direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C30.D4, C30⋊1SD16, C60.82D4, D20.36D6, Dic6⋊19D10, C60.153C23, C6⋊2(Q8⋊D5), C15⋊4(C2×SD16), (C2×Dic6)⋊7D5, (C2×D20).8S3, (C6×D20).7C2, (C2×C30).49D4, C30.81(C2×D4), (C2×C20).92D6, C10⋊2(D4.S3), (C10×Dic6)⋊9C2, (C2×C12).92D10, C4.6(C15⋊D4), C15⋊3C8⋊38C22, C20.26(C3⋊D4), C12.29(C5⋊D4), C20.92(C22×S3), C12.90(C22×D5), (C2×C60).188C22, (C5×Dic6)⋊25C22, (C3×D20).42C22, C22.20(C15⋊D4), C3⋊3(C2×Q8⋊D5), C5⋊3(C2×D4.S3), C4.126(C2×S3×D5), (C2×C15⋊3C8)⋊16C2, C6.75(C2×C5⋊D4), C2.9(C2×C15⋊D4), (C2×C4).198(S3×D5), C10.76(C2×C3⋊D4), (C2×C6).53(C5⋊D4), (C2×C10).53(C3⋊D4), SmallGroup(480,382)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C30.D4
G = < a,b,c,d | a2=b20=c6=1, d2=b15, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b9, dcd-1=b5c-1 >
Subgroups: 636 in 136 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C2×C8, SD16, C2×D4, C2×Q8, C20, C20, D10, C2×C10, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C3×D4, C22×C6, C3×D5, C30, C30, C2×SD16, C5⋊2C8, D20, D20, C2×C20, C2×C20, C5×Q8, C22×D5, C2×C3⋊C8, D4.S3, C2×Dic6, C6×D4, C5×Dic3, C60, C6×D5, C2×C30, C2×C5⋊2C8, Q8⋊D5, C2×D20, Q8×C10, C2×D4.S3, C15⋊3C8, C3×D20, C3×D20, C5×Dic6, C5×Dic6, C10×Dic3, C2×C60, D5×C2×C6, C2×Q8⋊D5, C30.D4, C2×C15⋊3C8, C6×D20, C10×Dic6, C2×C30.D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C3⋊D4, C22×S3, C2×SD16, C5⋊D4, C22×D5, D4.S3, C2×C3⋊D4, S3×D5, Q8⋊D5, C2×C5⋊D4, C2×D4.S3, C15⋊D4, C2×S3×D5, C2×Q8⋊D5, C30.D4, C2×C15⋊D4, C2×C30.D4
►On 240 points
Generators in S240
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 119)(22 120)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 159)(42 160)(43 141)(44 142)(45 143)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(81 230)(82 231)(83 232)(84 233)(85 234)(86 235)(87 236)(88 237)(89 238)(90 239)(91 240)(92 221)(93 222)(94 223)(95 224)(96 225)(97 226)(98 227)(99 228)(100 229)(121 197)(122 198)(123 199)(124 200)(125 181)(126 182)(127 183)(128 184)(129 185)(130 186)(131 187)(132 188)(133 189)(134 190)(135 191)(136 192)(137 193)(138 194)(139 195)(140 196)(161 208)(162 209)(163 210)(164 211)(165 212)(166 213)(167 214)(168 215)(169 216)(170 217)(171 218)(172 219)(173 220)(174 201)(175 202)(176 203)(177 204)(178 205)(179 206)(180 207)
(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 44 232 61 142 83)(2 43 233 80 143 82)(3 42 234 79 144 81)(4 41 235 78 145 100)(5 60 236 77 146 99)(6 59 237 76 147 98)(7 58 238 75 148 97)(8 57 239 74 149 96)(9 56 240 73 150 95)(10 55 221 72 151 94)(11 54 222 71 152 93)(12 53 223 70 153 92)(13 52 224 69 154 91)(14 51 225 68 155 90)(15 50 226 67 156 89)(16 49 227 66 157 88)(17 48 228 65 158 87)(18 47 229 64 159 86)(19 46 230 63 160 85)(20 45 231 62 141 84)(21 179 125 108 217 190)(22 178 126 107 218 189)(23 177 127 106 219 188)(24 176 128 105 220 187)(25 175 129 104 201 186)(26 174 130 103 202 185)(27 173 131 102 203 184)(28 172 132 101 204 183)(29 171 133 120 205 182)(30 170 134 119 206 181)(31 169 135 118 207 200)(32 168 136 117 208 199)(33 167 137 116 209 198)(34 166 138 115 210 197)(35 165 139 114 211 196)(36 164 140 113 212 195)(37 163 121 112 213 194)(38 162 122 111 214 193)(39 161 123 110 215 192)(40 180 124 109 216 191)
