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G = (C2×C60).C22order 480 = 25·3·5

5th non-split extension by C2×C60 of C22 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C605C43C2, Dic3⋊C44D5, (C2×C20).11D6, (C2×C12).11D10, (C2×C60).5C22, Dic155C46C2, (C22×D5).7D6, C156(C422C2), D10⋊C4.2S3, C30.28(C4○D4), C6.48(C4○D20), C10.6(C4○D12), (C2×C30).52C23, C6.8(Q82D5), (C2×Dic5).96D6, C52(C23.8D6), (Dic3×Dic5)⋊10C2, C2.8(D205S3), C6.65(D42D5), (C2×Dic3).13D10, D10⋊Dic3.6C2, C10.65(D42S3), C2.11(C12.28D10), (C6×Dic5).30C22, C2.11(C30.C23), (C2×Dic15).54C22, (C10×Dic3).32C22, C35(C4⋊C4⋊D5), (C2×C4).35(S3×D5), (D5×C2×C6).5C22, (C5×Dic3⋊C4)⋊4C2, C22.139(C2×S3×D5), (C2×C6).64(C22×D5), (C3×D10⋊C4).2C2, (C2×C10).64(C22×S3), SmallGroup(480,438)

Series: Derived Chief Lower central Upper central

C1C2×C30 — (C2×C60).C22
C1C5C15C30C2×C30D5×C2×C6D10⋊Dic3 — (C2×C60).C22
C15C2×C30 — (C2×C60).C22
C1C22C2×C4

Generators and relations for (C2×C60).C22
 G = < a,b,c,d | a2=b60=c2=1, d2=b30, ab=ba, dcd-1=ac=ca, ad=da, cbc=ab19, dbd-1=ab41 >

Subgroups: 556 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, C6 [×3], C6, C2×C4, C2×C4 [×5], C23, D5, C10 [×3], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×3], C15, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×3], C20 [×3], D10 [×3], C2×C10, C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12, C22×C6, C3×D5, C30 [×3], C422C2, C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C6.D4 [×2], C3×C22⋊C4, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], C60, C6×D5 [×3], C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4 [×2], C5×C4⋊C4, C23.8D6, C6×Dic5, C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, C4⋊C4⋊D5, Dic3×Dic5, D10⋊Dic3 [×2], Dic155C4, C3×D10⋊C4, C5×Dic3⋊C4, C605C4, (C2×C60).C22
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4 [×3], D10 [×3], C22×S3, C422C2, C22×D5, C4○D12, D42S3 [×2], S3×D5, C4○D20, D42D5, Q82D5, C23.8D6, C2×S3×D5, C4⋊C4⋊D5, D205S3, C12.28D10, C30.C23, (C2×C60).C22

