direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D27, C27⋊C14, C189⋊2C2, C63.2S3, C21.2D9, C9.(S3×C7), C3.(C7×D9), SmallGroup(378,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C7×D27 |
Generators and relations for C7×D27
G = < a,b,c | a7=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D27
►On 189 points
Generators in S189
(1 163 146 109 92 58 48)(2 164 147 110 93 59 49)(3 165 148 111 94 60 50)(4 166 149 112 95 61 51)(5 167 150 113 96 62 52)(6 168 151 114 97 63 53)(7 169 152 115 98 64 54)(8 170 153 116 99 65 28)(9 171 154 117 100 66 29)(10 172 155 118 101 67 30)(11 173 156 119 102 68 31)(12 174 157 120 103 69 32)(13 175 158 121 104 70 33)(14 176 159 122 105 71 34)(15 177 160 123 106 72 35)(16 178 161 124 107 73 36)(17 179 162 125 108 74 37)(18 180 136 126 82 75 38)(19 181 137 127 83 76 39)(20 182 138 128 84 77 40)(21 183 139 129 85 78 41)(22 184 140 130 86 79 42)(23 185 141 131 87 80 43)(24 186 142 132 88 81 44)(25 187 143 133 89 55 45)(26 188 144 134 90 56 46)(27 189 145 135 91 57 47)
(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)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(42 54)(43 53)(44 52)(45 51)(46 50)(47 49)(55 61)(56 60)(57 59)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(82 102)(83 101)(84 100)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(103 108)(104 107)(105 106)(110 135)(111 134)(112 133)(113 132)(114 131)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)(121 124)(122 123)(136 156)(137 155)(138 154)(139 153)(140 152)(141 151)(142 150)(143 149)(144 148)(145 147)(157 162)(158 161)(159 160)(164 189)(165 188)(166 187)(167 186)(168 185)(169 184)(170 183)(171 182)(172 181)(173 180)(174 179)(175 178)(176 177)
G:=sub<Sym(189)| (1,163,146,109,92,58,48)(2,164,147,110,93,59,49)(3,165,148,111,94,60,50)(4,166,149,112,95,61,51)(5,167,150,113,96,62,52)(6,168,151,114,97,63,53)(7,169,152,115,98,64,54)(8,170,153,116,99,65,28)(9,171,154,117,100,66,29)(10,172,155,118,101,67,30)(11,173,156,119,102,68,31)(12,174,157,120,103,69,32)(13,175,158,121,104,70,33)(14,176,159,122,105,71,34)(15,177,160,123,106,72,35)(16,178,161,124,107,73,36)(17,179,162,125,108,74,37)(18,180,136,126,82,75,38)(19,181,137,127,83,76,39)(20,182,138,128,84,77,40)(21,183,139,129,85,78,41)(22,184,140,130,86,79,42)(23,185,141,131,87,80,43)(24,186,142,132,88,81,44)(25,187,143,133,89,55,45)(26,188,144,134,90,56,46)(27,189,145,135,91,57,47), (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), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(55,61)(56,60)(57,59)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(103,108)(104,107)(105,106)(110,135)(111,134)(112,133)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)(136,156)(137,155)(138,154)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)(145,147)(157,162)(158,161)(159,160)(164,189)(165,188)(166,187)(167,186)(168,185)(169,184)(170,183)(171,182)(172,181)(173,180)(174,179)(175,178)(176,177)>;
G:=Group( (1,163,146,109,92,58,48)(2,164,147,110,93,59,49)(3,165,148,111,94,60,50)(4,166,149,112,95,61,51)(5,167,150,113,96,62,52)(6,168,151,114,97,63,53)(7,169,152,115,98,64,54)(8,170,153,116,99,65,28)(9,171,154,117,100,66,29)(10,172,155,118,101,67,30)(11,173,156,119,102,68,31)(12,174,157,120,103,69,32)(13,175,158,121,104,70,33)(14,176,159,122,105,71,34)(15,177,160,123,106,72,35)(16,178,161,124,107,73,36)(17,179,162,125,108,74,37)(18,180,136,126,82,75,38)(19,181,137,127,83,76,39)(20,182,138,128,84,77,40)(21,183,139,129,85,78,41