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G = SD16×C23order 368 = 24·23

Direct product of C23 and SD16

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

Aliases: SD16×C23, Q8⋊C46, C82C46, D4.C46, C1846C2, 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

C1C4 — SD16×C23
C1C2C4C92Q8×C23 — SD16×C23
C1C2C4 — SD16×C23
C1C46C92 — SD16×C23

Generators and relations for SD16×C23
 G = < a,b,c | a23=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

4C2
2C4
2C22
4C46
2C92
2C2×C46

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

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

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

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

161 conjugacy classes

class 1 2A2B4A4B8A8B23A···23V46A···46V46W···46AR92A···92V92W···92AR184A···184AR
order122448823···2346···4646···4692···9292···92184···184
size11424221···11···14···42···24···42···2

161 irreducible representations

dim111111112222
type+++++
imageC1C2C2C2C23C46C46C46D4SD16D4×C23SD16×C23
kernelSD16×C23C184D4×C23Q8×C23SD16C8D4Q8C46C23C2C1
# reps111122222222122244

Matrix representation of SD16×C23 in GL2(𝔽1289) generated by

2780
0278
,
1951094
195195
,
10
01288
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

Subgroup lattice of SD16×C23 in TeX

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