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## G = D8×C27order 432 = 24·33

### Direct product of C27 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — D8×C27
 Chief series C1 — C3 — C6 — C18 — C36 — C108 — D4×C27 — D8×C27
 Lower central C1 — C2 — C4 — D8×C27
 Upper central C1 — C54 — C108 — D8×C27

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)
(1 188 152 204 74 39 86 135)(2 189 153 205 75 40 87 109)(3 163 154 206 76 41 88 110)(4 164 155 207 77 42 89 111)(5 165 156 208 78 43 90 112)(6 166 157 209 79 44 91 113)(7 167 158 210 80 45 92 114)(8 168 159 211 81 46 93 115)(9 169 160 212 55 47 94 116)(10 170 161 213 56 48 95 117)(11 171 162 214 57 49 96 118)(12 172 136 215 58 50 97 119)(13 173 137 216 59 51 98 120)(14 174 138 190 60 52 99 121)(15 175 139 191 61 53 100 122)(16 176 140 192 62 54 101 123)(17 177 141 193 63 28 102 124)(18 178 142 194 64 29 103 125)(19 179 143 195 65 30 104 126)(20 180 144 196 66 31 105 127)(21 181 145 197 67 32 106 128)(22 182 146 198 68 33 107 129)(23 183 147 199 69 34 108 130)(24 184 148 200 70 35 82 131)(25 185 149 201 71 36 83 132)(26 186 150 202 72 37 84 133)(27 187 151 203 73 38 85 134)
(28 193)(29 194)(30 195)(31 196)(32 197)(33 198)(34 199)(35 200)(36 201)(37 202)(38 203)(39 204)(40 205)(41 206)(42 207)(43 208)(44 209)(45 210)(46 211)(47 212)(48 213)(49 214)(50 215)(51 216)(52 190)(53 191)(54 192)(82 148)(83 149)(84 150)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)(91 157)(92 158)(93 159)(94 160)(95 161)(96 162)(97 136)(98 137)(99 138)(100 139)(101 140)(102 141)(103 142)(104 143)(105 144)(106 145)(107 146)(108 147)(109 189)(110 163)(111 164)(112 165)(113 166)(114 167)(115 168)(116 169)(117 170)(118 171)(119 172)(120 173)(121 174)(122 175)(123 176)(124 177)(125 178)(126 179)(127 180)(128 181)(129 182)(130 183)(131 184)(132 185)(133 186)(134 187)(135 188)

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), (1,188,152,204,74,39,86,135)(2,189,153,205,75,40,87,109)(3,163,154,206,76,41,88,110)(4,164,155,207,77,42,89,111)(5,165,156,208,78,43,90,112)(6,166,157,209,79,44,91,113)(7,167,158,210,80,45,92,114)(8,168,159,211,81,46,93,115)(9,169,160,212,55,47,94,116)(10,170,161,213,56,48,95,117)(11,171,162,214,57,49,96,118)(12,172,136,215,58,50,97,119)(13,173,137,216,59,51,98,120)(14,174,138,190,60,52,99,121)(15,175,139,191,61,53,100,122)(16,176,140,192,62,54,101,123)(17,177,141,193,63,28,102,124)(18,178,142,194,64,29,103,125)(19,179,143,195,65,30,104,126)(20,180,144,196,66,31,105,127)(21,181,145,197,67,32,106,128)(22,182,146,198,68,33,107,129)(23,183,147,199,69,34,108,130)(24,184,148,200,70,35,82,131)(25,185,149,201,71,36,83,132)(26,186,150,202,72,37,84,133)(27,187,151,203,73,38,85,134), (28,193)(29,194)(30,195)(31,196)(32,197)(33,198)(34,199)(35,200)(36,201)(37,202)(38,203)(39,204)(40,205)(41,206)(42,207)(43,208)(44,209)(45,210)(46,211)(47,212)(48,213)(49,214)(50,215)(51,216)(52,190)(53,191)(54,192)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,189)(110,163)(111,164)(112,165)(113,166)(114,167)(115,168)(116,169)(117,170)(118,171)(119,172)(120,173)(121,174)(122,175)(123,176)(124,177)(125,178)(126,179)(127,180)(128,181)(129,182)(130,183)(131,184)(132,185)(133,186)(134,187)(135,188)>;

