direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D25, C25⋊C14, C175⋊2C2, C35.2D5, C5.(C7×D5), SmallGroup(350,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C7×D25 |
Generators and relations for C7×D25
G = < a,b,c | a7=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D25
►On 175 points
Generators in S175
(1 165 147 105 88 68 36)(2 166 148 106 89 69 37)(3 167 149 107 90 70 38)(4 168 150 108 91 71 39)(5 169 126 109 92 72 40)(6 170 127 110 93 73 41)(7 171 128 111 94 74 42)(8 172 129 112 95 75 43)(9 173 130 113 96 51 44)(10 174 131 114 97 52 45)(11 175 132 115 98 53 46)(12 151 133 116 99 54 47)(13 152 134 117 100 55 48)(14 153 135 118 76 56 49)(15 154 136 119 77 57 50)(16 155 137 120 78 58 26)(17 156 138 121 79 59 27)(18 157 139 122 80 60 28)(19 158 140 123 81 61 29)(20 159 141 124 82 62 30)(21 160 142 125 83 63 31)(22 161 143 101 84 64 32)(23 162 144 102 85 65 33)(24 163 145 103 86 66 34)(25 164 146 104 87 67 35)
(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)
(1 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(46 50)(47 49)(51 59)(52 58)(53 57)(54 56)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(76 99)(77 98)(78 97)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(101 108)(102 107)(103 106)(104 105)(109 125)(110 124)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)(126 142)(127 141)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(143 150)(144 149)(145 148)(146 147)(151 153)(154 175)(155 174)(156 173)(157 172)(158 171)(159 170)(160 169)(161 168)(162 167)(163 166)(164 165)
G:=sub<Sym(175)| (1,165,147,105,88,68,36)(2,166,148,106,89,69,37)(3,167,149,107,90,70,38)(4,168,150,108,91,71,39)(5,169,126,109,92,72,40)(6,170,127,110,93,73,41)(7,171,128,111,94,74,42)(8,172,129,112,95,75,43)(9,173,130,113,96,51,44)(10,174,131,114,97,52,45)(11,175,132,115,98,53,46)(12,151,133,116,99,54,47)(13,152,134,117,100,55,48)(14,153,135,118,76,56,49)(15,154,136,119,77,57,50)(16,155,137,120,78,58,26)(17,156,138,121,79,59,27)(18,157,139,122,80,60,28)(19,158,140,123,81,61,29)(20,159,141,124,82,62,30)(21,160,142,125,83,63,31)(22,161,143,101,84,64,32)(23,162,144,102,85,65,33)(24,163,145,103,86,66,34)(25,164,146,104,87,67,35), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(46,50)(47,49)(51,59)(52,58)(53,57)(54,56)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(101,108)(102,107)(103,106)(104,105)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(143,150)(144,149)(145,148)(146,147)(151,153)(154,175)(155,174)(156,173)(157,172)(158,171)(159,170)(160,169)(161,168)(162,167)(163,166)(164,165)>;
G:=Group( (1,165,147,105,88,68,36)(2,166,148,106,89,69,37)(3,167,149,107,90,70,38)(4,168,150,108,91,71,39)(5,169,126,109,92,72,40)(6,170,127,110,93,73,41)(7,171,128,111,94,74,42)(8,172,129,112,95,75,43)(9,173,130,113,96,51,44)(10,174,131,114,97,52,45)(11,175,132,115,98,53,46)(12,151,133,116,99,54,47)(13,152,134,117,100,55,48)(14,153,135,118,76,56,49)(15,154,136,119,77,57,50)(16,155,137,120,78,58,26)(17,156,138,121,79,59,27)(18,157,139,122,80,60,28)(19,158,140,123,81,61,29)(20,159,141,124,82,62,30)(21,160,142,125,83,63,31)(22,161,143,101,84,64,32)(23,162,144,102,85,65,33)(24,163,145,103,86,66,34)(25,164,146,104,87,67,35), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(46,50)(47,49)(51,59)(52,58)(53,57)(54,56)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(76,99)(77,98)(78,97)(79,96)(80,95)(81,94)(82,93)(83,92)(84,91)(85,90)(86,89)(87,88)(101,108)(102,107)(103,106)(104,105)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(143,150)(144,149)(145,148)(146,147)(151,153)(154,175)(155,174)(156,173)(157,172)(158,171)(159,170)(160,169)(161,168)(162,167)(163,166)(164,165) );
G=PermutationGroup([[(1,165,147,105,88,68,36),(2,166,148,106,89,69,37),(3,167,149,107,90,70,38),(4,168,150,108,91,71,39),(5,169,126,109,92,72,40),(6,170,127,110,93,73,41),(7,171,128,111,94,74,42),(8,172,129,112,95,75,43),(9,173,130,113,96,51,44),(10,174,131,114,97,52,45),(11,175,132,115,98,53,46),(12,151,133,116,99,54,47),(13,152,134,117,100,55,48),(14,153,135,118,76,56,49),(15,154,136,119,77,57,50),(16,155,137,120,78,58,26),(17,156,138,121,79,59,27),(18,157,139,122,80,60,28),(19,158,140,123,81,61,29),(20,159,141,124,82,62,30),(21,160,142,125,83,63,31),(22,161,143,101,84,64,32),(23,162,144,102,85,65,33),(24,163,145,103,86,66,34),(25,164,146,104,87,67,35)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(26,45),(27,44),(28,43),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(46,50),(47,49),(51,59),(52,58),(53,57),(54,56),(60,75),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(76,99),(77,98),(78,97),(79,96),(80,95),(81,94),(82,93),(83,92),(84,91),(85,90),(86,89),(87,88),(101,108),(102,107),(103,106),(104,105),(109,125),(110,124),(111,123),(112,122),(113,121),(114,120),(115,119),(116,118),(126,142),(127,141),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(143,150),(144,149),(145,148),(146,147),(151,153),(154,175),(155,174),(156,173),(157,172),(158,171),(159,170),(160,169),(161,168),(162,167),(163,166),(164,165)]])
98 conjugacy classes
| class | 1 | 2 | 5A | 5B | 7A | ··· | 7F | 14A | ··· | 14F | 25A | ··· | 25J | 35A | ··· | 35L | 175A | ··· | 175BH |
| order | 1 | 2 | 5 | 5 | 7 | ··· | 7 | 14 | ··· | 14 | 25 | ··· | 25 | 35 | ··· | 35 | 175 | ··· | 175 |
| size | 1 | 25 | 2 | 2 | 1 | ··· | 1 | 25 | ··· | 25 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C7 | C14 | D5 | D25 | C7×D5 | C7×D25 |
| kernel | C7×D25 | C175 | D25 | C25 | C35 | C7 | C5 | C1 |
| # reps | 1 | 1 | 6 | 6 | 2 | 10 | 12 | 60 |
Matrix representation of C7×D25 ►in GL2(𝔽701) generated by
| 550 | 0 |
| 0 | 550 |
| 178 | 472 |
| 229 | 304 |
| 304 | 676 |
| 472 | 397 |
G:=sub<GL(2,GF(701))| [550,0,0,550],[178,229,472,304],[304,472,676,397] >;
C7×D25 in GAP, Magma, Sage, TeX
C_7\times D_{25} % in TeX
G:=Group("C7xD25"); // GroupNames label
G:=SmallGroup(350,1);
// by ID
G=gap.SmallGroup(350,1);
# by ID
G:=PCGroup([4,-2,-7,-5,-5,1514,250,4483]);
// Polycyclic
G:=Group<a,b,c|a^7=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export