direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×D27, C216⋊3C2, C72.6S3, C24.7D9, D54.2C4, C4.13D54, C36.60D6, C12.60D18, Dic27.2C4, C108.13C22, C9.(S3×C8), C27⋊C8⋊6C2, C3.(C8×D9), C27⋊1(C2×C8), C6.5(C4×D9), C18.6(C4×S3), C54.1(C2×C4), C2.1(C4×D27), (C4×D27).3C2, SmallGroup(432,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C8×D27 |
Generators and relations for C8×D27
G = < a,b,c | a8=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C8×D27
►On 216 points
Generators in S216
(1 211 103 136 49 164 75 121)(2 212 104 137 50 165 76 122)(3 213 105 138 51 166 77 123)(4 214 106 139 52 167 78 124)(5 215 107 140 53 168 79 125)(6 216 108 141 54 169 80 126)(7 190 82 142 28 170 81 127)(8 191 83 143 29 171 55 128)(9 192 84 144 30 172 56 129)(10 193 85 145 31 173 57 130)(11 194 86 146 32 174 58 131)(12 195 87 147 33 175 59 132)(13 196 88 148 34 176 60 133)(14 197 89 149 35 177 61 134)(15 198 90 150 36 178 62 135)(16 199 91 151 37 179 63 109)(17 200 92 152 38 180 64 110)(18 201 93 153 39 181 65 111)(19 202 94 154 40 182 66 112)(20 203 95 155 41 183 67 113)(21 204 96 156 42 184 68 114)(22 205 97 157 43 185 69 115)(23 206 98 158 44 186 70 116)(24 207 99 159 45 187 71 117)(25 208 100 160 46 188 72 118)(26 209 101 161 47 189 73 119)(27 210 102 162 48 163 74 120)
(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)(190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)
(1 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 54)(24 53)(25 52)(26 51)(27 50)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(109 148)(110 147)(111 146)(112 145)(113 144)(114 143)(115 142)(116 141)(117 140)(118 139)(119 138)(120 137)(121 136)(122 162)(123 161)(124 160)(125 159)(126 158)(127 157)(128 156)(129 155)(130 154)(131 153)(132 152)(133 151)(134 150)(135 149)(163 212)(164 211)(165 210)(166 209)(167 208)(168 207)(169 206)(170 205)(171 204)(172 203)(173 202)(174 201)(175 200)(176 199)(177 198)(178 197)(179 196)(180 195)(181 194)(182 193)(183 192)(184 191)(185 190)(186 216)(187 215)(188 214)(189 213)
G:=sub<Sym(216)| (1,211,103,136,49,164,75,121)(2,212,104,137,50,165,76,122)(3,213,105,138,51,166,77,123)(4,214,106,139,52,167,78,124)(5,215,107,140,53,168,79,125)(6,216,108,141,54,169,80,126)(7,190,82,142,28,170,81,127)(8,191,83,143,29,171,55,128)(9,192,84,144,30,172,56,129)(10,193,85,145,31,173,57,130)(11,194,86,146,32,174,58,131)(12,195,87,147,33,175,59,132)(13,196,88,148,34,176,60,133)(14,197,89,149,35,177,61,134)(15,198,90,150,36,178,62,135)(16,199,91,151,37,179,63,109)(17,200,92,152,38,180,64,110)(18,201,93,153,39,181,65,111)(19,202,94,154,40,182,66,112)(20,203,95,155,41,183,67,113)(21,204,96,156,42,184,68,114)(22,205,97,157,43,185,69,115)(23,206,98,158,44,186,70,116)(24,207,99,159,45,187,71,117)(25,208,100,160,46,188,72,118)(26,209,101,161,47,189,73,119)(27,210,102,162,48,163,74,120), (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)(190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,54)(24,53)(25,52)(26,51)(27,50)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,140)(118,139)(119,138)(120,137)(121,136)(122,162)(123,161)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)(130,154)(131,153)(132,152)(133,151)(134,150)(135,149)(163,212)(164,211)(165,210)(166,209)(167,208)(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193)(183,192)(184,191)(185,190)(186,216)(187,215)(188,214)(189,213)>;
G:=Group( (1,211,103,136,49,164,75,121)(2,212,104,137,50,165,76,122)(3,213,105,138,51,166,77,123)(4,214,106,139,52,167,78,124)(5,215,107,140,53,168,79,125)(6,216,108,141,54,169,80,126)(7,190,82,142,28,170,81,127)(8,191,83,143,29,171,55,128)(9,192,84,144,30,172,56,129)(10,193,85,145,31,173,57,130)(11,194,86,146,32,174,58,131)(12,195,87,147,33,175,59,132)(13,196,88,148,34,176,60,133)(14,197,89,149,35,177,61,134)(15,198,90,150,36,178,62,135)(16,199,91,151,37,179,63,109)(17,200,92,152,38,180,64,110)(18,201,93,153,39,181,65,111)(19,202,94,154,40,182,66,112)(20,203,95,155,41,183,67,113)(21,204,96,156,42,184,68,114)(22,205,97,157,43,185,69,115