direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C26, C14⋊C26, C182⋊3C2, C91⋊4C22, C7⋊(C2×C26), SmallGroup(364,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C26 |
Generators and relations for D7×C26
G = < a,b,c | a26=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C26
►On 182 points
Generators in S182
(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)
(1 72 163 144 127 98 29)(2 73 164 145 128 99 30)(3 74 165 146 129 100 31)(4 75 166 147 130 101 32)(5 76 167 148 105 102 33)(6 77 168 149 106 103 34)(7 78 169 150 107 104 35)(8 53 170 151 108 79 36)(9 54 171 152 109 80 37)(10 55 172 153 110 81 38)(11 56 173 154 111 82 39)(12 57 174 155 112 83 40)(13 58 175 156 113 84 41)(14 59 176 131 114 85 42)(15 60 177 132 115 86 43)(16 61 178 133 116 87 44)(17 62 179 134 117 88 45)(18 63 180 135 118 89 46)(19 64 181 136 119 90 47)(20 65 182 137 120 91 48)(21 66 157 138 121 92 49)(22 67 158 139 122 93 50)(23 68 159 140 123 94 51)(24 69 160 141 124 95 52)(25 70 161 142 125 96 27)(26 71 162 143 126 97 28)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(105 180)(106 181)(107 182)(108 157)(109 158)(110 159)(111 160)(112 161)(113 162)(114 163)(115 164)(116 165)(117 166)(118 167)(119 168)(120 169)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 177)(129 178)(130 179)(131 144)(132 145)(133 146)(134 147)(135 148)(136 149)(137 150)(138 151)(139 152)(140 153)(141 154)(142 155)(143 156)
G:=sub<Sym(182)| (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), (1,72,163,144,127,98,29)(2,73,164,145,128,99,30)(3,74,165,146,129,100,31)(4,75,166,147,130,101,32)(5,76,167,148,105,102,33)(6,77,168,149,106,103,34)(7,78,169,150,107,104,35)(8,53,170,151,108,79,36)(9,54,171,152,109,80,37)(10,55,172,153,110,81,38)(11,56,173,154,111,82,39)(12,57,174,155,112,83,40)(13,58,175,156,113,84,41)(14,59,176,131,114,85,42)(15,60,177,132,115,86,43)(16,61,178,133,116,87,44)(17,62,179,134,117,88,45)(18,63,180,135,118,89,46)(19,64,181,136,119,90,47)(20,65,182,137,120,91,48)(21,66,157,138,121,92,49)(22,67,158,139,122,93,50)(23,68,159,140,123,94,51)(24,69,160,141,124,95,52)(25,70,161,142,125,96,27)(26,71,162,143,126,97,28), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(105,180)(106,181)(107,182)(108,157)(109,158)(110,159)(111,160)(112,161)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,169)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,144)(132,145)(133,146)(134,147)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,154)(142,155)(143,156)>;
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), (1,72,163,144,127,98,29)(2,73,164,145,128,99,30)(3,74,165,146,129,100,31)(4,75,166,147,130,101,32)(5,76,167,148,105,102,33)(6,77,168,149,106,103,34)(7,78,169,150,107,104,35)(8,53,170,151,108,79,36)(9,54,171,152,109,80,37)(10,55,172,153,110,81,38)(11,56,173,154,111,82,39)(12,57,174,155,112,83,40)(13,58,175,156,113,84,41)(14,59,176,131,114,85,42)(15,60,177,132,115,86,43)(16,61,178,133,116,87,44)(17,62,179,134,117,88,45)(18,63,180,135,118,89,46)(19,64,181,136,119,90,47)(20,65,182,137,120,91,48)(21,66,157,138,121,92,49)(22,67,158,139,122,93,50)(23,68,159,140,123,94,51)(24,69,160,141,124,95,52)(25,70,161,142,125,96,27)(26,71,162,143,126,97,28), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,37)(23,38)(24,39)(25,40)(26,41)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,79)(67,80)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(105,180)(106,181)(107,182)(108,157)(109,158)(110,159)(111,160)(112,161)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,169)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,144)(132,145)(133,146)(134,147)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,154)(142,155)(143,156) );
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)], [(1,72,163,144,127,98,29),(2,73,164,145,128,99,30),(3,74,165,146,129,100,31),(4,75,166,147,130,101,32),(5,76,167,148,105,102,33),(6,77,168,149,106,103,34),(7,78,169,150,107,104,35),(8,53,170,151,108,79,36),(9,54,171,152,109,80,37),(10,55,172,153,110,81,38),(11,56,173,154,111,82,39),(12,57,174,155,112,83,40),(13,58,175,156,113,84,41),(14,59,176,131,114,85,42),(15,60,177,132,115,86,43),(16,61,178,133,116,87,44),(17,62,179,134,117,88,45),(18,63,180,135,118,89,46),(19,64,181,136,119,90,47),(20,65,182,137,120,91,48),(21,66,157,138,121,92,49),(22,67,158,139,122,93,50),(23,68,159,140,123,94,51),(24,69,160,141,124,95,52),(25,70,161,142,125,96,27),(26,71,162,143,126,97,28)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,51),(11,52),(12,27),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,37),(23,38),(24,39),(25,40),(26,41),(53,92),(54,93),(55,94),(56,95),(57,96),(58,97),(59,98),(60,99),(61,100),(62,101),(63,102),(64,103),(65,104),(66,79),(67,80),(68,81),(69,82),(70,83),(71,84),(72,85),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(105,180),(106,181),(107,182),(108,157),(109,158),(110,159),(111,160),(112,161),(113,162),(114,163),(115,164),(116,165),(117,166),(118,167),(119,168),(120,169),(121,170),(122,171),(123,172),(124,173),(125,174),(126,175),(127,176),(128,177),(129,178),(130,179),(131,144),(132,145),(133,146),(134,147),(135,148),(136,149),(137,150),(138,151),(139,152),(140,153),(141,154),(142,155),(143,156)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | 7B | 7C | 13A | ··· | 13L | 14A | 14B | 14C | 26A | ··· | 26L | 26M | ··· | 26AJ | 91A | ··· | 91AJ | 182A | ··· | 182AJ |
| order | 1 | 2 | 2 | 2 | 7 | 7 | 7 | 13 | ··· | 13 | 14 | 14 | 14 | 26 | ··· | 26 | 26 | ··· | 26 | 91 | ··· | 91 | 182 | ··· | 182 |
| size | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | D7 | D14 | C13×D7 | D7×C26 |
| kernel | D7×C26 | C13×D7 | C182 | D14 | D7 | C14 | C26 | C13 | C2 | C1 |
| # reps | 1 | 2 | 1 | 12 | 24 | 12 | 3 | 3 | 36 | 36 |
Matrix representation of D7×C26 ►in GL2(𝔽547) generated by
| 197 | 0 |
| 0 | 197 |
| 179 | 1 |
| 546 | 0 |
| 0 | 546 |
| 546 | 0 |
G:=sub<GL(2,GF(547))| [197,0,0,197],[179,546,1,0],[0,546,546,0] >;
D7×C26 in GAP, Magma, Sage, TeX
D_7\times C_{26} % in TeX
G:=Group("D7xC26"); // GroupNames label
G:=SmallGroup(364,9);
// by ID
G=gap.SmallGroup(364,9);
# by ID
G:=PCGroup([4,-2,-2,-13,-7,4995]);
// Polycyclic
G:=Group<a,b,c|a^26=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export