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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, 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, C20, C20, D10, C2×C10, C2×C10, C4×S3, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C5×S3, D15, C30, C30, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, S3×C2×C4, S3×C2×C4, C2×D12, Q83S3, C6×Q8, C5×Dic3, C3×Dic5, C60, S3×C10, S3×C10, D30, D30, C2×C30, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C2×Q83S3, D30.C2, C5⋊D12, C3×Dic10, C6×Dic5, S3×C20, C10×Dic3, D60, C2×C60, S3×C2×C10, C22×D15, C2×C4○D20, D60⋊C2, C2×D30.C2, C2×C5⋊D12, C6×Dic10, S3×C2×C20, C2×D60, C2×D60⋊C2
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, Q83S3, S3×C23, S3×D5, C4○D20, C23×D5, C2×Q83S3, C2×S3×D5, C2×C4○D20, D60⋊C2, C22×S3×D5, C2×D60⋊C2

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

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

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

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

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

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