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## G = Dic3.Q16order 192 = 26·3

### 2nd non-split extension by Dic3 of Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic3.Q16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — Dic6⋊C4 — Dic3.Q16
 Lower central C3 — C6 — C2×C12 — Dic3.Q16
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for Dic3.Q16
G = < a,b,c,d | a6=c8=1, b2=a3, d2=a3c4, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 256 in 96 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, Q8⋊C4, C4⋊C8, C2.D8, C2.D8, C4×Q8, C4⋊Q8, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, C2×Dic6, C4.Q16, C6.Q16, C6.SD16, Dic3⋊C8, C2.Dic12, C3×C2.D8, Dic6⋊C4, C12⋊Q8, Dic3.Q16
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, Q16, C2×D4, C2×Q8, C4○D4, C22×S3, C22⋊Q8, C2×Q16, C8⋊C22, C4○D12, S3×D4, S3×Q8, C4.Q16, D6⋊Q8, D8⋊S3, S3×Q16, Dic3.Q16

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + - + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 Q8 D4 D6 D6 Q16 C4○D4 C4○D12 C8⋊C22 S3×Q8 S3×D4 D8⋊S3 S3×Q16 kernel Dic3.Q16 C6.Q16 C6.SD16 Dic3⋊C8 C2.Dic12 C3×C2.D8 Dic6⋊C4 C12⋊Q8 C2.D8 Dic6 C2×Dic3 C4⋊C4 C2×C8 Dic3 C12 C4 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 2 1 4 2 4 1 1 1 2 2

Matrix representation of Dic3.Q16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 28 0 0 0 0 31 70 0 0 0 0 0 0 7 46 0 0 0 0 37 66
,
 57 16 0 0 0 0 57 57 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 24 0 0 0 0 15 71
,
 50 45 0 0 0 0 45 23 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 47 3 0 0 0 0 42 26

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,31,0,0,0,0,28,70,0,0,0,0,0,0,7,37,0,0,0,0,46,66],[57,57,0,0,0,0,16,57,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,15,0,0,0,0,24,71],[50,45,0,0,0,0,45,23,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,47,42,0,0,0,0,3,26] >;

Dic3.Q16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(192,434);
// by ID

G=gap.SmallGroup(192,434);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,422,135,268,570,297,136,6278]);
// Polycyclic

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

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