direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D7×C27, C7⋊3C54, C189⋊3C2, C63.3C6, C21.3C18, C9.(C3×D7), C3.(C9×D7), (C9×D7).2C3, (C3×D7).2C9, SmallGroup(378,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C27 |
Generators and relations for D7×C27
G = < a,b,c | a27=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C27
►On 189 points
Generators in S189
(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)
(1 162 70 105 185 112 49)(2 136 71 106 186 113 50)(3 137 72 107 187 114 51)(4 138 73 108 188 115 52)(5 139 74 82 189 116 53)(6 140 75 83 163 117 54)(7 141 76 84 164 118 28)(8 142 77 85 165 119 29)(9 143 78 86 166 120 30)(10 144 79 87 167 121 31)(11 145 80 88 168 122 32)(12 146 81 89 169 123 33)(13 147 55 90 170 124 34)(14 148 56 91 171 125 35)(15 149 57 92 172 126 36)(16 150 58 93 173 127 37)(17 151 59 94 174 128 38)(18 152 60 95 175 129 39)(19 153 61 96 176 130 40)(20 154 62 97 177 131 41)(21 155 63 98 178 132 42)(22 156 64 99 179 133 43)(23 157 65 100 180 134 44)(24 158 66 101 181 135 45)(25 159 67 102 182 109 46)(26 160 68 103 183 110 47)(27 161 69 104 184 111 48)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(55 170)(56 171)(57 172)(58 173)(59 174)(60 175)(61 176)(62 177)(63 178)(64 179)(65 180)(66 181)(67 182)(68 183)(69 184)(70 185)(71 186)(72 187)(73 188)(74 189)(75 163)(76 164)(77 165)(78 166)(79 167)(80 168)(81 169)(109 159)(110 160)(111 161)(112 162)(113 136)(114 137)(115 138)(116 139)(117 140)(118 141)(119 142)(120 143)(121 144)(122 145)(123 146)(124 147)(125 148)(126 149)(127 150)(128 151)(129 152)(130 153)(131 154)(132 155)(133 156)(134 157)(135 158)
G:=sub<Sym(189)| (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), (1,162,70,105,185,112,49)(2,136,71,106,186,113,50)(3,137,72,107,187,114,51)(4,138,73,108,188,115,52)(5,139,74,82,189,116,53)(6,140,75,83,163,117,54)(7,141,76,84,164,118,28)(8,142,77,85,165,119,29)(9,143,78,86,166,120,30)(10,144,79,87,167,121,31)(11,145,80,88,168,122,32)(12,146,81,89,169,123,33)(13,147,55,90,170,124,34)(14,148,56,91,171,125,35)(15,149,57,92,172,126,36)(16,150,58,93,173,127,37)(17,151,59,94,174,128,38)(18,152,60,95,175,129,39)(19,153,61,96,176,130,40)(20,154,62,97,177,131,41)(21,155,63,98,178,132,42)(22,156,64,99,179,133,43)(23,157,65,100,180,134,44)(24,158,66,101,181,135,45)(25,159,67,102,182,109,46)(26,160,68,103,183,110,47)(27,161,69,104,184,111,48), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(55,170)(56,171)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,178)(64,179)(65,180)(66,181)(67,182)(68,183)(69,184)(70,185)(71,186)(72,187)(73,188)(74,189)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(109,159)(110,160)(111,161)(112,162)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(121,144)(122,145)(123,146)(124,147)(125,148)(126,149)(127,150)(128,151)(129,152)(130,153)(131,154)(132,155)(133,156)(134,157)(135,158)>;
