direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C23, Q8⋊C46, C8⋊2C46, D4.C46, C184⋊6C2, C46.15D4, C92.18C22, C4.2(C2×C46), (Q8×C23)⋊4C2, C2.4(D4×C23), (D4×C23).2C2, SmallGroup(368,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C23
G = < a,b,c | a23=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Smallest permutation representation of SD16×C23
►On 184 points
Generators in S184
(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)
(1 80 172 136 143 40 64 100)(2 81 173 137 144 41 65 101)(3 82 174 138 145 42 66 102)(4 83 175 116 146 43 67 103)(5 84 176 117 147 44 68 104)(6 85 177 118 148 45 69 105)(7 86 178 119 149 46 47 106)(8 87 179 120 150 24 48 107)(9 88 180 121 151 25 49 108)(10 89 181 122 152 26 50 109)(11 90 182 123 153 27 51 110)(12 91 183 124 154 28 52 111)(13 92 184 125 155 29 53 112)(14 70 162 126 156 30 54 113)(15 71 163 127 157 31 55 114)(16 72 164 128 158 32 56 115)(17 73 165 129 159 33 57 93)(18 74 166 130 160 34 58 94)(19 75 167 131 161 35 59 95)(20 76 168 132 139 36 60 96)(21 77 169 133 140 37 61 97)(22 78 170 134 141 38 62 98)(23 79 171 135 142 39 63 99)
(24 107)(25 108)(26 109)(27 110)(28 111)(29 112)(30 113)(31 114)(32 115)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 178)(48 179)(49 180)(50 181)(51 182)(52 183)(53 184)(54 162)(55 163)(56 164)(57 165)(58 166)(59 167)(60 168)(61 169)(62 170)(63 171)(64 172)(65 173)(66 174)(67 175)(68 176)(69 177)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(81 137)(82 138)(83 116)(84 117)(85 118)(86 119)(87 120)(88 121)(89 122)(90 123)(91 124)(92 125)
G:=sub<Sym(184)| (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), (1,80,172,136,143,40,64,100)(2,81,173,137,144,41,65,101)(3,82,174,138,145,42,66,102)(4,83,175,116,146,43,67,103)(5,84,176,117,147,44,68,104)(6,85,177,118,148,45,69,105)(7,86,178,119,149,46,47,106)(8,87,179,120,150,24,48,107)(9,88,180,121,151,25,49,108)(10,89,181,122,152,26,50,109)(11,90,182,123,153,27,51,110)(12,91,183,124,154,28,52,111)(13,92,184,125,155,29,53,112)(14,70,162,126,156,30,54,113)(15,71,163,127,157,31,55,114)(16,72,164,128,158,32,56,115)(17,73,165,129,159,33,57,93)(18,74,166,130,160,34,58,94)(19,75,167,131,161,35,59,95)(20,76,168,132,139,36,60,96)(21,77,169,133,140,37,61,97)(22,78,170,134,141,38,62,98)(23,79,171,135,142,39,63,99), (24,107)(25,108)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,178)(48,179)(49,180)(50,181)(51,182)(52,183)(53,184)(54,162)(55,163)(56,164)(57,165)(58,166)(59,167)(60,168)(61,169)(62,170)(63,171)(64,172)(65,173)(66,174)(67,175)(68,176)(69,177)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(81,137)(82,138)(83,116)(84,117)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,124)(92,125)>;