(1 214 16 209 11 204 6 219)(2 203 17 218 12 213 7 208)(3 212 18 207 13 202 8 217)(4 201 19 216 14 211 9 206)(5 210 20 205 15 220 10 215)(21 144 36 159 31 154 26 149)(22 153 37 148 32 143 27 158)(23 142 38 157 33 152 28 147)(24 151 39 146 34 141 29 156)(25 160 40 155 35 150 30 145)(41 109 56 104 51 119 46 114)(42 118 57 113 52 108 47 103)(43 107 58 102 53 117 48 112)(44 116 59 111 54 106 49 101)(45 105 60 120 55 115 50 110)(61 167 76 162 71 177 66 172)(62 176 77 171 72 166 67 161)(63 165 78 180 73 175 68 170)(64 174 79 169 74 164 69 179)(65 163 80 178 75 173 70 168)(81 200 96 195 91 190 86 185)(82 189 97 184 92 199 87 194)(83 198 98 193 93 188 88 183)(84 187 99 182 94 197 89 192)(85 196 100 191 95 186 90 181)(121 238 136 233 131 228 126 223)(122 227 137 222 132 237 127 232)(123 236 138 231 133 226 128 221)(124 225 139 240 134 235 129 230)(125 234 140 229 135 224 130 239)
G:=sub<Sym(240)| (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,119)(22,120)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,159)(42,160)(43,141)(44,142)(45,143)(46,144)(47,145)(48,146)(49,147)(50,148)(51,149)(52,150)(53,151)(54,152)(55,153)(56,154)(57,155)(58,156)(59,157)(60,158)(81,230)(82,231)(83,232)(84,233)(85,234)(86,235)(87,236)(88,237)(89,238)(90,239)(91,240)(92,221)(93,222)(94,223)(95,224)(96,225)(97,226)(98,227)(99,228)(100,229)(121,197)(122,198)(123,199)(124,200)(125,181)(126,182)(127,183)(128,184)(129,185)(130,186)(131,187)(132,188)(133,189)(134,190)(135,191)(136,192)(137,193)(138,194)(139,195)(140,196)(161,208)(162,209)(163,210)(164,211)(165,212)(166,213)(167,214)(168,215)(169,216)(170,217)(171,218)(172,219)(173,220)(174,201)(175,202)(176,203)(177,204)(178,205)(179,206)(180,207), (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), (1,44,232,61,142,83)(2,43,233,80,143,82)(3,42,234,79,144,81)(4,41,235,78,145,100)(5,60,236,77,146,99)(6,59,237,76,147,98)(7,58,238,75,148,97)(8,57,239,74,149,96)(9,56,240,73,150,95)(10,55,221,72,151,94)(11,54,222,71,152,93)(12,53,223,70,153,92)(13,52,224,69,154,91)(14,51,225,68,155,90)(15,50,226,67,156,89)(16,49,227,66,157,88)(17,48,228,65,158,87)(18,47,229,64,159,86)(19,46,230,63,160,85)(20,45,231,62,141,84)(21,179,125,108,217,190)(22,178,126,107,218,189)(23,177,127,106,219,188)(24,176,128,105,220,187)(25,175,129,104,201,186)(26,174,130,103,202,185)(27,173,131,102,203,184)(28,172,132,101,204,183)(29,171,133,120,205,182)(30,170,134,119,206,181)(31,169,135,118,207,200)(32,168,136,117,208,199)(33,167,137,116,209,198)(34,166,138,115,210,197)(35,165,139,114,211,196)(36,164,140,113,212,195)(37,163,121,112,213,194)(38,162,122,111,214,193)(39,161,123,110,215,192)(40,180,124,109,216,191), (1,214,16,209,11,204,6,219)(2,203,17,218,12,213,7,208)(3,212,18,207,13,202,8,217)(4,201,19,216,14,211,9,206)(5,210,20,205,15,220,10,215)(21,144,36,159,31,154,26,149)(22,153,37,148,32,143,27,158)(23,142,38,157,33,152,28,147)(24,151,39,146,34,141,29,156)(25,160,40,155,35,150,30,145)(41,109,56,104,51,119,46,114)(42,118,57,113,52,108,47,103)(43,107,58,102,53,117,48,112)(44,116,59,111,54,106,49,101)(45,105,60,120,55,115,50,110)(61,167,76,162,71,177,66,172)(62,176,77,171,72,166,67,161)(63,165,78,180,73,175,68,170)(64,174,79,169,74,164,69,179)(65,163,80,178,75,173,70,168)(81,200,96,195,91,190,86,185)(82,189,97,184,92,199,87,194)(83,198,98,193,93,188,88,183)(84,187,99,182,94,197,89,192)(85,196,100,191,95,186,90,181)(121,238,136,233,131,228,126,223)(122,227,137,222,132,237,127,232)(123,236,138,231,133,226,128,221)(124,225,139,240,134,235,129,230)(125,234,140,229,135,224,130,239)>;
