direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C30, C70⋊6C6, C210⋊5C2, C42⋊2C10, C14⋊3C30, C105⋊12C22, C7⋊3(C2×C30), C35⋊8(C2×C6), C21⋊3(C2×C10), SmallGroup(420,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C30 |
Generators and relations for D7×C30
G = < a,b,c | a30=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C30
►On 210 points
Generators in S210
(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)
(1 124 59 153 119 90 201)(2 125 60 154 120 61 202)(3 126 31 155 91 62 203)(4 127 32 156 92 63 204)(5 128 33 157 93 64 205)(6 129 34 158 94 65 206)(7 130 35 159 95 66 207)(8 131 36 160 96 67 208)(9 132 37 161 97 68 209)(10 133 38 162 98 69 210)(11 134 39 163 99 70 181)(12 135 40 164 100 71 182)(13 136 41 165 101 72 183)(14 137 42 166 102 73 184)(15 138 43 167 103 74 185)(16 139 44 168 104 75 186)(17 140 45 169 105 76 187)(18 141 46 170 106 77 188)(19 142 47 171 107 78 189)(20 143 48 172 108 79 190)(21 144 49 173 109 80 191)(22 145 50 174 110 81 192)(23 146 51 175 111 82 193)(24 147 52 176 112 83 194)(25 148 53 177 113 84 195)(26 149 54 178 114 85 196)(27 150 55 179 115 86 197)(28 121 56 180 116 87 198)(29 122 57 151 117 88 199)(30 123 58 152 118 89 200)
(1 186)(2 187)(3 188)(4 189)(5 190)(6 191)(7 192)(8 193)(9 194)(10 195)(11 196)(12 197)(13 198)(14 199)(15 200)(16 201)(17 202)(18 203)(19 204)(20 205)(21 206)(22 207)(23 208)(24 209)(25 210)(26 181)(27 182)(28 183)(29 184)(30 185)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 105)(61 140)(62 141)(63 142)(64 143)(65 144)(66 145)(67 146)(68 147)(69 148)(70 149)(71 150)(72 121)(73 122)(74 123)(75 124)(76 125)(77 126)(78 127)(79 128)(80 129)(81 130)(82 131)(83 132)(84 133)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(151 166)(152 167)(153 168)(154 169)(155 170)(156 171)(157 172)(158 173)(159 174)(160 175)(161 176)(162 177)(163 178)(164 179)(165 180)
G:=sub<Sym(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), (1,124,59,153,119,90,201)(2,125,60,154,120,61,202)(3,126,31,155,91,62,203)(4,127,32,156,92,63,204)(5,128,33,157,93,64,205)(6,129,34,158,94,65,206)(7,130,35,159,95,66,207)(8,131,36,160,96,67,208)(9,132,37,161,97,68,209)(10,133,38,162,98,69,210)(11,134,39,163,99,70,181)(12,135,40,164,100,71,182)(13,136,41,165,101,72,183)(14,137,42,166,102,73,184)(15,138,43,167,103,74,185)(16,139,44,168,104,75,186)(17,140,45,169,105,76,187)(18,141,46,170,106,77,188)(19,142,47,171,107,78,189)(20,143,48,172,108,79,190)(21,144,49,173,109,80,191)(22,145,50,174,110,81,192)(23,146,51,175,111,82,193)(24,147,52,176,112,83,194)(25,148,53,177,113,84,195)(26,149,54,178,114,85,196)(27,150,55,179,115,86,197)(28,121,56,180,116,87,198)(29,122,57,151,117,88,199)(30,123,58,152,118,89,200), (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,193)(9,194)(10,195)(11,196)(12,197)(13,198)(14,199)(15,200)(16,201)(17,202)(18,203)(19,204)(20,205)(21,206)(22,207)(23,208)(24,209)(25,210)(26,181)(27,182)(28,183)(29,184)(30,185)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(61,140)(62,141)(63,142)(64,143)(65,144)(66,145)(67,146)(68,147)(69,148)(70,149)(71,150)(72,121)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,130)(82,131)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(151,166)(152,167)(153,168)(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180)>;
G:=Group( (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), (1,124,59,153,119,90,201)(2,125,60,154,120,61,202)(3,126,31,155,91,62,203)(4,127,32,156,92,63,204)(5,128,33,157,93,64,205)(6,129,34,158,94,65,206)(7,130,35,159,95,66,207)(8,131,36,160,96,67,208)(9,132,37,161,97,68,209)(10,133,38,162,98,69,210)(11,134,39,163,99,70,181)(12,135,40,164,100,71,182)(13,136,41,165,101,72,183)(14,137,42,166,102,73,184)(15,138,43,167,103,74,185)(16,139,44,168,104,75,186)(17,140,45,169,105,76,187)(18,141,46,170,106,77,188)(19,142,47,171,107,78,189)(20,143,48,172,108,79,190)(21,144,49,173,109,80,191)(22,145,50,174,110,81,192)(23,146,51,175,111,82,193)(24,147,52,176,112,83,194)(25,148,53,177,113,84,195)(26,149,54,178,114,85,196)(27,150,55,179,115,86,197)(28,