metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊2Q16, Dic6.10D4, (C2×C8).14D6, C4.90(S3×D4), C2.8(S3×Q16), C4⋊C4.145D6, C3⋊1(C4⋊2Q16), (C2×Q8).35D6, C6.15(C2×Q16), C4.5(C4○D12), C12.114(C2×D4), Dic3⋊C8.3C2, Q8⋊C4.3S3, C12.15(C4○D4), C6.21(C4⋊D4), (C2×C24).14C22, (C2×Dic12).3C2, C6.SD16.2C2, C22.189(S3×D4), (C6×Q8).18C22, C2.24(Dic3⋊D4), (C2×C12).235C23, Dic3⋊Q8.3C2, Dic6⋊C4.3C2, (C2×Dic3).149D4, C2.14(D4.D6), C6.32(C8.C22), (C4×Dic3).19C22, (C2×Dic6).65C22, (C2×C6).248(C2×D4), (C2×C3⋊C8).30C22, (C2×C3⋊Q16).3C2, (C3×C4⋊C4).36C22, (C3×Q8⋊C4).3C2, (C2×C4).342(C22×S3), SmallGroup(192,354)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊Q16
G = < a,b,c,d | a6=c8=1, b2=a3, d2=c4, bab-1=cac-1=a-1, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >
Subgroups: 280 in 108 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, Q16, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, Q8⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, Dic12, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, C3⋊Q16, C3×C4⋊C4, C2×C24, C2×Dic6, C6×Q8, C4⋊2Q16, C6.SD16, Dic3⋊C8, C3×Q8⋊C4, Dic6⋊C4, C2×Dic12, C2×C3⋊Q16, Dic3⋊Q8, Dic3⋊Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, C22×S3, C4⋊D4, C2×Q16, C8.C22, C4○D12, S3×D4, C4⋊2Q16, Dic3⋊D4, D4.D6, S3×Q16, Dic3⋊Q16
►Regular action on 192 points
Generators in S192
(1 109 83 21 189 169)(2 170 190 22 84 110)(3 111 85 23 191 171)(4 172 192 24 86 112)(5 105 87 17 185 173)(6 174 186 18 88 106)(7 107 81 19 187 175)(8 176 188 20 82 108)(9 101 78 96 166 180)(10 181 167 89 79 102)(11 103 80 90 168 182)(12 183 161 91 73 104)(13 97 74 92 162 184)(14 177 163 93 75 98)(15 99 76 94 164 178)(16 179 165 95 77 100)(25 44 151 117 130 60)(26 61 131 118 152 45)(27 46 145 119 132 62)(28 63 133 120 146 47)(29 48 147 113 134 64)(30 57 135 114 148 41)(31 42 149 115 136 58)(32 59 129 116 150 43)(33 153 143 68 49 122)(34 123 50 69 144 154)(35 155 137 70 51 124)(36 125 52 71 138 156)(37 157 139 72 53 126)(38 127 54 65 140 158)(39 159 141 66 55 128)(40 121 56 67 142 160)
(1 38 21 65)(2 66 22 39)(3 40 23 67)(4 68 24 33)(5 34 17 69)(6 70 18 35)(7 36 19 71)(8 72 20 37)(9 146 96 63)(10 64 89 147)(11 148 90 57)(12 58 91 149)(13 150 92 59)(14 60 93 151)(15 152 94 61)(16 62 95 145)(25 163 117 98)(26 99 118 164)(27 165 119 100)(28 101 120 166)(29 167 113 102)(30 103 114 168)(31 161 115 104)(32 97 116 162)(41 80 135 182)(42 183 136 73)(43 74 129 184)(44 177 130 75)(45 76 131 178)(46 179 132 77)(47 78 133 180)(48 181 134 79)(49 192 153 112)(50 105 154 185)(51 186 155 106)(52 107 156 187)(53 188 157 108)(54 109 158 189)(55 190 159 110)(56 111 160 191)(81 138 175 125)(82 126 176 139)(83 140 169 127)(84 128 170 141)(85 142 171 121)(86 122 172 143)(87 144 173 123)(88 124 174 137)
