metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊4Q16, C3⋊2(C4×Q16), C3⋊Q16⋊2C4, C6.36(C4×D4), Q8.7(C4×S3), C2.1(S3×Q16), C4⋊C4.144D6, (C2×C8).205D6, C6.12(C2×Q16), C6.44(C4○D8), Q8⋊C4.9S3, (C2×Q8).122D6, Dic6.3(C2×C4), (Q8×Dic3).2C2, C6.Q16.1C2, C22.75(S3×D4), C12.11(C22×C4), (C8×Dic3).11C2, (C6×Q8).13C22, C12.156(C4○D4), C4.53(D4⋊2S3), (C2×C12).230C23, (C2×C24).234C22, (C2×Dic3).203D4, Dic6⋊C4.2C2, C2.3(Q8.7D6), C4⋊Dic3.80C22, C2.Dic12.10C2, (C2×Dic6).63C22, C2.20(Dic3⋊4D4), (C4×Dic3).225C22, C3⋊C8.7(C2×C4), C4.11(S3×C2×C4), (C3×Q8).2(C2×C4), (C2×C6).243(C2×D4), (C2×C3⋊Q16).2C2, (C3×C4⋊C4).31C22, (C2×C3⋊C8).215C22, (C2×C4).337(C22×S3), (C3×Q8⋊C4).10C2, SmallGroup(192,349)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊4Q16
G = < a,b,c,d | a6=c8=1, b2=a3, d2=c4, bab-1=cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 248 in 110 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, 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, C3×Q8, C4×C8, Q8⋊C4, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3⋊Q16, C3×C4⋊C4, C2×C24, C2×Dic6, C6×Q8, C4×Q16, C6.Q16, C8×Dic3, C2.Dic12, C3×Q8⋊C4, Dic6⋊C4, C2×C3⋊Q16, Q8×Dic3, Dic3⋊4Q16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, Q16, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C2×Q16, C4○D8, S3×C2×C4, S3×D4, D4⋊2S3, C4×Q16, Dic3⋊4D4, Q8.7D6, S3×Q16, Dic3⋊4Q16
►Regular action on 192 points
Generators in S192
(1 183 173 99 93 15)(2 16 94 100 174 184)(3 177 175 101 95 9)(4 10 96 102 176 178)(5 179 169 103 89 11)(6 12 90 104 170 180)(7 181 171 97 91 13)(8 14 92 98 172 182)(17 166 73 111 84 188)(18 189 85 112 74 167)(19 168 75 105 86 190)(20 191 87 106 76 161)(21 162 77 107 88 192)(22 185 81 108 78 163)(23 164 79 109 82 186)(24 187 83 110 80 165)(25 50 113 123 46 157)(26 158 47 124 114 51)(27 52 115 125 48 159)(28 160 41 126 116 53)(29 54 117 127 42 153)(30 154 43 128 118 55)(31 56 119 121 44 155)(32 156 45 122 120 49)(33 147 144 59 136 69)(34 70 129 60 137 148)(35 149 138 61 130 71)(36 72 131 62 139 150)(37 151 140 63 132 65)(38 66 133 64 141 152)(39 145 142 57 134 67)(40 68 135 58 143 146)
(1 132 99 151)(2 133 100 152)(3 134 101 145)(4 135 102 146)(5 136 103 147)(6 129 104 148)(7 130 97 149)(8 131 98 150)(9 67 175 142)(10 68 176 143)(11 69 169 144)(12 70 170 137)(13 71 171 138)(14 72 172 139)(15 65 173 140)(16 66 174 141)(17 160 111 116)(18 153 112 117)(19 154 105 118)(20 155 106 119)(21 156 107 120)(22 157 108 113)(23 158 109 114)(24 159 110 115)(25 81 123 163)(26 82 124 164)(27 83 125 165)(28 84 126 166)(29 85 127 167)(30 86 128 168)(31 87 121 161)(32 88 122 162)(33 179 59 89)(34 180 60 90)(35 181 61 91)(36 182 62 92)(37 183 63 93)(38 184 64 94)(39 177 57 95)(40 178 58 96)(41 73 53 188)(42 74 54 189)(43 75 55 190)(44 76 56 191)(45 77 49 192)(46 78 50 185)(47 79 51 186)(48 80 52 187)