G:=Group( (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,119)(22,120)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,159)(42,160)(43,141)(44,142)(45,143)(46,144)(47,145)(48,146)(49,147)(50,148)(51,149)(52,150)(53,151)(54,152)(55,153)(56,154)(57,155)(58,156)(59,157)(60,158)(81,230)(82,231)(83,232)(84,233)(85,234)(86,235)(87,236)(88,237)(89,238)(90,239)(91,240)(92,221)(93,222)(94,223)(95,224)(96,225)(97,226)(98,227)(99,228)(100,229)(121,197)(122,198)(123,199)(124,200)(125,181)(126,182)(127,183)(128,184)(129,185)(130,186)(131,187)(132,188)(133,189)(134,190)(135,191)(136,192)(137,193)(138,194)(139,195)(140,196)(161,208)(162,209)(163,210)(164,211)(165,212)(166,213)(167,214)(168,215)(169,216)(170,217)(171,218)(172,219)(173,220)(174,201)(175,202)(176,203)(177,204)(178,205)(179,206)(180,207), (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), (1,44,232,61,142,83)(2,43,233,80,143,82)(3,42,234,79,144,81)(4,41,235,78,145,100)(5,60,236,77,146,99)(6,59,237,76,147,98)(7,58,238,75,148,97)(8,57,239,74,149,96)(9,56,240,73,150,95)(10,55,221,72,151,94)(11,54,222,71,152,93)(12,53,223,70,153,92)(13,52,224,69,154,91)(14,51,225,68,155,90)(15,50,226,67,156,89)(16,49,227,66,157,88)(17,48,228,65,158,87)(18,47,229,64,159,86)(19,46,230,63,160,85)(20,45,231,62,141,84)(21,179,125,108,217,190)(22,178,126,107,218,189)(23,177,127,106,219,188)(24,176,128,105,220,187)(25,175,129,104,201,186)(26,174,130,103,202,185)(27,173,131,102,203,184)(28,172,132,101,204,183)(29,171,133,120,205,182)(30,170,134,119,206,181)(31,169,135,118,207,200)(32,168,136,117,208,199)(33,167,137,116,209,198)(34,166,138,115,210,197)(35,165,139,114,211,196)(36,164,140,113,212,195)(37,163,121,112,213,194)(38,162,122,111,214,193)(39,161,123,110,215,192)(40,180,124,109,216,191), (1,214,16,209,11,204,6,219)(2,203,17,218,12,213,7,208)(3,212,18,207,13,202,8,217)(4,201,19,216,14,211,9,206)(5,210,20,205,15,220,10,215)(21,144,36,159,31,154,26,149)(22,153,37,148,32,143,27,158)(23,142,38,157,33,152,28,147)(24,151,39,146,34,141,29,156)(25,160,40,155,35,150,30,145)(41,109,56,104,51,119,46,114)(42,118,57,113,52,108,47,103)(43,107,58,102,53,117,48,112)(44,116,59,111,54,106,49,101)(45,105,60,120,55,115,50,110)(61,167,76,162,71,177,66,172)(62,176,77,171,72,166,67,161)(63,165,78,180,73,175,68,170)(64,174,79,169,74,164,69,179)(65,163,80,178,75,173,70,168)(81,200,96,195,91,190,86,185)(82,189,97,184,92,199,87,194)(83,198,98,193,93,188,88,183)(84,187,99,182,94,197,89,192)(85,196,100,191,95,186,90,181)(121,238,136,233,131,228,126,223)(122,227,137,222,132,237,127,232)(123,236,138,231,133,226,128,221)(124,225,139,240,134,235,129,230)(125,234,140,229,135,224,130,239) );
G=PermutationGroup([[(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,119),(22,120),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,107),(30,108),(31,109),(32,110),(33,111),(34,112),(35,113),(36,114),(37,115),(38,116),(39,117),(40,118),(41,159),(42,160),(43,141),(44,142),(45,143),(46,144),(47,145),(48,146),(49,147),(50,148),(51,149),(52,150),(53,151),(54,152),(55,153),(56,154),(57,155),(58,156),(59,157),(60,158),(81,230),(82,231),(83,232),(84,233),(85,234),(86,235),(87,236),(88,237),(89,238),(90,239),(91,240),(92,221),(93,222),(94,223),(95,224),(96,225),(97,226),(98,227),(99,228),(100,229),(121,197),(122,198),(123,199),(124,200),(125,181),(126,182),(127,183),(128,184),(129,185),(130,186),(131,187),(132,188),(133,189),(134,190),(135,191),(136,192),(137,193),(138,194),(139,195),(140,196),(161,208),(162,209),(163,210),(164,211),(165,212),(166,213),(167,214),(168,215),(169,216),(170,217),(171,218),(172,219),(173,220),(174,201),(175,202),(176,203),(177,204),(178