direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C27, D4⋊C54, C8⋊1C54, C216⋊5C2, C72.7C6, C24.2C18, C54.14D4, C108.17C22, C3.(C9×D8), C9.(C3×D8), (C9×D8).C3, (C3×D8).C9, (D4×C27)⋊4C2, C4.1(C2×C54), (D4×C9).4C6, C6.14(D4×C9), C2.3(D4×C27), C36.40(C2×C6), (C3×D4).2C18, C18.30(C3×D4), C12.17(C2×C18), SmallGroup(432,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C27
G = < a,b,c | a27=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D8×C27
►On 216 points
Generators in S216
(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 78 45 105 179 207 162 123)(2 79 46 106 180 208 136 124)(3 80 47 107 181 209 137 125)(4 81 48 108 182 210 138 126)(5 55 49 82 183 211 139 127)(6 56 50 83 184 212 140 128)(7 57 51 84 185 213 141 129)(8 58 52 85 186 214 142 130)(9 59 53 86 187 215 143 131)(10 60 54 87 188 216 144 132)(11 61 28 88 189 190 145 133)(12 62 29 89 163 191 146 134)(13 63 30 90 164 192 147 135)(14 64 31 91 165 193 148 109)(15 65 32 92 166 194 149 110)(16 66 33 93 167 195 150 111)(17 67 34 94 168 196 151 112)(18 68 35 95 169 197 152 113)(19 69 36 96 170 198 153 114)(20 70 37 97 171 199 154 115)(21 71 38 98 172 200 155 116)(22 72 39 99 173 201 156 117)(23 73 40 100 174 202 157 118)(24 74 41 101 175 203 158 119)(25 75 42 102 176 204 159 120)(26 76 43 103 177 205 160 121)(27 77 44 104 178 206 161 122)
(28 145)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 157)(41 158)(42 159)(43 160)(44 161)(45 162)(46 136)(47 137)(48 138)(49 139)(50 140)(51 141)(52 142)(53 143)(54 144)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 211)(83 212)(84 213)(85 214)(86 215)(87 216)(88 190)(89 191)(90 192)(91 193)(92 194)(93 195)(94 196)(95 197)(96 198)(97 199)(98 200)(99 201)(100 202)(101 203)(102 204)(103 205)(104 206)(105 207)(106 208)(107 209)(108 210)
G:=sub<Sym(216)| (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), (1,78,45,105,179,207,162,123)(2,79,46,106,180,208,136,124)(3,80,47,107,181,209,137,125)(4,81,48,108,182,210,138,126)(5,55,49,82,183,211,139,127)(6,56,50,83,184,212,140,128)(7,57,51,84,185,213,141,129)(8,58,52,85,186,214,142,130)(9,59,53,86,187,215,143,131)(10,60,54,87,188,216,144,132)(11,61,28,88,189,190,145,133)(12,62,29,89,163,191,146,134)(13,63,30,90,164,192,147,135)(14,64,31,91,165,193,148,109)(15,65,32,92,166,194,149,110)(16,66,33,93,167,195,150,111)(17,67,34,94,168,196,151,112)(18,68,35,95,169,197,152,113)(19,69,36,96,170,198,153,114)(20,70,37,97,171,199,154,115)(21,71,38,98,172,200,155,116)(22,72,39,99,173,201,156,117)(23,73,40,100,174,202,157,118)(24,74,41,101,175,203,158,119)(25,75,42,102,176,204,159,120)(26,76,43,103,177,205,160,121)(27,77,44,104,178,206,161,122), (28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,157)(41,158)(42,159)(43,160)(44,161)(45,162)(46,136)(47,137)(48,138)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,211)(83,212)(84,213)(85,214)(86,215)(87,216)(88,190)(89,191)(90,192)(91,193)(92,194)(93,195)(94,196)(95,197)(96,198)(97,199)(98,200)(99,201)(100,202)(101,203)(102,204)(103,205)(104,206)(105,207)(106,208)(107,209)(108,210)>;
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)(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), (1,78,45,105,179,207,162,123)(2,79,46,106,180,208,136,124)(3,80,47,107,181,209,137,125)(4,81,48,108,182,210,138,126)(5,55,49,82,183,211,139,127)(6,56,50,83,184,212,140,128)(7,57,51,84,185,213,141,129)(8,58,52,85,186,214,142,130)(9,59,53,86,187,215,143,131)(10,60,54,87,188,216,144,132)(11,61,28,88,189,190,145,133)(12,62,29,89,163,191,146,134)(13,63,30,90,164,192,147,135)(14,64,31,91,165,193,148,109)(15,65,32,92,166,194,149,110)(16,66,33,93,167,195,150,111)(17,67,34,94,168,196,151,112)(18,68,35,95,169,197,152,113)(19,69,36,96,170,198,153,114)(20,70,37,97,171,199,154,115)(21,71,38