6 [×2], D6 [×10], C2×C6, C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×2], C20 [×2], D10 [×8], C2×C10, C2×C10 [×4], C4×S3 [×4], C4×S3 [×8], D12 [×12], C2×Dic3, C2×C12, C2×C12 [×2], C3×Q8 [×4], C22×S3, C22×S3 [×2], C5×S3 [×2], D15 [×4], C30, C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C2×C20 [×5], C22×D5 [×2], C22×C10, S3×C2×C4, S3×C2×C4 [×2], C2×D12 [×3], Q83S3 [×8], C6×Q8, C5×Dic3 [×2], C3×Dic5 [×4], C60 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×4], D30 [×4], C2×C30, C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, C2×Q83S3, D30.C2 [×8], C5⋊D12 [×8], C3×Dic10 [×4], C6×Dic5 [×2], S3×C20 [×4], C10×Dic3, D60 [×4], C2×C60, S3×C2×C10, C22×D15 [×2], C2×C4○D20, D60⋊C2 [×8], C2×D30.C2 [×2], C2×C5⋊D12 [×2], C6×Dic10, S3×C2×C20, C2×D60, C2×D60⋊C2
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, C4○D20 [×2], C23×D5, C2×Q83S3, C2×S3×D5 [×3], C2×C4○D20, D60⋊C2 [×2], C22×S3×D5, C2×D60⋊C2

Smallest permutation representation of C2×D60⋊C2
On 240 points
Generators in S240
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(121 233)(122 234)(123 235)(124 236)(125 237)(126 238)(127 239)(128 240)(129 181)(130 182)(131 183)(132 184)(133 185)(134 186)(135 187)(136 188)(137 189)(138 190)(139 191)(140 192)(141 193)(142 194)(143 195)(144 196)(145 197)(146 198)(147 199)(148 200)(149 201)(150 202)(151 203)(152 204)(153 205)(154 206)(155 207)(156 208)(157 209)(158 210)(159 211)(160 212)(161 213)(162 214)(163 215)(164 216)(165 217)(166 218)(167 219)(168 220)(169 221)(170 222)(171 223)(172 224)(173 225)(174 226)(175 227)(176 228)(177 229)(178 230)(179 231)(180 232)
(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 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 120)(15 119)(16 118)(17 117)(18 116)(19 115)(20 114)(21 113)(22 112)(23 111)(24 110)(25 109)(26 108)(27 107)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)(49 85)(50 84)(51 83)(52 82)(53 81)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)(121 208)(122 207)(123 206)(124 205)(125 204)(126 203)(127 202)(128 201)(129 200)(130 199)(131 198)(132 197)(133 196)(134 195)(135 194)(136 193)(137 192)(138 191)(139 190)(140 189)(141 188)(142 187)(143 186)(144 185)(145 184)(146 183)(147 182)(148 181)(149 240)(150 239)(151 238)(152 237)(153 236)(154 235)(155 234)(156 233)(157 232)(158 231)(159 230)(160 229)(161 228)(162 227)(163 226)(164 225)(165 224)(166 223)(167 222)(168 221)(169 220)(170 219)(171 218)(172 217)(173 216)(174 215)(175 214)(176 213)(177 212)(178 211)(179 210)(180 209)
(1 124)(2 165)(3 146)(4 127)(5 168)(6 149)(7 130)(8 171)(9 152)(10 133)(11 174)(12 155)(13 136)(14 177)(15 158)(16 139)(17 180)(18 161)(19 142)(20 123)(21 164)(22 145)(23 126)(24 167)(25 148)(26 129)(27 170)(28 151)(29 132)(30 173)(31 154)(32 135)(33 176)(34 157)(35 138)(36 179)(37 160)(38 141)(39 122)(40 163)(41 144)(42 125)(43 166)(44 147)(45 128)(46 169)(47 150)(48 131)(49 172)(50 153)(51 134)(52 175)(53 156)(54 137)(55 178)(56 159)(57 140)(58 121)(59 162)(60 143)(61 183)(62 224)(63 205)(64 186)(65 227)(66 208)(67 189)(68 230)(69 211)(70 192)(71 233)(72 214)(73 195)(74 236)(75 217)(76 198)(77 239)(78 220)(79 201)(80 182)(81 223)(82 204)(83 185)(84 226)(85 207)(86 188)(87 229)(88 210)(89 191)(90 232)(91 213)(92 194)(93 235)(94 216)(95 197)(96 238)(97 219)(98 200)(99 181)(100 222)(101 203)(102 184)(103 225)(104 206)(105 187)(106 228)(107 209)(108 190)(109 231)(110 212)(111 193)(112 234)(113 215)(114 196)(115 237)(116 218)(117 199)(118 240)(119 221)(120 202)

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

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

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

78 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 ··· 20H 20I ··· 20P 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 30 30 30 30 2 2 2 3 3 3 3 10 10 10 10 2 2 2 2 2 2 ··· 2 6 ··· 6 4 4 20 20 20 20 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

78 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 C4○D4 D10 D10 D10 D10 C4○D20 Q8⋊3S3 S3×D5 C2×S3×D5 C2×S3×D5 D60⋊C2 kernel C2×D60⋊C2 D60⋊C2 C2×D30.C2 C2×C5⋊D12 C6×Dic10 S3×C2×C20 C2×D60 C2×Dic10 S3×C2×C4 Dic10 C2×Dic5 C2×C20 C30 C4×S3 C2×Dic3 C2×C12 C22×S3 C6 C10 C2×C4 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 2 4 2 1 4 8 2 2 2 16 2 2 4 2 8

Matrix representation of C2×D60⋊C2 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 27 34 0 0 27 25 0 0 0 0 1 60 0 0 1 0
,
 27 34 0 0 36 34 0 0 0 0 60 0 0 0 60 1
,
 31 17 0 0 44 30 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[27,27,0,0,34,25,0,0,0,0,1,1,0,0,60,0],[27,36,0,0,34,34,0,0,0,0,60,60,0,0,0,1],[31,44,0,0,17,30,0,0,0,0,0,1,0,0,1,0] >;

C2×D60⋊C2 in GAP, Magma, Sage, TeX

C_2\times D_{60}\rtimes C_2
% in TeX

G:=Group("C2xD60:C2");
// GroupNames label

G:=SmallGroup(480,1081);
// by ID

G=gap.SmallGroup(480,1081);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,120,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^60=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^41,d*c*d=b^10*c>;
// generators/relations

׿
×
𝔽