,205),(179,206),(180,207)], [(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)], [(1,44,232,61,142,83),(2,43,233,80,143,82),(3,42,234,79,144,81),(4,41,235,78,145,100),(5,60,236,77,146,99),(6,59,237,76,147,98),(7,58,238,75,148,97),(8,57,239,74,149,96),(9,56,240,73,150,95),(10,55,221,72,151,94),(11,54,222,71,152,93),(12,53,223,70,153,92),(13,52,224,69,154,91),(14,51,225,68,155,90),(15,50,226,67,156,89),(16,49,227,66,157,88),(17,48,228,65,158,87),(18,47,229,64,159,86),(19,46,230,63,160,85),(20,45,231,62,141,84),(21,179,125,108,217,190),(22,178,126,107,218,189),(23,177,127,106,219,188),(24,176,128,105,220,187),(25,175,129,104,201,186),(26,174,130,103,202,185),(27,173,131,102,203,184),(28,172,132,101,204,183),(29,171,133,120,205,182),(30,170,134,119,206,181),(31,169,135,118,207,200),(32,168,136,117,208,199),(33,167,137,116,209,198),(34,166,138,115,210,197),(35,165,139,114,211,196),(36,164,140,113,212,195),(37,163,121,112,213,194),(38,162,122,111,214,193),(39,161,123,110,215,192),(40,180,124,109,216,191)], [(1,214,16,209,11,204,6,219),(2,203,17,218,12,213,7,208),(3,212,18,207,13,202,8,217),(4,201,19,216,14,211,9,206),(5,210,20,205,15,220,10,215),(21,144,36,159,31,154,26,149),(22,153,37,148,32,143,27,158),(23,142,38,157,33,152,28,147),(24,151,39,146,34,141,29,156),(25,160,40,155,35,150,30,145),(41,109,56,104,51,119,46,114),(42,118,57,113,52,108,47,103),(43,107,58,102,53,117,48,112),(44,116,59,111,54,106,49,101),(45,105,60,120,55,115,50,110),(61,167,76,162,71,177,66,172),(62,176,77,171,72,166,67,161),(63,165,78,180,73,175,68,170),(64,174,79,169,74,164,69,179),(65,163,80,178,75,173,70,168),(81,200,96,195,91,190,86,185),(82,189,97,184,92,199,87,194),(83,198,98,193,93,188,88,183),(84,187,99,182,94,197,89,192),(85,196,100,191,95,186,90,181),(121,238,136,233,131,228,126,223),(122,227,137,222,132,237,127,232),(123,236,138,231,133,226,128,221),(124,225,139,240,134,235,129,230),(125,234,140,229,135,224,130,239)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | SD16 | D10 | D10 | C3⋊D4 | C3⋊D4 | C5⋊D4 | C5⋊D4 | D4.S3 | S3×D5 | Q8⋊D5 | C15⋊D4 | C2×S3×D5 | C15⋊D4 | C30.D4 |
| kernel | C2×C30.D4 | C30.D4 | C2×C15⋊3C8 | C6×D20 | C10×Dic6 | C2×D20 | C60 | C2×C30 | C2×Dic6 | D20 | C2×C20 | C30 | Dic6 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C10 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of C2×C30.D4 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 190 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 239 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 226 | 0 | 0 | 0 | 0 | 0 |
| 202 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 51 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 31 | 180 | 0 | 0 | 0 | 0 |
| 158 | 210 | 0 | 0 | 0 | 0 |
| 0 | 0 | 76 | 49 | 0 | 0 |
| 0 | 0 | 69 | 165 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 203 |
| 0 | 0 | 0 | 0 | 19 | 38 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,240,190,0,0,0,0,0,0,240,1,0,0,0,0,239,1],[226,202,0,0,0,0,0,225,0,0,0,0,0,0,1,51,0,0,0,0,0,240,0,0,0,0,0,0,240,1,0,0,0,0,0,1],[31,158,0,0,0,0,180,210,0,0,0,0,0,0,76,69,0,0,0,0,49,165,0,0,0,0,0,0,0,19,0,0,0,0,203,38] >;
C2×C30.D4 in GAP, Magma, Sage, TeX
C_2\times C_{30}.D_4 % in TeX
G:=Group("C2xC30.D4"); // GroupNames label
G:=SmallGroup(480,382);
// by ID
G=gap.SmallGroup(480,382);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^6=1,d^2=b^15,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^9,d*c*d^-1=b^5*c^-1>;
// generators/relations