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## G = SD16×C27order 432 = 24·33

### Direct product of C27 and SD16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — SD16×C27
 Chief series C1 — C3 — C6 — C18 — C36 — C108 — Q8×C27 — SD16×C27
 Lower central C1 — C2 — C4 — SD16×C27
 Upper central C1 — C54 — C108 — SD16×C27

Generators and relations for SD16×C27
G = < a,b,c | a27=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Smallest permutation representation of SD16×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 32 149 189 205 119 81 104)(2 33 150 163 206 120 55 105)(3 34 151 164 207 121 56 106)(4 35 152 165 208 122 57 107)(5 36 153 166 209 123 58 108)(6 37 154 167 210 124 59 82)(7 38 155 168 211 125 60 83)(8 39 156 169 212 126 61 84)(9 40 157 170 213 127 62 85)(10 41 158 171 214 128 63 86)(11 42 159 172 215 129 64 87)(12 43 160 173 216 130 65 88)(13 44 161 174 190 131 66 89)(14 45 162 175 191 132 67 90)(15 46 136 176 192 133 68 91)(16 47 137 177 193 134 69 92)(17 48 138 178 194 135 70 93)(18 49 139 179 195 109 71 94)(19 50 140 180 196 110 72 95)(20 51 141 181 197 111 73 96)(21 52 142 182 198 112 74 97)(22 53 143 183 199 113 75 98)(23 54 144 184 200 114 76 99)(24 28 145 185 201 115 77 100)(25 29 146 186 202 116 78 101)(26 30 147 187 203 117 79 102)(27 31 148 188 204 118 80 103)
(28 185)(29 186)(30 187)(31 188)(32 189)(33 163)(34 164)(35 165)(36 166)(37 167)(38 168)(39 169)(40 170)(41 171)(42 172)(43 173)(44 174)(45 175)(46 176)(47 177)(48 178)(49 179)(50 180)(51 181)(52 182)(53 183)(54 184)(55 150)(56 151)(57 152)(58 153)(59 154)(60 155)(61 156)(62 157)(63 158)(64 159)(65 160)(66 161)(67 162)(68 136)(69 137)(70 138)(71 139)(72 140)(73 141)(74 142)(75 143)(76 144)(77 145)(78 146)(79 147)(80 148)(81 149)(82 124)(83 125)(84 126)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 133)(92 134)(93 135)(94 109)(95 110)(96 111)(97 112)(98 113)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(106 121)(107 122)(108 123)

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

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

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

189 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 6A 6B 6C 6D 8A 8B 9A ··· 9F 12A 12B 12C 12D 18A ··· 18F 18G ··· 18L 24A 24B 24C 24D 27A ··· 27R 36A ··· 36F 36G ··· 36L 54A ··· 54R 54S ··· 54AJ 72A ··· 72L 108A ··· 108R 108S ··· 108AJ 216A ··· 216AJ order 1 2 2 3 3 4 4 6 6 6 6 8 8 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 24 24 24 24 27 ··· 27 36 ··· 36 36 ··· 36 54 ··· 54 54 ··· 54 72 ··· 72 108 ··· 108 108 ··· 108 216 ··· 216 size 1 1 4 1 1 2 4 1 1 4 4 2 2 1 ··· 1 2 2 4 4 1 ··· 1 4 ··· 4 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2

189 irreducible representations

 dim 1 1 1 1 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 C2 C3 C6 C6 C6 C9 C18 C18 C18 C27 C54 C54 C54 D4 SD16 C3×D4 C3×SD16 D4×C9 C9×SD16 D4×C27 SD16×C27 kernel SD16×C27 C216 D4×C27 Q8×C27 C9×SD16 C72 D4×C9 Q8×C9 C3×SD16 C24 C3×D4 C3×Q8 SD16 C8 D4 Q8 C54 C27 C18 C9 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 6 6 6 6 18 18 18 18 1 2 2 4 6 12 18 36

Matrix representation of SD16×C27 in GL2(𝔽433) generated by

 161 0 0 161
,
 182 251 182 182
,
 1 0 0 432
G:=sub<GL(2,GF(433))| [161,0,0,161],[182,182,251,182],[1,0,0,432] >;

SD16×C27 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_{27}
% in TeX

G:=Group("SD16xC27");
// GroupNames label

G:=SmallGroup(432,26);
// by ID

G=gap.SmallGroup(432,26);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,504,197,142,2355,1186,192,242]);
// 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^3>;
// generators/relations

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