direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C60.C4, C30⋊3(C2×C8), (C6×D5)⋊5C8, C6⋊2(D5⋊C8), D10⋊3(C3⋊C8), C15⋊5(C22×C8), C60.58(C2×C4), (C2×C60).15C4, (C4×D5).95D6, (C2×C12).15F5, C12.60(C2×F5), (D5×C12).12C4, (C4×D5).8Dic3, C15⋊C8⋊14C22, C6.32(C22×F5), C30.70(C22×C4), C20.20(C2×Dic3), (C2×C20).11Dic3, (C2×Dic5).206D6, D10.12(C2×Dic3), (C22×D5).7Dic3, C10.1(C22×Dic3), Dic5.14(C2×Dic3), (D5×C12).123C22, (C3×Dic5).62C23, Dic5.48(C22×S3), (C6×Dic5).265C22, D5⋊(C2×C3⋊C8), C10⋊1(C2×C3⋊C8), C3⋊3(C2×D5⋊C8), C5⋊1(C22×C3⋊C8), C4.19(C2×C3⋊F5), (C3×D5)⋊4(C2×C8), (C2×C4×D5).18S3, (D5×C2×C6).13C4, C2.1(C22×C3⋊F5), (D5×C2×C12).32C2, (C2×C15⋊C8)⋊13C2, (C2×C4).11(C3⋊F5), (C2×C6).43(C2×F5), (C2×C30).37(C2×C4), C22.16(C2×C3⋊F5), (C6×D5).56(C2×C4), (C3×Dic5).64(C2×C4), (C2×C10).13(C2×Dic3), SmallGroup(480,1060)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C2×C60.C4 |
Subgroups: 524 in 152 conjugacy classes, 81 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×6], C5, C6, C6 [×2], C6 [×4], C8 [×4], C2×C4, C2×C4 [×5], C23, D5 [×4], C10, C10 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C15, C2×C8 [×6], C22×C4, Dic5 [×2], C20 [×2], D10 [×6], C2×C10, C3⋊C8 [×4], C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×4], C30, C30 [×2], C22×C8, C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8 [×6], C22×C12, C3×Dic5 [×2], C60 [×2], C6×D5 [×6], C2×C30, D5⋊C8 [×4], C2×C5⋊C8 [×2], C2×C4×D5, C22×C3⋊C8, C15⋊C8 [×4], D5×C12 [×4], C6×Dic5, C2×C60, D5×C2×C6, C2×D5⋊C8, C60.C4 [×4], C2×C15⋊C8 [×2], D5×C2×C12, C2×C60.C4
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C8 [×4], C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C2×C8 [×6], C22×C4, F5, C3⋊C8 [×4], C2×Dic3 [×6], C22×S3, C22×C8, C2×F5 [×3], C2×C3⋊C8 [×6], C22×Dic3, C3⋊F5, D5⋊C8 [×2], C22×F5, C22×C3⋊C8, C2×C3⋊F5 [×3], C2×D5⋊C8, C60.C4 [×2], C22×C3⋊F5, C2×C60.C4
Generators and relations
G = < a,b,c | a2=b60=1, c4=b30, ab=ba, ac=ca, cbc-1=b17 >
►On 240 points
Generators in S240
(1 150)(2 151)(3 152)(4 153)(5 154)(6 155)(7 156)(8 157)(9 158)(10 159)(11 160)(12 161)(13 162)(14 163)(15 164)(16 165)(17 166)(18 167)(19 168)(20 169)(21 170)(22 171)(23 172)(24 173)(25 174)(26 175)(27 176)(28 177)(29 178)(30 179)(31 180)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 140)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 211)(62 212)(63 213)(64 214)(65 215)(66 216)(67 217)(68 218)(69 219)(70 220)(71 221)(72 222)(73 223)(74 224)(75 225)(76 226)(77 227)(78 228)(79 229)(80 230)(81 231)(82 232)(83 233)(84 234)(85 235)(86 236)(87 237)(88 238)(89 239)(90 240)(91 181)(92 182)(93 183)(94 184)(95 185)(96 186)(97 187)(98 188)(99 189)(100 190)(101 191)(102 192)(103 193)(104 194)(105 195)(106 196)(107 197)(108 198)(109 199)(110 200)(111 201)(112 202)(113 203)(114 204)(115 205)(116 206)(117 207)(118 208)(119 209)(120 210)
(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)
(1 197 135 92 31 227 165 62)(2 190 124 109 32 220 154 79)(3 183 173 66 33 213 143 96)(4 236 162 83 34 206 132 113)(5 229 151 100 35 199 121 70)(6 222 140 117 36 192 170 87)(7 215 129 74 37 185 159 104)(8 208 178 91 38 238 148 61)(9 201 167 108 39 231 137 78)(10 194 156 65 40 224 126 95)(11 187 145 82 41 217 175 112)(12 240 134 99 42 210 164 69)(13 233 123 116 43 203 153 86)(14 226 172 73 44 196 142 103)(15 219 161 90 45 189 131 120)(16 212 150 107 46 182 180 77)(17 205 139 64 47 235 169 94)(18 198 128 81 48 228 158 111)(19 191 177 98 49 221 147 68)(20 184 166 115 50 214 136 85)(21 237 155 72 51 207 125 102)(22 230 144 89 52 200 174 119)(23 223 133 106 53 193 163 76)(24 216 122 63 54 186 152 93)(25 209 171 80 55 239 141 110)(26 202 160 97 56 232 130 67)(27 195 149 114 57 225 179 84)(28 188 138 71 58 218 168 101)(29 181 127 88 59 211 157 118)(30 234 176 105 60 204 146 75)
G:=sub<Sym(240)| (1,150)(2,151)(3,152)(4,153)(5,154)(6,155)(7,156)(8,157)(9,158)(10,159)(11,160)(12,161)(13,162)(14,163)(15,164)(16,165)(17,166)(18,167)(19,168)(20,169)(21,170)(22,171)(23,172)(24,173)(25,174)(26,175)(27,176)(28,177)(29,178)(30,179)(31,180)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,211)(62,212)(63,213)(64,214)(65,215)(66,216)(67,217)(68,218)(69,219)(70,220)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,227)(78,228)(79,229)(80,230)(81,231)(82,232)(83,233)(84,234)(85,235)(86,236)(87,237)(88,238)(89,239)(90,240)(91,181)(92,182)(93,183)(94,184)(95,185)(96,186)(97,187)(98,188)(99,189)(100,190)(101,191)(102,192)(103,193)(104,194)(105,195)(106,196)(107,197)(108,198)(109,199)(110,200)(111,201)(112,202)(113,203)(114,204)(115,205)(116,206)(117,207)(118,208)(119,209)(120,210), (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), (1,197,135,92,31,227,165,62)(2,190,124,109,32,220,154,79)(3,183,173,66,33,213,143,96)(4,236,162,83,34,206,132,113)(5,229,151,100,35,199,121,70)(6,222,140,117,36,192,170,87)(7,215,129,74,37,185,159,104)(8,208,178,91,38,238,148,61)(9,201,167,108,39,231,137,78)(10,194,156,65,40,224,126,95)(11,187,145,82,41,217,175,112)(12,240,134,99,42,210,164,69)(13,233,123,116,43,203,153,86)(14,226,172,73,44,196,142,103)(15,219,161,90,45,189,131,120)(16,212,150,107,46,182,180,77)(17,205,139,64,47,235,169,94)(18,198,128,81,48,228,158,111)(19,191,177,98,49,221,147,68)(20,184,166,115,50,214,136,85)(21,237,155,72,51,207,125,102)(22,230,144,89,52,200,174,119)(23,223,133,106,53,193,163,76)(24,216,122,63,54,186,152,93)(25,209,171,80,55,239,141,110)(26,202,160,97,56,232,130,67)(27,195,149,114,57,225,179,84)(28,188,138,71,58,218,168,101)(29,181,127,88,59,211,157,118)(30,234,176,105,60,204,146,75)>;
G:=Group( (1,150)(2,151)(3,152)(4,153)(5,154)(6,155)(7,156)(8,157)(9,158)(10,159)(11,160)(12,161)(13,162)(14,163)(15,164)(16,165)(17,166)(18,167)(19,168)(20,169)(21,170)(22,171)(23,172)(24,173)(25,174)(26,175)(27,176)(28,177)(29,178)(30,179)(31,180)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,211)(62,212)(63,213)(64,214)(65,215)(66,216)(67,217)(68,218)(69,219)(70,220)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,227)(78,228)(79,229)(80,230)(81,231)(82,232)(83,233)(84,234)(85,235)(86,236)(87,237)(88,238)(89,239)(90,240)(91,181)(92,182)(93,183)(94,184)(95,185)(96,186)(97,187)(98,188)(99,189)(100,190)(101,191)(102,192)(103,193)(104,194)(105,195)(106,196)(107,197)(108,198)(109,199)(110,200)(111,201)(112,202)(113,203)(114,204)(115,205)(116,206)(117,207)(118,208)(119,209)(120,210), (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), (1,197,135,92,31,227,165,62)(2,190,124,109,32,220,154,79)(3,183,173,66,33,213,143,96)(4,236,162,83,34,206,132,113)(5,229,151,100,35,199,121,70)(6,222,140,117,36,192,170,87)(7,215,129,74,37,185,159,104)(8,208,178,91,38,238,148,61)(9,201,167,108,39,231,137,78)(10,194,156,65,40,224,126,95)(11,187,145,82,41,217,175,112)(12,240,134,99,42,210,164,69)(13,233,123,116,43,203,153,86)(14,226,172,73,44,196,142,103)(15,219,161,90,45,189,131,120)(16,212,150,107,46,182,180,77)(17,205,139,64,47,235,169,94)(18,198,128,81,48,228,158,111)(19,191,177,98,49,221,147,68)(20,184,166,115,50,214,136,85)(21,237,155,72,51,207,125,102)(22,230,144,89,52,200,174,119)(23,223,133,106,53,193,163,76)(24,216,122,63,54,186,152,93)(25,209,171,80,55,239,141,110)(26,202,160,97,56,232,130,67)(27,195,149,114,57,225,179,84)(28,188,138,71,58,218,168,101)(29,181,127,88,59,211,157,118)(30,234,176,105,60,204,146,75) );