,98,172,200,155,116)(22,72,39,99,173,201,156,117)(23,73,40,100,174,202,157,118)(24,74,41,101,175,203,158,119)(25,75,42,102,176,204,159,120)(26,76,43,103,177,205,160,121)(27,77,44,104,178,206,161,122), (28,145)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,157)(41,158)(42,159)(43,160)(44,161)(45,162)(46,136)(47,137)(48,138)(49,139)(50,140)(51,141)(52,142)(53,143)(54,144)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,121)(77,122)(78,123)(79,124)(80,125)(81,126)(82,211)(83,212)(84,213)(85,214)(86,215)(87,216)(88,190)(89,191)(90,192)(91,193)(92,194)(93,195)(94,196)(95,197)(96,198)(97,199)(98,200)(99,201)(100,202)(101,203)(102,204)(103,205)(104,206)(105,207)(106,208)(107,209)(108,210) );
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),(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)], [(1,78,45,105,179,207,162,123),(2,79,46,106,180,208,136,124),(3,80,47,107,181,209,137,125),(4,81,48,108,182,210,138,126),(5,55,49,82,183,211,139,127),(6,56,50,83,184,212,140,128),(7,57,51,84,185,213,141,129),(8,58,52,85,186,214,142,130),(9,59,53,86,187,215,143,131),(10,60,54,87,188,216,144,132),(11,61,28,88,189,190,145,133),(12,62,29,89,163,191,146,134),(13,63,30,90,164,192,147,135),(14,64,31,91,165,193,148,109),(15,65,32,92,166,194,149,110),(16,66,33,93,167,195,150,111),(17,67,34,94,168,196,151,112),(18,68,35,95,169,197,152,113),(19,69,36,96,170,198,153,114),(20,70,37,97,171,199,154,115),(21,71,38,98,172,200,155,116),(22,72,39,99,173,201,156,117),(23,73,40,100,174,202,157,118),(24,74,41,101,175,203,158,119),(25,75,42,102,176,204,159,120),(26,76,43,103,177,205,160,121),(27,77,44,104,178,206,161,122)], [(28,145),(29,146),(30,147),(31,148),(32,149),(33,150),(34,151),(35,152),(36,153),(37,154),(38,155),(39,156),(40,157),(41,158),(42,159),(43,160),(44,161),(45,162),(46,136),(47,137),(48,138),(49,139),(50,140),(51,141),(52,142),(53,143),(54,144),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,109),(65,110),(66,111),(67,112),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(76,121),(77,122),(78,123),(79,124),(80,125),(81,126),(82,211),(83,212),(84,213),(85,214),(86,215),(87,216),(88,190),(89,191),(90,192),(91,193),(92,194),(93,195),(94,196),(95,197),(96,198),(97,199),(98,200),(99,201),(100,202),(101,203),(102,204),(103,205),(104,206),(105,207),(106,208),(107,209),(108,210)]])
189 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 9A | ··· | 9F | 12A | 12B | 18A | ··· | 18F | 18G | ··· | 18R | 24A | 24B | 24C | 24D | 27A | ··· | 27R | 36A | ··· | 36F | 54A | ··· | 54R | 54S | ··· | 54BB | 72A | ··· | 72L | 108A | ··· | 108R | 216A | ··· | 216AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 54 | ··· | 54 | 72 | ··· | 72 | 108 | ··· | 108 | 216 | ··· | 216 |
| size | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
189 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | C9 | C18 | C18 | C27 | C54 | C54 | D4 | D8 | C3×D4 | C3×D8 | D4×C9 | C9×D8 | D4×C27 | D8×C27 |
| kernel | D8×C27 | C216 | D4×C27 | C9×D8 | C72 | D4×C9 | C3×D8 | C24 | C3×D4 | D8 | C8 | D4 | C54 | C27 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 6 | 12 | 18 | 18 | 36 | 1 | 2 | 2 | 4 | 6 | 12 | 18 | 36 |
Matrix representation of D8×C27 ►in GL2(𝔽433) generated by
| 374 | 0 |
| 0 | 374 |
| 265 | 393 |
| 261 | 374 |
| 1 | 0 |
| 138 | 432 |
G:=sub<GL(2,GF(433))| [374,0,0,374],[265,261,393,374],[1,138,0,432] >;
D8×C27 in GAP, Magma, Sage, TeX
D_8\times C_{27} % in TeX
G:=Group("D8xC27"); // GroupNames label
G:=SmallGroup(432,25);
// by ID
G=gap.SmallGroup(432,25);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,197,142,2355,1186,192,242]);
// Polycyclic
G:=Group<a,b,c|a^27=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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