(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 95 5 91)(2 94 6 90)(3 93 7 89)(4 92 8 96)(9 24 13 20)(10 23 14 19)(11 22 15 18)(12 21 16 17)(25 125 29 121)(26 124 30 128)(27 123 31 127)(28 122 32 126)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 64 40 60)(41 55 45 51)(42 54 46 50)(43 53 47 49)(44 52 48 56)(65 145 69 149)(66 152 70 148)(67 151 71 147)(68 150 72 146)(73 109 77 105)(74 108 78 112)(75 107 79 111)(76 106 80 110)(81 102 85 98)(82 101 86 97)(83 100 87 104)(84 99 88 103)(113 142 117 138)(114 141 118 137)(115 140 119 144)(116 139 120 143)(129 157 133 153)(130 156 134 160)(131 155 135 159)(132 154 136 158)(161 169 165 173)(162 176 166 172)(163 175 167 171)(164 174 168 170)(177 187 181 191)(178 186 182 190)(179 185 183 189)(180 192 184 188)
G:=sub<Sym(192)| (1,109,83,21,189,169)(2,170,190,22,84,110)(3,111,85,23,191,171)(4,172,192,24,86,112)(5,105,87,17,185,173)(6,174,186,18,88,106)(7,107,81,19,187,175)(8,176,188,20,82,108)(9,101,78,96,166,180)(10,181,167,89,79,102)(11,103,80,90,168,182)(12,183,161,91,73,104)(13,97,74,92,162,184)(14,177,163,93,75,98)(15,99,76,94,164,178)(16,179,165,95,77,100)(25,44,151,117,130,60)(26,61,131,118,152,45)(27,46,145,119,132,62)(28,63,133,120,146,47)(29,48,147,113,134,64)(30,57,135,114,148,41)(31,42,149,115,136,58)(32,59,129,116,150,43)(33,153,143,68,49,122)(34,123,50,69,144,154)(35,155,137,70,51,124)(36,125,52,71,138,156)(37,157,139,72,53,126)(38,127,54,65,140,158)(39,159,141,66,55,128)(40,121,56,67,142,160), (1,38,21,65)(2,66,22,39)(3,40,23,67)(4,68,24,33)(5,34,17,69)(6,70,18,35)(7,36,19,71)(8,72,20,37)(9,146,96,63)(10,64,89,147)(11,148,90,57)(12,58,91,149)(13,150,92,59)(14,60,93,151)(15,152,94,61)(16,62,95,145)(25,163,117,98)(26,99,118,164)(27,165,119,100)(28,101,120,166)(29,167,113,102)(30,103,114,168)(31,161,115,104)(32,97,116,162)(41,80,135,182)(42,183,136,73)(43,74,129,184)(44,177,130,75)(45,76,131,178)(46,179,132,77)(47,78,133,180)(48,181,134,79)(49,192,153,112)(50,105,154,185)(51,186,155,106)(52,107,156,187)(53,188,157,108)(54,109,158,189)(55,190,159,110)(56,111,160,191)(81,138,175,125)(82,126,176,139)(83,140,169,127)(84,128,170,141)(85,142,171,121)(86,122,172,143)(87,144,173,123)(88,124,174,137), (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), (1,95,5,91)(2,94,6,90)(3,93,7,89)(4,92,8,96)(9,24,13,20)(10,23,14,19)(11,22,15,18)(12,21,16,17)(25,125,29,121)(26,124,30,128)(27,123,31,127)(28,122,32,126)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60)(41,55,45,51)(42,54,46,50)(43,53,47,49)(44,52,48,56)(65,145,69,149)(66,152,70,148)(67,151,71,147)(68,150,72,146)(73,109,77,105)(74,108,78,112)(75,107,79,111)(76,106,80,110)(81,102,85,98)(82,101,86,97)(83,100,87,104)(84,99,88,103)(113,142,117,138)(114,141,118,137)(115,140,119,144)(116,139,120,143)(129,157,133,153)(130,156,134,160)(131,155,135,159)(132,154,136,158)(161,169,165,173)(162,176,166,172)(163,175,167,171)(164,174,168,170)(177,187,181,191)(178,186,182,190)(179,185,183,189)(180,192,184,188)>;
