metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D27, D54.C4, C216⋊4C2, C72.7S3, C24.8D9, C4.14D54, C36.61D6, Dic27.C4, C27⋊1M4(2), C12.61D18, C108.14C22, C27⋊C8⋊4C2, C6.6(C4×D9), C9.(C8⋊S3), C18.7(C4×S3), C54.2(C2×C4), C3.(C8⋊D9), C2.3(C4×D27), (C4×D27).2C2, SmallGroup(432,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊D27
G = < a,b,c | a8=b27=c2=1, ab=ba, cac=a5, cbc=b-1 >
Smallest permutation representation of C8⋊D27
►On 216 points
Generators in S216
(1 200 83 152 29 170 64 124)(2 201 84 153 30 171 65 125)(3 202 85 154 31 172 66 126)(4 203 86 155 32 173 67 127)(5 204 87 156 33 174 68 128)(6 205 88 157 34 175 69 129)(7 206 89 158 35 176 70 130)(8 207 90 159 36 177 71 131)(9 208 91 160 37 178 72 132)(10 209 92 161 38 179 73 133)(11 210 93 162 39 180 74 134)(12 211 94 136 40 181 75 135)(13 212 95 137 41 182 76 109)(14 213 96 138 42 183 77 110)(15 214 97 139 43 184 78 111)(16 215 98 140 44 185 79 112)(17 216 99 141 45 186 80 113)(18 190 100 142 46 187 81 114)(19 191 101 143 47 188 55 115)(20 192 102 144 48 189 56 116)(21 193 103 145 49 163 57 117)(22 194 104 146 50 164 58 118)(23 195 105 147 51 165 59 119)(24 196 106 148 52 166 60 120)(25 197 107 149 53 167 61 121)(26 198 108 150 54 168 62 122)(27 199 82 151 28 169 63 123)
(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)
(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 30)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(74 81)(75 80)(76 79)(77 78)(82 84)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(109 140)(110 139)(111 138)(112 137)(113 136)(114 162)(115 161)(116 160)(117 159)(118 158)(119 157)(120 156)(121 155)(122 154)(123 153)(124 152)(125 151)(126 150)(127 149)(128 148)(129 147)(130 146)(131 145)(132 144)(133 143)(134 142)(135 141)(163 207)(164 206)(165 205)(166 204)(167 203)(168 202)(169 201)(170 200)(171 199)(172 198)(173 197)(174 196)(175 195)(176 194)(177 193)(178 192)(179 191)(180 190)(181 216)(182 215)(183 214)(184 213)(185 212)(186 211)(187 210)(188 209)(189 208)
G:=sub<Sym(216)| (1,200,83,152,29,170,64,124)(2,201,84,153,30,171,65,125)(3,202,85,154,31,172,66,126)(4,203,86,155,32,173,67,127)(5,204,87,156,33,174,68,128)(6,205,88,157,34,175,69,129)(7,206,89,158,35,176,70,130)(8,207,90,159,36,177,71,131)(9,208,91,160,37,178,72,132)(10,209,92,161,38,179,73,133)(11,210,93,162,39,180,74,134)(12,211,94,136,40,181,75,135)(13,212,95,137,41,182,76,109)(14,213,96,138,42,183,77,110)(15,214,97,139,43,184,78,111)(16,215,98,140,44,185,79,112)(17,216,99,141,45,186,80,113)(18,190,100,142,46,187,81,114)(19,191,101,143,47,188,55,115)(20,192,102,144,48,189,56,116)(21,193,103,145,49,163,57,117)(22,194,104,146,50,164,58,118)(23,195,105,147,51,165,59,119)(24,196,106,148,52,166,60,120)(25,197,107,149,53,167,61,121)(26,198,108,150,54,168,62,122)(27,199,82,151,28,169,63,123), (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), (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)(28,30)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(74,81)(75,80)(76,79)(77,78)(82,84)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(109,140)(110,139)(111,138)(112,137)(113,136)(114,162)(115,161)(116,160)(117,159)(118,158)(119,157)(120,156)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)(169,201)(170,200)(171,199)(172,198)(173,197)(174,196)(175,195)(176,194)(177,193)(178,192)(179,191)(180,190)(181,216)(182,215)(183,214)(184,213)(185,212)(186,211)(187,210)(188,209)(189,208)>;
G:=Group( (1,200,83,152,29,170,64,124)(2,201,84,153,30,171,65,125)(3,202,85,154,31,172,66,126)(4,203,86,155,32,173,67,127)(5,204,87,156,33,174,68,128)(6,205,88,157,34,175,69,129)(7,206,89,158,35,176,70,130)(8,207,90,159,36,177,71,131)(9,208,91,160,37,178,72,132)(10,209,92,161,38,179,73,133)(11,210,93,162,39,180,74,134)(12,211,94,136,40,181,75,135)(13,212,95,137,41,182,76,109)(14,213,96,138,42,183,77,110)(15,214,97,139,43,184,78,111)(16,215,98,140,44,185,79,112)(17,216,99,141,45,186,80,113)(18,190,100,142,46,187,81,114)(19,191,101,143,47,188,55,115)(20,192,102,144,48,189,56,116)(21,193,103,145,49,163,57,