)(23,206,98,158,44,186,70,116)(24,207,99,159,45,187,71,117)(25,208,100,160,46,188,72,118)(26,209,101,161,47,189,73,119)(27,210,102,162,48,163,74,120), (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)(190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,54)(24,53)(25,52)(26,51)(27,50)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(109,148)(110,147)(111,146)(112,145)(113,144)(114,143)(115,142)(116,141)(117,140)(118,139)(119,138)(120,137)(121,136)(122,162)(123,161)(124,160)(125,159)(126,158)(127,157)(128,156)(129,155)(130,154)(131,153)(132,152)(133,151)(134,150)(135,149)(163,212)(164,211)(165,210)(166,209)(167,208)(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193)(183,192)(184,191)(185,190)(186,216)(187,215)(188,214)(189,213) );
G=PermutationGroup([[(1,211,103,136,49,164,75,121),(2,212,104,137,50,165,76,122),(3,213,105,138,51,166,77,123),(4,214,106,139,52,167,78,124),(5,215,107,140,53,168,79,125),(6,216,108,141,54,169,80,126),(7,190,82,142,28,170,81,127),(8,191,83,143,29,171,55,128),(9,192,84,144,30,172,56,129),(10,193,85,145,31,173,57,130),(11,194,86,146,32,174,58,131),(12,195,87,147,33,175,59,132),(13,196,88,148,34,176,60,133),(14,197,89,149,35,177,61,134),(15,198,90,150,36,178,62,135),(16,199,91,151,37,179,63,109),(17,200,92,152,38,180,64,110),(18,201,93,153,39,181,65,111),(19,202,94,154,40,182,66,112),(20,203,95,155,41,183,67,113),(21,204,96,156,42,184,68,114),(22,205,97,157,43,185,69,115),(23,206,98,158,44,186,70,116),(24,207,99,159,45,187,71,117),(25,208,100,160,46,188,72,118),(26,209,101,161,47,189,73,119),(27,210,102,162,48,163,74,120)], [(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),(190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,28),(23,54),(24,53),(25,52),(26,51),(27,50),(55,96),(56,95),(57,94),(58,93),(59,92),(60,91),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,108),(71,107),(72,106),(73,105),(74,104),(75,103),(76,102),(77,101),(78,100),(79,99),(80,98),(81,97),(109,148),(110,147),(111,146),(112,145),(113,144),(114,143),(115,142),(116,141),(117,140),(118,139),(119,138),(120,137),(121,136),(122,162),(123,161),(124,160),(125,159),(126,158),(127,157),(128,156),(129,155),(130,154),(131,153),(132,152),(133,151),(134,150),(135,149),(163,212),(164,211),(165,210),(166,209),(167,208),(168,207),(169,206),(170,205),(171,204),(172,203),(173,202),(174,201),(175,200),(176,199),(177,198),(178,197),(179,196),(180,195),(181,194),(182,193),(183,192),(184,191),(185,190),(186,216),(187,215),(188,214),(189,213)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 27A | ··· | 27I | 36A | ··· | 36F | 54A | ··· | 54I | 72A | ··· | 72L | 108A | ··· | 108R | 216A | ··· | 216AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 72 | ··· | 72 | 108 | ··· | 108 | 216 | ··· | 216 |
| size | 1 | 1 | 27 | 27 | 2 | 1 | 1 | 27 | 27 | 2 | 1 | 1 | 1 | 1 | 27 | 27 | 27 | 27 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D6 | D9 | C4×S3 | D18 | S3×C8 | D27 | C4×D9 | D54 | C8×D9 | C4×D27 | C8×D27 |
| kernel | C8×D27 | C27⋊C8 | C216 | C4×D27 | Dic27 | D54 | D27 | C72 | C36 | C24 | C18 | C12 | C9 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 3 | 2 | 3 | 4 | 9 | 6 | 9 | 12 | 18 | 36 |
Matrix representation of C8×D27 ►in GL4(𝔽433) generated by
| 148 | 0 | 0 | 0 |
| 0 | 148 | 0 | 0 |
| 0 | 0 | 179 | 0 |
| 0 | 0 | 0 | 179 |
| 10 | 1 | 0 | 0 |
| 322 | 422 | 0 | 0 |
| 0 | 0 | 271 | 130 |
| 0 | 0 | 303 | 401 |
| 422 | 432 | 0 | 0 |
| 120 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(433))| [148,0,0,0,0,148,0,0,0,0,179,0,0,0,0,179],[10,322,0,0,1,422,0,0,0,0,271,303,0,0,130,401],[422,120,0,0,432,11,0,0,0,0,0,1,0,0,1,0] >;
C8×D27 in GAP, Magma, Sage, TeX
C_8\times D_{27} % in TeX
G:=Group("C8xD27"); // GroupNames label
G:=SmallGroup(432,5);
// by ID
G=gap.SmallGroup(432,5);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,36,58,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^8=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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