G:=Group( (1,109,83,21,189,169)(2,170,190,22,84,110)(3,111,85,23,191,171)(4,172,192,24,86,112)(5,105,87,17,185,173)(6,174,186,18,88,106)(7,107,81,19,187,175)(8,176,188,20,82,108)(9,101,78,96,166,180)(10,181,167,89,79,102)(11,103,80,90,168,182)(12,183,161,91,73,104)(13,97,74,92,162,184)(14,177,163,93,75,98)(15,99,76,94,164,178)(16,179,165,95,77,100)(25,44,151,117,130,60)(26,61,131,118,152,45)(27,46,145,119,132,62)(28,63,133,120,146,47)(29,48,147,113,134,64)(30,57,135,114,148,41)(31,42,149,115,136,58)(32,59,129,116,150,43)(33,153,143,68,49,122)(34,123,50,69,144,154)(35,155,137,70,51,124)(36,125,52,71,138,156)(37,157,139,72,53,126)(38,127,54,65,140,158)(39,159,141,66,55,128)(40,121,56,67,142,160), (1,38,21,65)(2,66,22,39)(3,40,23,67)(4,68,24,33)(5,34,17,69)(6,70,18,35)(7,36,19,71)(8,72,20,37)(9,146,96,63)(10,64,89,147)(11,148,90,57)(12,58,91,149)(13,150,92,59)(14,60,93,151)(15,152,94,61)(16,62,95,145)(25,163,117,98)(26,99,118,164)(27,165,119,100)(28,101,120,166)(29,167,113,102)(30,103,114,168)(31,161,115,104)(32,97,116,162)(41,80,135,182)(42,183,136,73)(43,74,129,184)(44,177,130,75)(45,76,131,178)(46,179,132,77)(47,78,133,180)(48,181,134,79)(49,192,153,112)(50,105,154,185)(51,186,155,106)(52,107,156,187)(53,188,157,108)(54,109,158,189)(55,190,159,110)(56,111,160,191)(81,138,175,125)(82,126,176,139)(83,140,169,127)(84,128,170,141)(85,142,171,121)(86,122,172,143)(87,144,173,123)(88,124,174,137), (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), (1,95,5,91)(2,94,6,90)(3,93,7,89)(4,92,8,96)(9,24,13,20)(10,23,14,19)(11,22,15,18)(12,21,16,17)(25,125,29,121)(26,124,30,128)(27,123,31,127)(28,122,32,126)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,64,40,60)(41,55,45,51)(42,54,46,50)(43,53,47,49)(44,52,48,56)(65,145,69,149)(66,152,70,148)(67,151,71,147)(68,150,72,146)(73,109,77,105)(74,108,78,112)(75,107,79,111)(76,106,80,110)(81,102,85,98)(82,101,86,97)(83,100,87,104)(84,99,88,103)(113,142,117,138)(114,141,118,137)(115,140,119,144)(116,139,120,143)(129,157,133,153)(130,156,134,160)(131,155,135,159)(132,154,136,158)(161,169,165,173)(162,176,166,172)(163,175,167,171)(164,174,168,170)(177,187,181,191)(178,186,182,190)(179,185,183,189)(180,192,184,188) );
G=PermutationGroup([[(1,109,83,21,189,169),(2,170,190,22,84,110),(3,111,85,23,191,171),(4,172,192,24,86,112),(5,105,87,17,185,173),(6,174,186,18,88,106),(7,107,81,19,187,175),(8,176,188,20,82,108),(9,101,78,96,166,180),(10,181,167,89,79,102),(11,103,80,90,168,182),(12,183,161,91,73,104),(13,97,74,92,162,184),(14,177,163,93,75,98),(15,99,76,94,164,178),(16,179,165,95,77,100),(25,44,151,117,130,60),(26,61,131,118,152,45),(27,46,145,119,132,62),(28,63,133,120,146,47),(29,48,147,113,134,64),(30,57,135,114,148,41),(31,42,149,115,136,58),(32,59,129,116,150,43),(33,153,143,68,49,122),(34,123,50,69,144,154),(35,155,137,70,51,124),(36,125,52,71,138,156),(37,157,139,72,53,126),(38,127,54,65,140,158),(39,