121,56,180,116,87,198)(29,122,57,151,117,88,199)(30,123,58,152,118,89,200), (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,193)(9,194)(10,195)(11,196)(12,197)(13,198)(14,199)(15,200)(16,201)(17,202)(18,203)(19,204)(20,205)(21,206)(22,207)(23,208)(24,209)(25,210)(26,181)(27,182)(28,183)(29,184)(30,185)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,91)(47,92)(48,93)(49,94)(50,95)(51,96)(52,97)(53,98)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(61,140)(62,141)(63,142)(64,143)(65,144)(66,145)(67,146)(68,147)(69,148)(70,149)(71,150)(72,121)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,130)(82,131)(83,132)(84,133)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(151,166)(152,167)(153,168)(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)(162,177)(163,178)(164,179)(165,180) );
G=PermutationGroup([[(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)], [(1,124,59,153,119,90,201),(2,125,60,154,120,61,202),(3,126,31,155,91,62,203),(4,127,32,156,92,63,204),(5,128,33,157,93,64,205),(6,129,34,158,94,65,206),(7,130,35,159,95,66,207),(8,131,36,160,96,67,208),(9,132,37,161,97,68,209),(10,133,38,162,98,69,210),(11,134,39,163,99,70,181),(12,135,40,164,100,71,182),(13,136,41,165,101,72,183),(14,137,42,166,102,73,184),(15,138,43,167,103,74,185),(16,139,44,168,104,75,186),(17,140,45,169,105,76,187),(18,141,46,170,106,77,188),(19,142,47,171,107,78,189),(20,143,48,172,108,79,190),(21,144,49,173,109,80,191),(22,145,50,174,110,81,192),(23,146,51,175,111,82,193),(24,147,52,176,112,83,194),(25,148,53,177,113,84,195),(26,149,54,178,114,85,196),(27,150,55,179,115,86,197),(28,121,56,180,116,87,198),(29,122,57,151,117,88,199),(30,123,58,152,118,89,200)], [(1,186),(2,187),(3,188),(4,189),(5,190),(6,191),(7,192),(8,193),(9,194),(10,195),(11,196),(12,197),(13,198),(14,199),(15,200),(16,201),(17,202),(18,203),(19,204),(20,205),(21,206),(22,207),(23,208),(24,209),(25,210),(26,181),(27,182),(28,183),(29,184),(30,185),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,113),(39,114),(40,115),(41,116),(42,117),(43,118),(44,119),(45,120),(46,91),(47,92),(48,93),(49,94),(50,95),(51,96),(52,97),(53,98),(54,99),(55,100),(56,101),(57,102),(58,103),(59,104),(60,105),(61,140),(62,141),(63,142),(64,143),(65,144),(66,145),(67,146),(68,147),(69,148),(70,149),(71,150),(72,121),(73,122),(74,123),(75,124),(76,125),(77,126),(78,127),(79,128),(80,129),(81,130),(82,131),(83,132),(84,133),(85,134),(86,135),(87,136),(88,137),(89,138),(90,139),(151,166),(152,167),(153,168),(154,169),(155,170),(156,171),(157,172),(158,173),(159,174),(160,175),(161,176),(162,177),(163,178),(164,179),(165,180)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 14A | 14B | 14C | 15A | ··· | 15H | 21A | ··· | 21F | 30A | ··· | 30H | 30I | ··· | 30X | 35A | ··· | 35L | 42A | ··· | 42F | 70A | ··· | 70L | 105A | ··· | 105X | 210A | ··· | 210X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 14 | 14 | 14 | 15 | ··· | 15 | 21 | ··· | 21 | 30 | ··· | 30 | 30 | ··· | 30 | 35 | ··· | 35 | 42 | ··· | 42 | 70 | ··· | 70 | 105 | ··· | 105 | 210 | ··· | 210 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | D7 | D14 | C3×D7 | C5×D7 | C6×D7 | C10×D7 | D7×C15 | D7×C30 |
| kernel | D7×C30 | D7×C15 | C210 | C10×D7 | C6×D7 | C5×D7 | C70 | C3×D7 | C42 | D14 | D7 | C14 | C30 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 4 | 8 | 16 | 8 | 3 | 3 | 6 | 12 | 6 | 12 | 24 | 24 |
Matrix representation of D7×C30 ►in GL3(𝔽211) generated by
| 210 | 0 | 0 |
| 0 | 137 | 0 |
| 0 | 0 | 137 |
| 1 | 0 | 0 |
| 0 | 81 | 1 |
| 0 | 210 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(211))| [210,0,0,0,137,0,0,0,137],[1,0,0,0,81,210,0,1,0],[1,0,0,0,0,1,0,1,0] >;
D7×C30 in GAP, Magma, Sage, TeX
D_7\times C_{30} % in TeX
G:=Group("D7xC30"); // GroupNames label
G:=SmallGroup(420,34);
// by ID
G=gap.SmallGroup(420,34);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-7,9004]);
// Polycyclic
G:=Group<a,b,c|a^30=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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