(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)
(1 159 5 155)(2 158 6 154)(3 157 7 153)(4 156 8 160)(9 25 13 29)(10 32 14 28)(11 31 15 27)(12 30 16 26)(17 146 21 150)(18 145 22 149)(19 152 23 148)(20 151 24 147)(33 191 37 187)(34 190 38 186)(35 189 39 185)(36 188 40 192)(41 178 45 182)(42 177 46 181)(43 184 47 180)(44 183 48 179)(49 92 53 96)(50 91 54 95)(51 90 55 94)(52 89 56 93)(57 78 61 74)(58 77 62 73)(59 76 63 80)(60 75 64 79)(65 83 69 87)(66 82 70 86)(67 81 71 85)(68 88 72 84)(97 117 101 113)(98 116 102 120)(99 115 103 119)(100 114 104 118)(105 133 109 129)(106 132 110 136)(107 131 111 135)(108 130 112 134)(121 173 125 169)(122 172 126 176)(123 171 127 175)(124 170 128 174)(137 168 141 164)(138 167 142 163)(139 166 143 162)(140 165 144 161)
G:=sub<Sym(192)| (1,183,173,99,93,15)(2,16,94,100,174,184)(3,177,175,101,95,9)(4,10,96,102,176,178)(5,179,169,103,89,11)(6,12,90,104,170,180)(7,181,171,97,91,13)(8,14,92,98,172,182)(17,166,73,111,84,188)(18,189,85,112,74,167)(19,168,75,105,86,190)(20,191,87,106,76,161)(21,162,77,107,88,192)(22,185,81,108,78,163)(23,164,79,109,82,186)(24,187,83,110,80,165)(25,50,113,123,46,157)(26,158,47,124,114,51)(27,52,115,125,48,159)(28,160,41,126,116,53)(29,54,117,127,42,153)(30,154,43,128,118,55)(31,56,119,121,44,155)(32,156,45,122,120,49)(33,147,144,59,136,69)(34,70,129,60,137,148)(35,149,138,61,130,71)(36,72,131,62,139,150)(37,151,140,63,132,65)(38,66,133,64,141,152)(39,145,142,57,134,67)(40,68,135,58,143,146), (1,132,99,151)(2,133,100,152)(3,134,101,145)(4,135,102,146)(5,136,103,147)(6,129,104,148)(7,130,97,149)(8,131,98,150)(9,67,175,142)(10,68,176,143)(11,69,169,144)(12,70,170,137)(13,71,171,138)(14,72,172,139)(15,65,173,140)(16,66,174,141)(17,160,111,116)(18,153,112,117)(19,154,105,118)(20,155,106,119)(21,156,107,120)(22,157,108,113)(23,158,109,114)(24,159,110,115)(25,81,123,163)(26,82,124,164)(27,83,125,165)(28,84,126,166)(29,85,127,167)(30,86,128,168)(31,87,121,161)(32,88,122,162)(33,179,59,89)(34,180,60,90)(35,181,61,91)(36,182,62,92)(37,183,63,93)(38,184,64,94)(39,177,57,95)(40,178,58,96)(41,73,53,188)(42,74,54,189)(43,75,55,190)(44,76,56,191)(45,77,49,192)(46,78,50,185)(47,79,51,186)(48,80,52,187), (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), (1,159,5,155)(2,158,6,154)(3,157,7,153)(4,156,8,160)(9,25,13,29)(10,32,14,28)(11,31,15,27)(12,30,16,26)(17,146,21,150)(18,145,22,149)(19,152,23,148)(20,151,24,147)(33,191,37,187)(34,190,38,186)(35,189,39,185)(36,188,40,192)(41,178,45,182)(42,177,46,181)(43,184,47,180)(44,183,48,179)(49,92,53,96)(50