159,141,66,55,128),(40,121,56,67,142,160)], [(1,38,21,65),(2,66,22,39),(3,40,23,67),(4,68,24,33),(5,34,17,69),(6,70,18,35),(7,36,19,71),(8,72,20,37),(9,146,96,63),(10,64,89,147),(11,148,90,57),(12,58,91,149),(13,150,92,59),(14,60,93,151),(15,152,94,61),(16,62,95,145),(25,163,117,98),(26,99,118,164),(27,165,119,100),(28,101,120,166),(29,167,113,102),(30,103,114,168),(31,161,115,104),(32,97,116,162),(41,80,135,182),(42,183,136,73),(43,74,129,184),(44,177,130,75),(45,76,131,178),(46,179,132,77),(47,78,133,180),(48,181,134,79),(49,192,153,112),(50,105,154,185),(51,186,155,106),(52,107,156,187),(53,188,157,108),(54,109,158,189),(55,190,159,110),(56,111,160,191),(81,138,175,125),(82,126,176,139),(83,140,169,127),(84,128,170,141),(85,142,171,121),(86,122,172,143),(87,144,173,123),(88,124,174,137)], [(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)], [(1,95,5,91),(2,94,6,90),(3,93,7,89),(4,92,8,96),(9,24,13,20),(10,23,14,19),(11,22,15,18),(12,21,16,17),(25,125,29,121),(26,124,30,128),(27,123,31,127),(28,122,32,126),(33,59,37,63),(34,58,38,62),(35,57,39,61),(36,64,40,60),(41,55,45,51),(42,54,46,50),(43,53,47,49),(44,52,48,56),(65,145,69,149),(66,152,70,148),(67,151,71,147),(68,150,72,146),(73,109,77,105),(74,108,78,112),(75,107,79,111),(76,106,80,110),(81,102,85,98),(82,101,86,97),(83,100,87,104),(84,99,88,103),(113,142,117,138),(114,141,118,137),(115,140,119,144),(116,139,120,143),(129,157,133,153),(130,156,134,160),(131,155,135,159),(132,154,136,158),(161,169,165,173),(162,176,166,172),(163,175,167,171),(164,174,168,170),(177,187,181,191),(178,186,182,190),(179,185,183,189),(180,192,184,188)]])
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 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | Q16 | C4○D4 | C4○D12 | C8.C22 | S3×D4 | S3×D4 | D4.D6 | S3×Q16 |
| kernel | Dic3⋊Q16 | C6.SD16 | Dic3⋊C8 | C3×Q8⋊C4 | Dic6⋊C4 | C2×Dic12 | C2×C3⋊Q16 | Dic3⋊Q8 | Q8⋊C4 | Dic6 | C2×Dic3 | C4⋊C4 | C2×C8 | C2×Q8 | Dic3 | C12 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Dic3⋊Q16 ►in GL6(𝔽73)
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 32 | 0 | 0 |
| 0 | 0 | 41 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 25 |
| 0 | 0 | 0 | 0 | 35 | 32 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 71 | 10 |
| 0 | 0 | 0 | 0 | 36 | 2 |
G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,1,41,0,0,0,0,32,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,35,0,0,0,0,25,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,71,36,0,0,0,0,10,2] >;
Dic3⋊Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes Q_{16} % in TeX
G:=Group("Dic3:Q16"); // GroupNames label
G:=SmallGroup(192,354);
// by ID
G=gap.SmallGroup(192,354);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,422,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^8=1,b^2=a^3,d^2=c^4,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations