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## G = Dic3.1Q16order 192 = 26·3

### 1st non-split extension by Dic3 of Q16 acting via Q16/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Dic3.1Q16
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C12⋊Q8 — Dic3.1Q16
 Lower central C3 — C6 — C2×C12 — Dic3.1Q16
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Dic3.1Q16
G = < a,b,c,d | a6=c8=1, b2=a3, d2=c4, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a3c-1 >

Subgroups: 264 in 98 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, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×C8, Q8⋊C4, Q8⋊C4, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, C2×Dic6, C6×Q8, C4.SD16, C6.SD16, C8×Dic3, C2.Dic12, Q82Dic3, C3×Q8⋊C4, C12⋊Q8, Dic3⋊Q8, Dic3.1Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, Q16, C2×D4, C4○D4, C22×S3, C4.4D4, C2×SD16, C2×Q16, C4○D12, S3×D4, D42S3, C4.SD16, C23.11D6, S3×SD16, S3×Q16, Dic3.1Q16

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

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

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

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

36 conjugacy classes

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

36 irreducible representations

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

Matrix representation of Dic3.1Q16 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 1 71 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 23 55 0 0 0 0 5 50
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 0 41 0 0 0 0 16 41 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 72 0 0 0 0 0 0 40 31 0 0 0 0 19 33 0 0 0 0 0 0 72 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,23,5,0,0,0,0,55,50],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,16,0,0,0,0,41,41,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,40,19,0,0,0,0,31,33,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

Dic3.1Q16 in GAP, Magma, Sage, TeX

{\rm Dic}_3._1Q_{16}
% in TeX

G:=Group("Dic3.1Q16");
// GroupNames label

G:=SmallGroup(192,351);
// by ID

G=gap.SmallGroup(192,351);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,422,135,184,570,297,136,6278]);
// Polycyclic

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

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