117)(22,194,104,146,50,164,58,118)(23,195,105,147,51,165,59,119)(24,196,106,148,52,166,60,120)(25,197,107,149,53,167,61,121)(26,198,108,150,54,168,62,122)(27,199,82,151,28,169,63,123), (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), (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)(28,30)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(74,81)(75,80)(76,79)(77,78)(82,84)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(109,140)(110,139)(111,138)(112,137)(113,136)(114,162)(115,161)(116,160)(117,159)(118,158)(119,157)(120,156)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)(169,201)(170,200)(171,199)(172,198)(173,197)(174,196)(175,195)(176,194)(177,193)(178,192)(179,191)(180,190)(181,216)(182,215)(183,214)(184,213)(185,212)(186,211)(187,210)(188,209)(189,208) );
G=PermutationGroup([[(1,200,83,152,29,170,64,124),(2,201,84,153,30,171,65,125),(3,202,85,154,31,172,66,126),(4,203,86,155,32,173,67,127),(5,204,87,156,33,174,68,128),(6,205,88,157,34,175,69,129),(7,206,89,158,35,176,70,130),(8,207,90,159,36,177,71,131),(9,208,91,160,37,178,72,132),(10,209,92,161,38,179,73,133),(11,210,93,162,39,180,74,134),(12,211,94,136,40,181,75,135),(13,212,95,137,41,182,76,109),(14,213,96,138,42,183,77,110),(15,214,97,139,43,184,78,111),(16,215,98,140,44,185,79,112),(17,216,99,141,45,186,80,113),(18,190,100,142,46,187,81,114),(19,191,101,143,47,188,55,115),(20,192,102,144,48,189,56,116),(21,193,103,145,49,163,57,117),(22,194,104,146,50,164,58,118),(23,195,105,147,51,165,59,119),(24,196,106,148,52,166,60,120),(25,197,107,149,53,167,61,121),(26,198,108,150,54,168,62,122),(27,199,82,151,28,169,63,123)], [(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)], [(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),(28,30),(31,54),(32,53),(33,52),(34,51),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43),(55,73),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(74,81),(75,80),(76,79),(77,78),(82,84),(85,108),(86,107),(87,106),(88,105),(89,104),(90,103),(91,102),(92,101),(93,100),(94,99),(95,98),(96,97),(109,140),(110,139),(111,138),(112,137),(113,136),(114,162),(115,161),(116,160),(117,159),(118,158),(119,157),(120,156),(121,155),(122,154),(123,153),(124,152),(125,151),(126,150),(127,149),(128,148),(129,147),(130,146),(131,145),(132,144),(133,143),(134,142),(135,141),(163,207),(164,206),(165,205),(166,204),(167,203),(168,202),(169,201),(170,200),(171,199),(172,198),(173,197),(174,196),(175,195),(176,194),(177,193),(178,192),(179,191),(180,190),(181,216),(182,215),(183,214),(184,213),(185,212),(186,211),(187,210),(188,209),(189,208)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 8C | 8D | 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 | 3 | 4 | 4 | 4 | 6 | 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 | 54 | 2 | 1 | 1 | 54 | 2 | 2 | 2 | 54 | 54 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | M4(2) | D9 | C4×S3 | D18 | C8⋊S3 | D27 | C4×D9 | D54 | C8⋊D9 | C4×D27 | C8⋊D27 |
| kernel | C8⋊D27 | C27⋊C8 | C216 | C4×D27 | Dic27 | D54 | C72 | C36 | C27 | C24 | C18 | C12 | C9 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 3 | 2 | 3 | 4 | 9 | 6 | 9 | 12 | 18 | 36 |
Matrix representation of C8⋊D27 ►in GL4(𝔽433) generated by
| 179 | 0 | 0 | 0 |
| 0 | 179 | 0 | 0 |
| 0 | 0 | 36 | 169 |
| 0 | 0 | 145 | 397 |
| 263 | 25 | 0 | 0 |
| 408 | 238 | 0 | 0 |
| 0 | 0 | 315 | 184 |
| 0 | 0 | 122 | 129 |
| 1 | 0 | 0 | 0 |
| 432 | 432 | 0 | 0 |
| 0 | 0 | 118 | 249 |
| 0 | 0 | 351 | 315 |
G:=sub<GL(4,GF(433))| [179,0,0,0,0,179,0,0,0,0,36,145,0,0,169,397],[263,408,0,0,25,238,0,0,0,0,315,122,0,0,184,129],[1,432,0,0,0,432,0,0,0,0,118,351,0,0,249,315] >;
C8⋊D27 in GAP, Magma, Sage, TeX
C_8\rtimes D_{27} % in TeX
G:=Group("C8:D27"); // GroupNames label
G:=SmallGroup(432,6);
// by ID
G=gap.SmallGroup(432,6);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,58,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^8=b^27=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations
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