direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C27, C8⋊2C54, D4.C54, C216⋊6C2, Q8⋊2C54, C72.10C6, C24.6C18, C54.15D4, C108.18C22, C3.(C9×SD16), C9.(C3×SD16), C4.2(C2×C54), (Q8×C27)⋊4C2, (D4×C9).5C6, C2.4(D4×C27), C6.15(D4×C9), (C3×SD16).C9, (C9×SD16).C3, C36.41(C2×C6), (D4×C27).2C2, (C3×D4).3C18, C18.31(C3×D4), (Q8×C9).10C6, (C3×Q8).6C18, C12.18(C2×C18), SmallGroup(432,26)
Series: Derived ►Chief ►Lower central ►Upper central
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)
(1 102 66 187 153 47 204 126)(2 103 67 188 154 48 205 127)(3 104 68 189 155 49 206 128)(4 105 69 163 156 50 207 129)(5 106 70 164 157 51 208 130)(6 107 71 165 158 52 209 131)(7 108 72 166 159 53 210 132)(8 82 73 167 160 54 211 133)(9 83 74 168 161 28 212 134)(10 84 75 169 162 29 213 135)(11 85 76 170 136 30 214 109)(12 86 77 171 137 31 215 110)(13 87 78 172 138 32 216 111)(14 88 79 173 139 33 190 112)(15 89 80 174 140 34 191 113)(16 90 81 175 141 35 192 114)(17 91 55 176 142 36 193 115)(18 92 56 177 143 37 194 116)(19 93 57 178 144 38 195 117)(20 94 58 179 145 39 196 118)(21 95 59 180 146 40 197 119)(22 96 60 181 147 41 198 120)(23 97 61 182 148 42 199 121)(24 98 62 183 149 43 200 122)(25 99 63 184 150 44 201 123)(26 100 64 185 151 45 202 124)(27 101 65 186 152 46 203 125)
(28 134)(29 135)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(41 120)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 128)(50 129)(51 130)(52 131)(53 132)(54 133)(55 193)(56 194)(57 195)(58 196)(59 197)(60 198)(61 199)(62 200)(63 201)(64 202)(65 203)(66 204)(67 205)(68 206)(69 207)(70 208)(71 209)(72 210)(73 211)(74 212)(75 213)(76 214)(77 215)(78 216)(79 190)(80 191)(81 192)(82 167)(83 168)(84 169)(85 170)(86 171)(87 172)(88 173)(89 174)(90 175)(91 176)(92 177)(93 178)(94 179)(95 180)(96 181)(97 182)(98 183)(99 184)(100 185)(101 186)(102 187)(103 188)(104 189)(105 163)(106 164)(107 165)(108 166)
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), (1,102,66,187,153,47,204,126)(2,103,67,188,154,48,205,127)(3,104,68,189,155,49,206,128)(4,105,69,163,156,50,207,129)(5,106,70,164,157,51,208,130)(6,107,71,165,158,52,209,131)(7,108,72,166,159,53,210,132)(8,82,73,167,160,54,211,133)(9,83,74,168,161,28,212,134)(10,84,75,169,162,29,213,135)(11,85,76,170,136,30,214,109)(12,86,77,171,137,31,215,110)(13,87,78,172,138,32,216,111)(14,88,79,173,139,33,190,112)(15,89,80,174,140,34,191,113)(16,90,81,175,141,35,192,114)(17,91,55,176,142,36,193,115)(18,92,56,177,143,37,194,116)(19,93,57,178,144,38,195,117)(20,94,58,179,145,39,196,118)(21,95,59,180,146,40,197,119)(22,96,60,181,147,41,198,120)(23,97,61,182,148,42,199,121)(24,98,62,183,149,43,200,122)(25,99,63,184,150,44,201,123)(26,100,64,185,151,45,202,124)(27,101,65,186,152,46,203,125), (28,134)(29,135)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,193)(56,194)(57,195)(58,196)(59,197)(60,198)(61,199)(62,200)(63,201)(64,202)(65,203)(66,204)(67,205)(68,206)(69,207)(70,208)(71,209)(72,210)(73,211)(74,212)(75,213)(76,214)(77,215)(78,216)(79,190)(80,191)(81,192)(82,167)(83,168)(84,169)(85,170)(86,171)(87,172)(88,173)(89,174)(90,175)(91,176)(92,177)(93,178)(94,179)(95,180)(96,181)(97,182)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,163)(106,164)(107,165)(108,166)>;