G=PermutationGroup([(1,150),(2,151),(3,152),(4,153),(5,154),(6,155),(7,156),(8,157),(9,158),(10,159),(11,160),(12,161),(13,162),(14,163),(15,164),(16,165),(17,166),(18,167),(19,168),(20,169),(21,170),(22,171),(23,172),(24,173),(25,174),(26,175),(27,176),(28,177),(29,178),(30,179),(31,180),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,127),(39,128),(40,129),(41,130),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,137),(49,138),(50,139),(51,140),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(61,211),(62,212),(63,213),(64,214),(65,215),(66,216),(67,217),(68,218),(69,219),(70,220),(71,221),(72,222),(73,223),(74,224),(75,225),(76,226),(77,227),(78,228),(79,229),(80,230),(81,231),(82,232),(83,233),(84,234),(85,235),(86,236),(87,237),(88,238),(89,239),(90,240),(91,181),(92,182),(93,183),(94,184),(95,185),(96,186),(97,187),(98,188),(99,189),(100,190),(101,191),(102,192),(103,193),(104,194),(105,195),(106,196),(107,197),(108,198),(109,199),(110,200),(111,201),(112,202),(113,203),(114,204),(115,205),(116,206),(117,207),(118,208),(119,209),(120,210)], [(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)], [(1,197,135,92,31,227,165,62),(2,190,124,109,32,220,154,79),(3,183,173,66,33,213,143,96),(4,236,162,83,34,206,132,113),(5,229,151,100,35,199,121,70),(6,222,140,117,36,192,170,87),(7,215,129,74,37,185,159,104),(8,208,178,91,38,238,148,61),(9,201,167,108,39,231,137,78),(10,194,156,65,40,224,126,95),(11,187,145,82,41,217,175,112),(12,240,134,99,42,210,164,69),(13,233,123,116,43,203,153,86),(14,226,172,73,44,196,142,103),(15,219,161,90,45,189,131,120),(16,212,150,107,46,182,180,77),(17,205,139,64,47,235,169,94),(18,198,128,81,48,228,158,111),(19,191,177,98,49,221,147,68),(20,184,166,115,50,214,136,85),(21,237,155,72,51,207,125,102),(22,230,144,89,52,200,174,119),(23,223,133,106,53,193,163,76),(24,216,122,63,54,186,152,93),(25,209,171,80,55,239,141,110),(26,202,160,97,56,232,130,67),(27,195,149,114,57,225,179,84),(28,188,138,71,58,218,168,101),(29,181,127,88,59,211,157,118),(30,234,176,105,60,204,146,75)])
Matrix representation ►G ⊆ 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 177 | 177 | 0 | 0 | 0 | 0 |
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 196 | 175 | 66 | 45 |
| 0 | 0 | 196 | 130 | 0 | 111 |
| 0 | 0 | 0 | 130 | 196 | 45 |
| 0 | 0 | 66 | 175 | 196 | 0 |
| 3 | 91 | 0 | 0 | 0 | 0 |
| 88 | 238 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 136 | 74 | 167 |
| 0 | 0 | 167 | 0 | 74 | 62 |
| 0 | 0 | 62 | 74 | 0 | 167 |
| 0 | 0 | 167 | 74 | 136 | 0 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[177,64,0,0,0,0,177,0,0,0,0,0,0,0,196,196,0,66,0,0,175,130,130,175,0,0,66,0,196,196,0,0,45,111,45,0],[3,88,0,0,0,0,91,238,0,0,0,0,0,0,0,167,62,167,0,0,136,0,74,74,0,0,74,74,0,136,0,0,167,62,167,0] >;
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8P | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | S3 | Dic3 | D6 | D6 | Dic3 | Dic3 | C3⋊C8 | F5 | C2×F5 | C2×F5 | C3⋊F5 | D5⋊C8 | C2×C3⋊F5 | C2×C3⋊F5 | C60.C4 |
| kernel | C2×C60.C4 | C60.C4 | C2×C15⋊C8 | D5×C2×C12 | D5×C12 | C2×C60 | D5×C2×C6 | C6×D5 | C2×C4×D5 | C4×D5 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | D10 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 4 | 2 | 2 | 16 | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{60}.C_4 % in TeX
G:=Group("C2xC60.C4"); // GroupNames label
G:=SmallGroup(480,1060);
// by ID
G=gap.SmallGroup(480,1060);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,100,80,2693,14118,2379]);
// Polycyclic
G:=Group<a,b,c|a^2=b^60=1,c^4=b^30,a*b=b*a,a*c=c*a,c*b*c^-1=b^17>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