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), (1,162,70,105,185,112,49)(2,136,71,106,186,113,50)(3,137,72,107,187,114,51)(4,138,73,108,188,115,52)(5,139,74,82,189,116,53)(6,140,75,83,163,117,54)(7,141,76,84,164,118,28)(8,142,77,85,165,119,29)(9,143,78,86,166,120,30)(10,144,79,87,167,121,31)(11,145,80,88,168,122,32)(12,146,81,89,169,123,33)(13,147,55,90,170,124,34)(14,148,56,91,171,125,35)(15,149,57,92,172,126,36)(16,150,58,93,173,127,37)(17,151,59,94,174,128,38)(18,152,60,95,175,129,39)(19,153,61,96,176,130,40)(20,154,62,97,177,131,41)(21,155,63,98,178,132,42)(22,156,64,99,179,133,43)(23,157,65,100,180,134,44)(24,158,66,101,181,135,45)(25,159,67,102,182,109,46)(26,160,68,103,183,110,47)(27,161,69,104,184,111,48), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(55,170)(56,171)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,178)(64,179)(65,180)(66,181)(67,182)(68,183)(69,184)(70,185)(71,186)(72,187)(73,188)(74,189)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(109,159)(110,160)(111,161)(112,162)(113,136)(114,137)(115,138)(116,139)(117,140)(118,141)(119,142)(120,143)(121,144)(122,145)(123,146)(124,147)(125,148)(126,149)(127,150)(128,151)(129,152)(130,153)(131,154)(132,155)(133,156)(134,157)(135,158) );
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)], [(1,162,70,105,185,112,49),(2,136,71,106,186,113,50),(3,137,72,107,187,114,51),(4,138,73,108,188,115,52),(5,139,74,82,189,116,53),(6,140,75,83,163,117,54),(7,141,76,84,164,118,28),(8,142,77,85,165,119,29),(9,143,78,86,166,120,30),(10,144,79,87,167,121,31),(11,145,80,88,168,122,32),(12,146,81,89,169,123,33),(13,147,55,90,170,124,34),(14,148,56,91,171,125,35),(15,149,57,92,172,126,36),(16,150,58,93,173,127,37),(17,151,59,94,174,128,38),(18,152,60,95,175,129,39),(19,153,61,96,176,130,40),(20,154,62,97,177,131,41),(21,155,63,98,178,132,42),(22,156,64,99,179,133,43),(23,157,65,100,180,134,44),(24,158,66,101,181,135,45),(25,159,67,102,182,109,46),(26,160,68,103,183,110,47),(27,161,69,104,184,111,48)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(55,170),(56,171),(57,172),(58,173),(59,174),(60,175),(61,176),(62,177),(63,178),(64,179),(65,180),(66,181),(67,182),(68,183),(69,184),(70,185),(71,186),(72,187),(73,188),(74,189),(75,163),(76,164),(77,165),(78,166),(79,167),(80,168),(81,169),(109,159),(110,160),(111,161),(112,162),(113,136),(114,137),(115,138),(116,139),(117,140),(118,141),(119,142),(120,143),(121,144),(122,145),(123,146),(124,147),(125,148),(126,149),(127,150),(128,151),(129,152),(130,153),(131,154),(132,155),(133,156),(134,157),(135,158)]])
135 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 7C | 9A | ··· | 9F | 18A | ··· | 18F | 21A | ··· | 21F | 27A | ··· | 27R | 54A | ··· | 54R | 63A | ··· | 63R | 189A | ··· | 189BB |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 7 | 7 | 9 | ··· | 9 | 18 | ··· | 18 | 21 | ··· | 21 | 27 | ··· | 27 | 54 | ··· | 54 | 63 | ··· | 63 | 189 | ··· | 189 |
| size | 1 | 7 | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 | D7 | C3×D7 | C9×D7 | D7×C27 |
| kernel | D7×C27 | C189 | C9×D7 | C63 | C3×D7 | C21 | D7 | C7 | C27 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 3 | 6 | 18 | 54 |
Matrix representation of D7×C27 ►in GL2(𝔽379) generated by
| 294 | 0 |
| 0 | 294 |
| 0 | 1 |
| 378 | 205 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(379))| [294,0,0,294],[0,378,1,205],[0,1,1,0] >;
D7×C27 in GAP, Magma, Sage, TeX
D_7\times C_{27} % in TeX
G:=Group("D7xC27"); // GroupNames label
G:=SmallGroup(378,4);
// by ID
G=gap.SmallGroup(378,4);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,36,57,8104]);
// Polycyclic
G:=Group<a,b,c|a^27=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export