,91,54,95)(51,90,55,94)(52,89,56,93)(57,78,61,74)(58,77,62,73)(59,76,63,80)(60,75,64,79)(65,83,69,87)(66,82,70,86)(67,81,71,85)(68,88,72,84)(97,117,101,113)(98,116,102,120)(99,115,103,119)(100,114,104,118)(105,133,109,129)(106,132,110,136)(107,131,111,135)(108,130,112,134)(121,173,125,169)(122,172,126,176)(123,171,127,175)(124,170,128,174)(137,168,141,164)(138,167,142,163)(139,166,143,162)(140,165,144,161)>;
G:=Group( (1,183,173,99,93,15)(2,16,94,100,174,184)(3,177,175,101,95,9)(4,10,96,102,176,178)(5,179,169,103,89,11)(6,12,90,104,170,180)(7,181,171,97,91,13)(8,14,92,98,172,182)(17,166,73,111,84,188)(18,189,85,112,74,167)(19,168,75,105,86,190)(20,191,87,106,76,161)(21,162,77,107,88,192)(22,185,81,108,78,163)(23,164,79,109,82,186)(24,187,83,110,80,165)(25,50,113,123,46,157)(26,158,47,124,114,51)(27,52,115,125,48,159)(28,160,41,126,116,53)(29,54,117,127,42,153)(30,154,43,128,118,55)(31,56,119,121,44,155)(32,156,45,122,120,49)(33,147,144,59,136,69)(34,70,129,60,137,148)(35,149,138,61,130,71)(36,72,131,62,139,150)(37,151,140,63,132,65)(38,66,133,64,141,152)(39,145,142,57,134,67)(40,68,135,58,143,146), (1,132,99,151)(2,133,100,152)(3,134,101,145)(4,135,102,146)(5,136,103,147)(6,129,104,148)(7,130,97,149)(8,131,98,150)(9,67,175,142)(10,68,176,143)(11,69,169,144)(12,70,170,137)(13,71,171,138)(14,72,172,139)(15,65,173,140)(16,66,174,141)(17,160,111,116)(18,153,112,117)(19,154,105,118)(20,155,106,119)(21,156,107,120)(22,157,108,113)(23,158,109,114)(24,159,110,115)(25,81,123,163)(26,82,124,164)(27,83,125,165)(28,84,126,166)(29,85,127,167)(30,86,128,168)(31,87,121,161)(32,88,122,162)(33,179,59,89)(34,180,60,90)(35,181,61,91)(36,182,62,92)(37,183,63,93)(38,184,64,94)(39,177,57,95)(40,178,58,96)(41,73,53,188)(42,74,54,189)(43,75,55,190)(44,76,56,191)(45,77,49,192)(46,78,50,185)(47,79,51,186)(48,80,52,187), (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), (1,159,5,155)(2,158,6,154)(3,157,7,153)(4,156,8,160)(9,25,13,29)(10,32,14,28)(11,31,15,27)(12,30,16,26)(17,146,21,150)(18,145,22,149)(19,152,23,148)(20,151,24,147)(33,191,37,187)(34,190,38,186)(35,189,39,185)(36,188,40,192)(41,178,45,182)(42,177,46,181)(43,184,47,180)(44,183,48,179)(49,92,53,96)(50,91,54,95)(51,90,55,94)(52,89,56,93)(57,78,61,74)(58,77,62,73)(59,76,63,80)(60,75,64,79)(65,83,69,87)(66,82,70,86)(67,81,71,85)(68,88,72,84)(97,117,101,113)(98,116,102,120)(99,115,103,119)(100,114,104,118)(105,133,109,129)(106,132,110,136)(107,131,111,135)(108,130,112,134)(121,173,125,169)(122,172,126,176)(123,171,127,175)(124,170,128,174)(137,168,141,164)(138,167,142,163)(139,166,143,162)(140,165,144,161) );