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), (1,102,66,187,153,47,204,126)(2,103,67,188,154,48,205,127)(3,104,68,189,155,49,206,128)(4,105,69,163,156,50,207,129)(5,106,70,164,157,51,208,130)(6,107,71,165,158,52,209,131)(7,108,72,166,159,53,210,132)(8,82,73,167,160,54,211,133)(9,83,74,168,161,28,212,134)(10,84,75,169,162,29,213,135)(11,85,76,170,136,30,214,109)(12,86,77,171,137,31,215,110)(13,87,78,172,138,32,216,111)(14,88,79,173,139,33,190,112)(15,89,80,174,140,34,191,113)(16,90,81,175,141,35,192,114)(17,91,55,176,142,36,193,115)(18,92,56,177,143,37,194,116)(19,93,57,178,144,38,195,117)(20,94,58,179,145,39,196,118)(21,95,59,180,146,40,197,119)(22,96,60,181,147,41,198,120)(23,97,61,182,148,42,199,121)(24,98,62,183,149,43,200,122)(25,99,63,184,150,44,201,123)(26,100,64,185,151,45,202,124)(27,101,65,186,152,46,203,125), (28,134)(29,135)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,120)(42,121)(43,122)(44,123)(45,124)(46,125)(47,126)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,193)(56,194)(57,195)(58,196)(59,197)(60,198)(61,199)(62,200)(63,201)(64,202)(65,203)(66,204)(67,205)(68,206)(69,207)(70,208)(71,209)(72,210)(73,211)(74,212)(75,213)(76,214)(77,215)(78,216)(79,190)(80,191)(81,192)(82,167)(83,168)(84,169)(85,170)(86,171)(87,172)(88,173)(89,174)(90,175)(91,176)(92,177)(93,178)(94,179)(95,180)(96,181)(97,182)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,163)(106,164)(107,165)(108,166) );
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)], [(1,102,66,187,153,47,204,126),(2,103,67,188,154,48,205,127),(3,104,68,189,155,49,206,128),(4,105,69,163,156,50,207,129),(5,106,70,164,157,51,208,130),(6,107,71,165,158,52,209,131),(7,108,72,166,159,53,210,132),(8,82,73,167,160,54,211,133),(9,83,74,168,161,28,212,134),(10,84,75,169,162,29,213,135),(11,85,76,170,136,30,214,109),(12,86,77,171,137,31,215,110),(13,87,78,172,138,32,216,111),(14,88,79,173,139,33,190,112),(15,89,80,174,140,34,191,113),(16,90,81,175,141,35,192,114),(17,91,55,176,142,36,193,115),(18,92,56,177,143,37,194,116),(19,93,57,178,144,38,195,117),(20,94,58,179,145,39,196,118),(21,95,59,180,146,40,197,119),(22,96,60,181,147,41,198,120),(23,97,61,182,148,42,199,121),(24,98,62,183,149,43,200,122),(25,99,63,184,150,44,201,123),(26,100,64,185,151,45,202,124),(27,101,65,186,152,46,203,125)], [(28,134),(29,135),(30,109),(31,110),(32,111),(33,112),(34,113),(35,114),(36,115),(37,116),(38,117),(39,118),(40,119),(41,120),(42,121),(43,122),(44,123),(45,124),(46,125),(47,126),(48,127),(49,128),(50,129),(51,130),(52,131),(53,132),(54,133),(55,193),(56,194),(57,195),(58,196),(59,197),(60,198),(61,199),(62,200),(63,201),(64,202),(65,203),(66,204),(67,205),(68,206),(69,207),(70,208),(71,209),(72,210),(73,211),(74,212),(75,213),(76,214),(77,215),(78,216),(79,190),(80,191),(81,192),(82,167),(83,168),(84,169),(85,170),(86,171),(87,172),(88,173),(89,174),(90,175),(91,176),(92,177),(93,178),(94,179),(95,180),(96,181),(97,182),(98,183),(99,184),(100,185),(101,186),(102,187),(103,188),(104,189),(105,163),(106,164),(107,165),(108,166)]])
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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