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## G = Q16⋊Dic3order 192 = 26·3

### 3rd semidirect product of Q16 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q16⋊Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×Dic3 — Q8×Dic3 — Q16⋊Dic3
 Lower central C3 — C6 — C12 — Q16⋊Dic3
 Upper central C1 — C22 — C2×C4 — C2×Q16

Generators and relations for Q16⋊Dic3
G = < a,b,c,d | a8=c6=1, b2=a4, d2=c3, bab-1=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 232 in 108 conjugacy classes, 57 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C2×Q16, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4⋊Dic3, C4⋊Dic3, C2×C24, C3×Q16, C6×Q8, Q16⋊C4, C24⋊C4, C8⋊Dic3, Q82Dic3, Q8×Dic3, C6×Q16, Q16⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, C2×Dic3, C22×S3, C4×D4, C8.C22, S3×D4, D42S3, C22×Dic3, Q16⋊C4, Q16⋊S3, D4×Dic3, Q16⋊Dic3

Smallest permutation representation of Q16⋊Dic3
Regular action on 192 points
Generators in S192
(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)
(1 64 5 60)(2 63 6 59)(3 62 7 58)(4 61 8 57)(9 182 13 178)(10 181 14 177)(11 180 15 184)(12 179 16 183)(17 149 21 145)(18 148 22 152)(19 147 23 151)(20 146 24 150)(25 38 29 34)(26 37 30 33)(27 36 31 40)(28 35 32 39)(41 55 45 51)(42 54 46 50)(43 53 47 49)(44 52 48 56)(65 125 69 121)(66 124 70 128)(67 123 71 127)(68 122 72 126)(73 188 77 192)(74 187 78 191)(75 186 79 190)(76 185 80 189)(81 116 85 120)(82 115 86 119)(83 114 87 118)(84 113 88 117)(89 132 93 136)(90 131 94 135)(91 130 95 134)(92 129 96 133)(97 168 101 164)(98 167 102 163)(99 166 103 162)(100 165 104 161)(105 153 109 157)(106 160 110 156)(107 159 111 155)(108 158 112 154)(137 170 141 174)(138 169 142 173)(139 176 143 172)(140 175 144 171)
(1 130 127 149 31 49)(2 131 128 150 32 50)(3 132 121 151 25 51)(4 133 122 152 26 52)(5 134 123 145 27 53)(6 135 124 146 28 54)(7 136 125 147 29 55)(8 129 126 148 30 56)(9 74 156 161 137 87)(10 75 157 162 138 88)(11 76 158 163 139 81)(12 77 159 164 140 82)(13 78 160 165 141 83)(14 79 153 166 142 84)(15 80 154 167 143 85)(16 73 155 168 144 86)(17 36 47 60 91 71)(18 37 48 61 92 72)(19 38 41 62 93 65)(20 39 42 63 94 66)(21 40 43 64 95 67)(22 33 44 57 96 68)(23 34 45 58 89 69)(24 35 46 59 90 70)(97 175 115 179 192 111)(98 176 116 180 185 112)(99 169 117 181 186 105)(100 170 118 182 187 106)(101 171 119 183 188 107)(102 172 120 184 189 108)(103 173 113 177 190 109)(104 174 114 178 191 110)
(1 83 149 160)(2 88 150 157)(3 85 151 154)(4 82 152 159)(5 87 145 156)(6 84 146 153)(7 81 147 158)(8 86 148 155)(9 123 161 53)(10 128 162 50)(11 125 163 55)(12 122 164 52)(13 127 165 49)(14 124 166 54)(15 121 167 51)(16 126 168 56)(17 110 60 114)(18 107 61 119)(19 112 62 116)(20 109 63 113)(21 106 64 118)(22 111 57 115)(23 108 58 120)(24 105 59 117)(25 80 132 143)(26 77 133 140)(27 74 134 137)(28 79 135 142)(29 76 136 139)(30 73 129 144)(31 78 130 141)(32 75 131 138)(33 192 96 175)(34 189 89 172)(35 186 90 169)(36 191 91 174)(37 188 92 171)(38 185 93 176)(39 190 94 173)(40 187 95 170)(41 180 65 98)(42 177 66 103)(43 182 67 100)(44 179 68 97)(45 184 69 102)(46 181 70 99)(47 178 71 104)(48 183 72 101)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 12 24 24 24 24 size 1 1 1 1 2 2 2 4 4 4 4 6 6 6 6 12 12 12 12 2 2 2 4 4 12 12 4 4 8 8 8 8 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + - - + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D6 Dic3 D6 C4○D4 C8.C22 D4⋊2S3 S3×D4 Q16⋊S3 kernel Q16⋊Dic3 C24⋊C4 C8⋊Dic3 Q8⋊2Dic3 Q8×Dic3 C6×Q16 C3×Q16 C2×Q16 C2×Dic3 C2×C8 Q16 C2×Q8 C12 C6 C4 C22 C2 # reps 1 1 1 2 2 1 8 1 2 1 4 2 2 2 1 1 4

Matrix representation of Q16⋊Dic3 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 62 51 0 0 0 0 22 11 0 0 42 11 62 51 0 0 62 31 22 11
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 43 11 34 22 0 0 62 32 51 12 0 0 60 22 30 62 0 0 51 38 11 41
,
 0 72 0 0 0 0 1 1 0 0 0 0 0 0 0 72 0 0 0 0 1 1 0 0 0 0 0 0 0 72 0 0 0 0 1 1
,
 55 2 0 0 0 0 20 18 0 0 0 0 0 0 22 2 41 64 0 0 53 51 23 32 0 0 16 41 63 66 0 0 25 57 3 10

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,42,62,0,0,0,0,11,31,0,0,62,22,62,22,0,0,51,11,51,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,62,60,51,0,0,11,32,22,38,0,0,34,51,30,11,0,0,22,12,62,41],[0,1,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,72,1],[55,20,0,0,0,0,2,18,0,0,0,0,0,0,22,53,16,25,0,0,2,51,41,57,0,0,41,23,63,3,0,0,64,32,66,10] >;

Q16⋊Dic3 in GAP, Magma, Sage, TeX

Q_{16}\rtimes {\rm Dic}_3
% in TeX

G:=Group("Q16:Dic3");
// GroupNames label

G:=SmallGroup(192,743);
// by ID

G=gap.SmallGroup(192,743);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,758,219,184,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^6=1,b^2=a^4,d^2=c^3,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

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