)(22,184,140,130,86,79,42)(23,185,141,131,87,80,43)(24,186,142,132,88,81,44)(25,187,143,133,89,55,45)(26,188,144,134,90,56,46)(27,189,145,135,91,57,47), (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), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(42,54)(43,53)(44,52)(45,51)(46,50)(47,49)(55,61)(56,60)(57,59)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(82,102)(83,101)(84,100)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(103,108)(104,107)(105,106)(110,135)(111,134)(112,133)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)(136,156)(137,155)(138,154)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)(145,147)(157,162)(158,161)(159,160)(164,189)(165,188)(166,187)(167,186)(168,185)(169,184)(170,183)(171,182)(172,181)(173,180)(174,179)(175,178)(176,177) );
G=PermutationGroup([[(1,163,146,109,92,58,48),(2,164,147,110,93,59,49),(3,165,148,111,94,60,50),(4,166,149,112,95,61,51),(5,167,150,113,96,62,52),(6,168,151,114,97,63,53),(7,169,152,115,98,64,54),(8,170,153,116,99,65,28),(9,171,154,117,100,66,29),(10,172,155,118,101,67,30),(11,173,156,119,102,68,31),(12,174,157,120,103,69,32),(13,175,158,121,104,70,33),(14,176,159,122,105,71,34),(15,177,160,123,106,72,35),(16,178,161,124,107,73,36),(17,179,162,125,108,74,37),(18,180,136,126,82,75,38),(19,181,137,127,83,76,39),(20,182,138,128,84,77,40),(21,183,139,129,85,78,41),(22,184,140,130,86,79,42),(23,185,141,131,87,80,43),(24,186,142,132,88,81,44),(25,187,143,133,89,55,45),(26,188,144,134,90,56,46),(27,189,145,135,91,57,47)], [(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)], [(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(28,41),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(42,54),(43,53),(44,52),(45,51),(46,50),(47,49),(55,61),(56,60),(57,59),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(82,102),(83,101),(84,100),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(103,108),(104,107),(105,106),(110,135),(111,134),(112,133),(113,132),(114,131),(115,130),(116,129),(117,128),(118,127),(119,126),(120,125),(121,124),(122,123),(136,156),(137,155),(138,154),(139,153),(140,152),(141,151),(142,150),(143,149),(144,148),(145,147),(157,162),(158,161),(159,160),(164,189),(165,188),(166,187),(167,186),(168,185),(169,184),(170,183),(171,182),(172,181),(173,180),(174,179),(175,178),(176,177)]])
105 conjugacy classes
| class | 1 | 2 | 3 | 7A | ··· | 7F | 9A | 9B | 9C | 14A | ··· | 14F | 21A | ··· | 21F | 27A | ··· | 27I | 63A | ··· | 63R | 189A | ··· | 189BB |
| order | 1 | 2 | 3 | 7 | ··· | 7 | 9 | 9 | 9 | 14 | ··· | 14 | 21 | ··· | 21 | 27 | ··· | 27 | 63 | ··· | 63 | 189 | ··· | 189 |
| size | 1 | 27 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 27 | ··· | 27 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C7 | C14 | S3 | D9 | S3×C7 | D27 | C7×D9 | C7×D27 |
| kernel | C7×D27 | C189 | D27 | C27 | C63 | C21 | C9 | C7 | C3 | C1 |
| # reps | 1 | 1 | 6 | 6 | 1 | 3 | 6 | 9 | 18 | 54 |
Matrix representation of C7×D27 ►in GL2(𝔽379) generated by
| 125 | 0 |
| 0 | 125 |
| 30 | 25 |
| 354 | 55 |
| 183 | 153 |
| 336 | 196 |
G:=sub<GL(2,GF(379))| [125,0,0,125],[30,354,25,55],[183,336,153,196] >;
C7×D27 in GAP, Magma, Sage, TeX
C_7\times D_{27} % in TeX
G:=Group("C7xD27"); // GroupNames label
G:=SmallGroup(378,3);
// by ID
G=gap.SmallGroup(378,3);
# by ID
G:=PCGroup([5,-2,-7,-3,-3,-3,1052,237,4203,138,6304]);
// Polycyclic
G:=Group<a,b,c|a^7=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export