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), (1,80,172,136,143,40,64,100)(2,81,173,137,144,41,65,101)(3,82,174,138,145,42,66,102)(4,83,175,116,146,43,67,103)(5,84,176,117,147,44,68,104)(6,85,177,118,148,45,69,105)(7,86,178,119,149,46,47,106)(8,87,179,120,150,24,48,107)(9,88,180,121,151,25,49,108)(10,89,181,122,152,26,50,109)(11,90,182,123,153,27,51,110)(12,91,183,124,154,28,52,111)(13,92,184,125,155,29,53,112)(14,70,162,126,156,30,54,113)(15,71,163,127,157,31,55,114)(16,72,164,128,158,32,56,115)(17,73,165,129,159,33,57,93)(18,74,166,130,160,34,58,94)(19,75,167,131,161,35,59,95)(20,76,168,132,139,36,60,96)(21,77,169,133,140,37,61,97)(22,78,170,134,141,38,62,98)(23,79,171,135,142,39,63,99), (24,107)(25,108)(26,109)(27,110)(28,111)(29,112)(30,113)(31,114)(32,115)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,178)(48,179)(49,180)(50,181)(51,182)(52,183)(53,184)(54,162)(55,163)(56,164)(57,165)(58,166)(59,167)(60,168)(61,169)(62,170)(63,171)(64,172)(65,173)(66,174)(67,175)(68,176)(69,177)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(81,137)(82,138)(83,116)(84,117)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,124)(92,125) );
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)], [(1,80,172,136,143,40,64,100),(2,81,173,137,144,41,65,101),(3,82,174,138,145,42,66,102),(4,83,175,116,146,43,67,103),(5,84,176,117,147,44,68,104),(6,85,177,118,148,45,69,105),(7,86,178,119,149,46,47,106),(8,87,179,120,150,24,48,107),(9,88,180,121,151,25,49,108),(10,89,181,122,152,26,50,109),(11,90,182,123,153,27,51,110),(12,91,183,124,154,28,52,111),(13,92,184,125,155,29,53,112),(14,70,162,126,156,30,54,113),(15,71,163,127,157,31,55,114),(16,72,164,128,158,32,56,115),(17,73,165,129,159,33,57,93),(18,74,166,130,160,34,58,94),(19,75,167,131,161,35,59,95),(20,76,168,132,139,36,60,96),(21,77,169,133,140,37,61,97),(22,78,170,134,141,38,62,98),(23,79,171,135,142,39,63,99)], [(24,107),(25,108),(26,109),(27,110),(28,111),(29,112),(30,113),(31,114),(32,115),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,178),(48,179),(49,180),(50,181),(51,182),(52,183),(53,184),(54,162),(55,163),(56,164),(57,165),(58,166),(59,167),(60,168),(61,169),(62,170),(63,171),(64,172),(65,173),(66,174),(67,175),(68,176),(69,177),(70,126),(71,127),(72,128),(73,129),(74,130),(75,131),(76,132),(77,133),(78,134),(79,135),(80,136),(81,137),(82,138),(83,116),(84,117),(85,118),(86,119),(87,120),(88,121),(89,122),(90,123),(91,124),(92,125)]])
161 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 23A | ··· | 23V | 46A | ··· | 46V | 46W | ··· | 46AR | 92A | ··· | 92V | 92W | ··· | 92AR | 184A | ··· | 184AR |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 46 | ··· | 46 | 92 | ··· | 92 | 92 | ··· | 92 | 184 | ··· | 184 |
| size | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
161 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C23 | C46 | C46 | C46 | D4 | SD16 | D4×C23 | SD16×C23 |
| kernel | SD16×C23 | C184 | D4×C23 | Q8×C23 | SD16 | C8 | D4 | Q8 | C46 | C23 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 22 | 22 | 22 | 22 | 1 | 2 | 22 | 44 |
Matrix representation of SD16×C23 ►in GL2(𝔽1289) generated by
| 278 | 0 |
| 0 | 278 |
| 195 | 1094 |
| 195 | 195 |
| 1 | 0 |
| 0 | 1288 |
G:=sub<GL(2,GF(1289))| [278,0,0,278],[195,195,1094,195],[1,0,0,1288] >;
SD16×C23 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_{23} % in TeX
G:=Group("SD16xC23"); // GroupNames label
G:=SmallGroup(368,25);
// by ID
G=gap.SmallGroup(368,25);
# by ID
G:=PCGroup([5,-2,-2,-23,-2,-2,920,941,5523,2768,58]);
// Polycyclic
G:=Group<a,b,c|a^23=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations
Export