(C2xC60).C2^2 - GroupNames

G = (C₂×C₆₀).C₂² order 480 = 2⁵·3·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₆₀⋊₅C₄⋊₃C₂, Dic₃⋊C₄⋊₄D₅, (C₂×C₂₀).₁₁D₆, (C₂×C₁₂).₁₁D₁₀, (C₂×C₆₀).₅C₂², Dic₁₅⋊₅C₄⋊₆C₂, (C₂²×D₅).₇D₆, C₁₅⋊₆(C₄²⋊₂C₂), D₁₀⋊C₄.₂S₃, C₃₀.₂₈(C₄○D₄), C₆.₄₈(C₄○D₂₀), C₁₀.₆(C₄○D₁₂), (C₂×C₃₀).₅₂C₂³, C₆.₈(Q₈⋊₂D₅), (C₂×Dic₅).₉₆D₆, C₅⋊₂(C₂³.₈D₆), (Dic₃×Dic₅)⋊₁₀C₂, C₂.₈(D₂₀⋊₅S₃), C₆.₆₅(D₄⋊₂D₅), (C₂×Dic₃).₁₃D₁₀, D₁₀⋊Dic₃.₆C₂, C₁₀.₆₅(D₄⋊₂S₃), C₂.₁₁(C₁₂.₂₈D₁₀), (C₆×Dic₅).₃₀C₂², C₂.₁₁(C₃₀.C₂³), (C₂×Dic₁₅).₅₄C₂², (C₁₀×Dic₃).₃₂C₂², C₃⋊₅(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

Derived series

C₁ — C₂×C₃₀ — (C₂×C₆₀).C₂²

Chief series

C₁ — C₅ — C₁₅ — C₃₀ — C₂×C₃₀ — D₅×C₂×C₆ — D₁₀⋊Dic₃ — (C₂×C₆₀).C₂²

Lower central

C₁₅ — C₂×C₃₀ — (C₂×C₆₀).C₂²

Upper central

C₁ — C₂² — C₂×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, C₂, C₃, C₄, C₂², C₂², C₅, C₆, C₆, C₂×C₄, C₂×C₄, C₂³, D₅, C₁₀, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₁₅, C₄², C₂²⋊C₄, C₄⋊C₄, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₂²×C₆, C₃×D₅, C₃₀, C₄²⋊₂C₂, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₂²×D₅, C₄×Dic₃, Dic₃⋊C₄, Dic₃⋊C₄, C₄⋊Dic₃, C₆.D₄, C₃×C₂²⋊C₄, C₅×Dic₃, C₃×Dic₅, Dic₁₅, C₆₀, C₆×D₅, C₂×C₃₀, C₄×Dic₅, C₁₀.D₄, C₄⋊Dic₅, D₁₀⋊C₄, D₁₀⋊C₄, C₅×C₄⋊C₄, C₂³.₈D₆, C₆×Dic₅, C₁₀×Dic₃, C₂×Dic₁₅, C₂×C₆₀, D₅×C₂×C₆, C₄⋊C₄⋊D₅, Dic₃×Dic₅, D₁₀⋊Dic₃, Dic₁₅⋊₅C₄, C₃×D₁₀⋊C₄, C₅×Dic₃⋊C₄, C₆₀⋊₅C₄, (C₂×C₆₀).C₂²
Quotients: C₁, C₂, C₂², S₃, C₂³, D₅, D₆, C₄○D₄, D₁₀, C₂²×S₃, C₄²⋊₂C₂, C₂²×D₅, C₄○D₁₂, D₄⋊₂S₃, S₃×D₅, C₄○D₂₀, D₄⋊₂D₅, Q₈⋊₂D₅, C₂³.₈D₆, C₂×S₃×D₅, C₄⋊C₄⋊D₅, D₂₀⋊₅S₃, C₁₂.₂₈D₁₀, C₃₀.C₂³, (C₂×C₆₀).C₂²

Smallest permutation representation of (C₂×C₆₀).C₂²
►On 240 points

Generators in S₂₄₀

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

(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)

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

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

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

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

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

60 conjugacy classes

class	1	2A	2B	2C	2D	3	4A	4B	4C	4D	4E	4F	4G	4H	4I	5A	5B	6A	6B	6C	6D	6E	10A	···	10F	12A	12B	12C	12D	15A	15B	20A	20B	20C	20D	20E	···	20L	30A	···	30F	60A	···	60H
order	1	2	2	2	2	3	4	4	4	4	4	4	4	4	4	5	5	6	6	6	6	6	10	···	10	12	12	12	12	15	15	20	20	20	20	20	···	20	30	···	30	60	···	60
size	1	1	1	1	20	2	4	6	6	10	10	12	30	30	60	2	2	2	2	2	20	20	2	···	2	4	4	20	20	4	4	4	4	4	4	12	···	12	4	···	4	4	···	4

60 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+		+	+			-	+	-	+	+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	S₃	D₅	D₆	D₆	D₆	C₄○D₄	D₁₀	D₁₀	C₄○D₁₂	C₄○D₂₀	D₄⋊₂S₃	S₃×D₅	D₄⋊₂D₅	Q₈⋊₂D₅	C₂×S₃×D₅	D₂₀⋊₅S₃	C₁₂.₂₈D₁₀	C₃₀.C₂³
kernel	(C₂×C₆₀).C₂²	Dic₃×Dic₅	D₁₀⋊Dic₃	Dic₁₅⋊₅C₄	C₃×D₁₀⋊C₄	C₅×Dic₃⋊C₄	C₆₀⋊₅C₄	D₁₀⋊C₄	Dic₃⋊C₄	C₂×Dic₅	C₂×C₂₀	C₂²×D₅	C₃₀	C₂×Dic₃	C₂×C₁₂	C₁₀	C₆	C₁₀	C₂×C₄	C₆	C₆	C₂²	C₂	C₂	C₂
# reps	1	1	2	1	1	1	1	1	2	1	1	1	6	4	2	4	8	2	2	2	2	2	4	4	4

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

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	0	0	0	0	0
28	50	0	0	0	0
0	0	14	0	0	0
0	0	31	48	0	0
0	0	0	0	54	34
0	0	0	0	56	59

1	0	0	0	0	0
58	60	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	17	1
0	0	0	0	17	44

16	31	0	0	0	0
35	45	0	0	0	0
0	0	53	5	0	0
0	0	24	8	0	0
0	0	0	0	50	0
0	0	0	0	0	50

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

G = (C2×C60).C22 order 480 = 25·3·5

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

G = (C₂×C₆₀).C₂² order 480 = 2⁵·3·5

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