Smallest permutation representation of (C2×C60).C22
On 240 points
Generators in S240
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(121 212)(122 213)(123 214)(124 215)(125 216)(126 217)(127 218)(128 219)(129 220)(130 221)(131 222)(132 223)(133 224)(134 225)(135 226)(136 227)(137 228)(138 229)(139 230)(140 231)(141 232)(142 233)(143 234)(144 235)(145 236)(146 237)(147 238)(148 239)(149 240)(150 181)(151 182)(152 183)(153 184)(154 185)(155 186)(156 187)(157 188)(158 189)(159 190)(160 191)(161 192)(162 193)(163 194)(164 195)(165 196)(166 197)(167 198)(168 199)(169 200)(170 201)(171 202)(172 203)(173 204)(174 205)(175 206)(176 207)(177 208)(178 209)(179 210)(180 211)
(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)
(2 62)(3 39)(4 100)(5 17)(6 78)(7 55)(8 116)(9 33)(10 94)(12 72)(13 49)(14 110)(15 27)(16 88)(18 66)(19 43)(20 104)(22 82)(23 59)(24 120)(25 37)(26 98)(28 76)(29 53)(30 114)(32 92)(34 70)(35 47)(36 108)(38 86)(40 64)(42 102)(44 80)(45 57)(46 118)(48 96)(50 74)(52 112)(54 90)(56 68)(58 106)(60 84)(61 85)(65 101)(67 79)(69 117)(71 95)(75 111)(77 89)(81 105)(87 99)(91 115)(97 109)(107 119)(121 151)(122 201)(123 129)(124 239)(125 167)(126 217)(127 145)(128 195)(130 233)(131 161)(132 211)(133 139)(134 189)(135 177)(136 227)(137 155)(138 205)(140 183)(141 171)(142 221)(143 149)(144 199)(146 237)(147 165)(148 215)(150 193)(152 231)(153 159)(154 209)(156 187)(157 175)(158 225)(160 203)(162 181)(163 169)(164 219)(166 197)(168 235)(170 213)(172 191)(173 179)(174 229)(176 207)(178 185)(180 223)(182 212)(184 190)(186 228)(188 206)(192 222)(194 200)(196 238)(198 216)(202 232)(204 210)(208 226)(214 220)(218 236)(224 230)(234 240)
(1 237 31 207)(2 127 32 157)(3 199 33 229)(4 149 34 179)(5 221 35 191)(6 171 36 141)(7 183 37 213)(8 133 38 163)(9 205 39 235)(10 155 40 125)(11 227 41 197)(12 177 42 147)(13 189 43 219)(14 139 44 169)(15 211 45 181)(16 161 46 131)(17 233 47 203)(18 123 48 153)(19 195 49 225)(20 145 50 175)(21 217 51 187)(22 167 52 137)(23 239 53 209)(24 129 54 159)(25 201 55 231)(26 151 56 121)(27 223 57 193)(28 173 58 143)(29 185 59 215)(30 135 60 165)(61 164 91 134)(62 236 92 206)(63 126 93 156)(64 198 94 228)(65 148 95 178)(66 220 96 190)(67 170 97 140)(68 182 98 212)(69 132 99 162)(70 204 100 234)(71 154 101 124)(72 226 102 196)(73 176 103 146)(74 188 104 218)(75 138 105 168)(76 210 106 240)(77 160 107 130)(78 232 108 202)(79 122 109 152)(80 194 110 224)(81 144 111 174)(82 216 112 186)(83 166 113 136)(84 238 114 208)(85 128 115 158)(86 200 116 230)(87 150 117 180)(88 222 118 192)(89 172 119 142)(90 184 120 214)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I5A5B6A6B6C6D6E10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L30A···30F60A···60H
order122223444444444556666610···101212121215152020202020···2030···3060···60
size11112024661010123030602222220202···244202044444412···124···44···4

60 irreducible representations

dim1111111222222222244444444
type++++++++++++++-+-++-+-
imageC1C2C2C2C2C2C2S3D5D6D6D6C4○D4D10D10C4○D12C4○D20D42S3S3×D5D42D5Q82D5C2×S3×D5D205S3C12.28D10C30.C23
kernel(C2×C60).C22Dic3×Dic5D10⋊Dic3Dic155C4C3×D10⋊C4C5×Dic3⋊C4C605C4D10⋊C4Dic3⋊C4C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C10C6C10C2×C4C6C6C22C2C2C2
# reps1121111121116424822222444

Matrix representation of (C2×C60).C22 in GL6(𝔽61)

6000000
0600000
001000
000100
000010
000001
,
1100000
28500000
0014000
00314800
00005434
00005659
,
100000
58600000
001000
000100
0000171
00001744
,
16310000
35450000
0053500
0024800
0000500
0000050

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,28,0,0,0,0,0,50,0,0,0,0,0,0,14,31,0,0,0,0,0,48,0,0,0,0,0,0,54,56,0,0,0,0,34,59],[1,58,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,17,0,0,0,0,1,44],[16,35,0,0,0,0,31,45,0,0,0,0,0,0,53,24,0,0,0,0,5,8,0,0,0,0,0,0,50,0,0,0,0,0,0,50] >;

(C2×C60).C22 in GAP, Magma, Sage, TeX

(C_2\times C_{60}).C_2^2
% in TeX

G:=Group("(C2xC60).C2^2");
// GroupNames label

G:=SmallGroup(480,438);
// by ID

G=gap.SmallGroup(480,438);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,590,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^60=c^2=1,d^2=b^30,a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,c*b*c=a*b^19,d*b*d^-1=a*b^41>;
// generators/relations

׿
×
𝔽