direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D7×C25, C7⋊C50, C175⋊3C2, C35.C10, C5.(C5×D7), (C5×D7).C5, SmallGroup(350,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C25 |
Generators and relations for D7×C25
G = < a,b,c | a25=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C25
►On 175 points
Generators in S175
(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)
(1 136 43 117 61 160 97)(2 137 44 118 62 161 98)(3 138 45 119 63 162 99)(4 139 46 120 64 163 100)(5 140 47 121 65 164 76)(6 141 48 122 66 165 77)(7 142 49 123 67 166 78)(8 143 50 124 68 167 79)(9 144 26 125 69 168 80)(10 145 27 101 70 169 81)(11 146 28 102 71 170 82)(12 147 29 103 72 171 83)(13 148 30 104 73 172 84)(14 149 31 105 74 173 85)(15 150 32 106 75 174 86)(16 126 33 107 51 175 87)(17 127 34 108 52 151 88)(18 128 35 109 53 152 89)(19 129 36 110 54 153 90)(20 130 37 111 55 154 91)(21 131 38 112 56 155 92)(22 132 39 113 57 156 93)(23 133 40 114 58 157 94)(24 134 41 115 59 158 95)(25 135 42 116 60 159 96)
(1 97)(2 98)(3 99)(4 100)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(126 175)(127 151)(128 152)(129 153)(130 154)(131 155)(132 156)(133 157)(134 158)(135 159)(136 160)(137 161)(138 162)(139 163)(140 164)(141 165)(142 166)(143 167)(144 168)(145 169)(146 170)(147 171)(148 172)(149 173)(150 174)
G:=sub<Sym(175)| (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), (1,136,43,117,61,160,97)(2,137,44,118,62,161,98)(3,138,45,119,63,162,99)(4,139,46,120,64,163,100)(5,140,47,121,65,164,76)(6,141,48,122,66,165,77)(7,142,49,123,67,166,78)(8,143,50,124,68,167,79)(9,144,26,125,69,168,80)(10,145,27,101,70,169,81)(11,146,28,102,71,170,82)(12,147,29,103,72,171,83)(13,148,30,104,73,172,84)(14,149,31,105,74,173,85)(15,150,32,106,75,174,86)(16,126,33,107,51,175,87)(17,127,34,108,52,151,88)(18,128,35,109,53,152,89)(19,129,36,110,54,153,90)(20,130,37,111,55,154,91)(21,131,38,112,56,155,92)(22,132,39,113,57,156,93)(23,133,40,114,58,157,94)(24,134,41,115,59,158,95)(25,135,42,116,60,159,96), (1,97)(2,98)(3,99)(4,100)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(126,175)(127,151)(128,152)(129,153)(130,154)(131,155)(132,156)(133,157)(134,158)(135,159)(136,160)(137,161)(138,162)(139,163)(140,164)(141,165)(142,166)(143,167)(144,168)(145,169)(146,170)(147,171)(148,172)(149,173)(150,174)>;
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), (1,136,43,117,61,160,97)(2,137,44,118,62,161,98)(3,138,45,119,63,162,99)(4,139,46,120,64,163,100)(5,140,47,121,65,164,76)(6,141,48,122,66,165,77)(7,142,49,123,67,166,78)(8,143,50,124,68,167,79)(9,144,26,125,69,168,80)(10,145,27,101,70,169,81)(11,146,28,102,71,170,82)(12,147,29,103,72,171,83)(13,148,30,104,73,172,84)(14,149,31,105,74,173,85)(15,150,32,106,75,174,86)(16,126,33,107,51,175,87)(17,127,34,108,52,151,88)(18,128,35,109,53,152,89)(19,129,36,110,54,153,90)(20,130,37,111,55,154,91)(21,131,38,112,56,155,92)(22,132,39,113,57,156,93)(23,133,40,114,58,157,94)(24,134,41,115,59,158,95)(25,135,42,116,60,159,96), (1,97)(2,98)(3,99)(4,100)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(126,175)(127,151)(128,152)(129,153)(130,154)(131,155)(132,156)(133,157)(134,158)(135,159)(136,160)(137,161)(138,162)(139,163)(140,164)(141,165)(142,166)(143,167)(144,168)(145,169)(146,170)(147,171)(148,172)(149,173)(150,174) );
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)], [(1,136,43,117,61,160,97),(2,137,44,118,62,161,98),(3,138,45,119,63,162,99),(4,139,46,120,64,163,100),(5,140,47,121,65,164,76),(6,141,48,122,66,165,77),(7,142,49,123,67,166,78),(8,143,50,124,68,167,79),(9,144,26,125,69,168,80),(10,145,27,101,70,169,81),(11,146,28,102,71,170,82),(12,147,29,103,72,171,83),(13,148,30,104,73,172,84),(14,149,31,105,74,173,85),(15,150,32,106,75,174,86),(16,126,33,107,51,175,87),(17,127,34,108,52,151,88),(18,128,35,109,53,152,89),(19,129,36,110,54,153,90),(20,130,37,111,55,154,91),(21,131,38,112,56,155,92),(22,132,39,113,57,156,93),(23,133,40,114,58,157,94),(24,134,41,115,59,158,95),(25,135,42,116,60,159,96)], [(1,97),(2,98),(3,99),(4,100),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,84),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,75),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(126,175),(127,151),(128,152),(129,153),(130,154),(131,155),(132,156),(133,157),(134,158),(135,159),(136,160),(137,161),(138,162),(139,163),(140,164),(141,165),(142,166),(143,167),(144,168),(145,169),(146,170),(147,171),(148,172),(149,173),(150,174)]])
125 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 25A | ··· | 25T | 35A | ··· | 35L | 50A | ··· | 50T | 175A | ··· | 175BH |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 25 | ··· | 25 | 35 | ··· | 35 | 50 | ··· | 50 | 175 | ··· | 175 |
| size | 1 | 7 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 7 | 7 | 7 | 7 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 2 | ··· | 2 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C5 | C10 | C25 | C50 | D7 | C5×D7 | D7×C25 |
| kernel | D7×C25 | C175 | C5×D7 | C35 | D7 | C7 | C25 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 20 | 20 | 3 | 12 | 60 |
Matrix representation of D7×C25 ►in GL3(𝔽701) generated by
| 627 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 313 | 1 |
| 0 | 559 | 172 |
| 700 | 0 | 0 |
| 0 | 172 | 700 |
| 0 | 141 | 529 |
G:=sub<GL(3,GF(701))| [627,0,0,0,1,0,0,0,1],[1,0,0,0,313,559,0,1,172],[700,0,0,0,172,141,0,700,529] >;
D7×C25 in GAP, Magma, Sage, TeX
D_7\times C_{25} % in TeX
G:=Group("D7xC25"); // GroupNames label
G:=SmallGroup(350,2);
// by ID
G=gap.SmallGroup(350,2);
# by ID
G:=PCGroup([4,-2,-5,-5,-7,45,4803]);
// Polycyclic
G:=Group<a,b,c|a^25=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export