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), (1,188,152,204,74,39,86,135)(2,189,153,205,75,40,87,109)(3,163,154,206,76,41,88,110)(4,164,155,207,77,42,89,111)(5,165,156,208,78,43,90,112)(6,166,157,209,79,44,91,113)(7,167,158,210,80,45,92,114)(8,168,159,211,81,46,93,115)(9,169,160,212,55,47,94,116)(10,170,161,213,56,48,95,117)(11,171,162,214,57,49,96,118)(12,172,136,215,58,50,97,119)(13,173,137,216,59,51,98,120)(14,174,138,190,60,52,99,121)(15,175,139,191,61,53,100,122)(16,176,140,192,62,54,101,123)(17,177,141,193,63,28,102,124)(18,178,142,194,64,29,103,125)(19,179,143,195,65,30,104,126)(20,180,144,196,66,31,105,127)(21,181,145,197,67,32,106,128)(22,182,146,198,68,33,107,129)(23,183,147,199,69,34,108,130)(24,184,148,200,70,35,82,131)(25,185,149,201,71,36,83,132)(26,186,150,202,72,37,84,133)(27,187,151,203,73,38,85,134), (28,193)(29,194)(30,195)(31,196)(32,197)(33,198)(34,199)(35,200)(36,201)(37,202)(38,203)(39,204)(40,205)(41,206)(42,207)(43,208)(44,209)(45,210)(46,211)(47,212)(48,213)(49,214)(50,215)(51,216)(52,190)(53,191)(54,192)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,189)(110,163)(111,164)(112,165)(113,166)(114,167)(115,168)(116,169)(117,170)(118,171)(119,172)(120,173)(121,174)(122,175)(123,176)(124,177)(125,178)(126,179)(127,180)(128,181)(129,182)(130,183)(131,184)(132,185)(133,186)(134,187)(135,188) );

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)], [(1,188,152,204,74,39,86,135),(2,189,153,205,75,40,87,109),(3,163,154,206,76,41,88,110),(4,164,155,207,77,42,89,111),(5,165,156,208,78,43,90,112),(6,166,157,209,79,44,91,113),(7,167,158,210,80,45,92,114),(8,168,159,211,81,46,93,115),(9,169,160,212,55,47,94,116),(10,170,161,213,56,48,95,117),(11,171,162,214,57,49,96,118),(12,172,136,215,58,50,97,119),(13,173,137,216,59,51,98,120),(14,174,138,190,60,52,99,121),(15,175,139,191,61,53,100,122),(16,176,140,192,62,54,101,123),(17,177,141,193,63,28,102,124),(18,178,142,194,64,29,103,125),(19,179,143,195,65,30,104,126),(20,180,144,196,66,31,105,127),(21,181,145,197,67,32,106,128),(22,182,146,198,68,33,107,129),(23,183,147,199,69,34,108,130),(24,184,148,200,70,35,82,131),(25,185,149,201,71,36,83,132),(26,186,150,202,72,37,84,133),(27,187,151,203,73,38,85,134)], [(28,193),(29,194),(30,195),(31,196),(32,197),(33,198),(34,199),(35,200),(36,201),(37,202),(38,203),(39,204),(40,205),(41,206),(42,207),(43,208),(44,209),(45,210),(46,211),(47,212),(48,213),(49,214),(50,215),(51,216),(52,190),(53,191),(54,192),(82,148),(83,149),(84,150),(85,151),(86,152),(87,153),(88,154),(89,155),(90,156),(91,157),(92,158),(93,159),(94,160),(95,161),(96,162),(97,136),(98,137),(99,138),(100,139),(101,140),(102,141),(103,142),(104,143),(105,144),(106,145),(107,146),(108,147),(109,189),(110,163),(111,164),(112,165),(113,166),(114,167),(115,168),(116,169),(117,170),(118,171),(119,172),(120,173),(121,174),(122,175),(123,176),(124,177),(125,178),(126,179),(127,180),(128,181),(129,182),(130,183),(131,184),(132,185),(133,186),(134,187),(135,188)])

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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