G=PermutationGroup([[(1,183,173,99,93,15),(2,16,94,100,174,184),(3,177,175,101,95,9),(4,10,96,102,176,178),(5,179,169,103,89,11),(6,12,90,104,170,180),(7,181,171,97,91,13),(8,14,92,98,172,182),(17,166,73,111,84,188),(18,189,85,112,74,167),(19,168,75,105,86,190),(20,191,87,106,76,161),(21,162,77,107,88,192),(22,185,81,108,78,163),(23,164,79,109,82,186),(24,187,83,110,80,165),(25,50,113,123,46,157),(26,158,47,124,114,51),(27,52,115,125,48,159),(28,160,41,126,116,53),(29,54,117,127,42,153),(30,154,43,128,118,55),(31,56,119,121,44,155),(32,156,45,122,120,49),(33,147,144,59,136,69),(34,70,129,60,137,148),(35,149,138,61,130,71),(36,72,131,62,139,150),(37,151,140,63,132,65),(38,66,133,64,141,152),(39,145,142,57,134,67),(40,68,135,58,143,146)], [(1,132,99,151),(2,133,100,152),(3,134,101,145),(4,135,102,146),(5,136,103,147),(6,129,104,148),(7,130,97,149),(8,131,98,150),(9,67,175,142),(10,68,176,143),(11,69,169,144),(12,70,170,137),(13,71,171,138),(14,72,172,139),(15,65,173,140),(16,66,174,141),(17,160,111,116),(18,153,112,117),(19,154,105,118),(20,155,106,119),(21,156,107,120),(22,157,108,113),(23,158,109,114),(24,159,110,115),(25,81,123,163),(26,82,124,164),(27,83,125,165),(28,84,126,166),(29,85,127,167),(30,86,128,168),(31,87,121,161),(32,88,122,162),(33,179,59,89),(34,180,60,90),(35,181,61,91),(36,182,62,92),(37,183,63,93),(38,184,64,94),(39,177,57,95),(40,178,58,96),(41,73,53,188),(42,74,54,189),(43,75,55,190),(44,76,56,191),(45,77,49,192),(46,78,50,185),(47,79,51,186),(48,80,52,187)], [(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)], [(1,159,5,155),(2,158,6,154),(3,157,7,153),(4,156,8,160),(9,25,13,29),(10,32,14,28),(11,31,15,27),(12,30,16,26),(17,146,21,150),(18,145,22,149),(19,152,23,148),(20,151,24,147),(33,191,37,187),(34,190,38,186),(35,189,39,185),(36,188,40,192),(41,178,45,182),(42,177,46,181),(43,184,47,180),(44,183,48,179),(49,92,53,96),(50,91,54,95),(51,90,55,94),(52,89,56,93),(57,78,61,74),(58,77,62,73),(59,76,63,80),(60,75,64,79),(65,83,69,87),(66,82,70,86),(67,81,71,85),(68,88,72,84),(97,117,101,113),(98,116,102,120),(99,115,103,119),(100,114,104,118),(105,133,109,129),(106,132,110,136),(107,131,111,135),(108,130,112,134),(121,173,125,169),(122,172,126,176),(123,171,127,175),(124,170,128,174),(137,168,141,164),(138,167,142,163),(139,166,143,162),(140,165,144,161)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 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 | 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 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 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 | C4 | S3 | D4 | D6 | D6 | D6 | Q16 | C4○D4 | C4×S3 | C4○D8 | D4⋊2S3 | S3×D4 | Q8.7D6 | S3×Q16 |
| kernel | Dic3⋊4Q16 | C6.Q16 | C8×Dic3 | C2.Dic12 | C3×Q8⋊C4 | Dic6⋊C4 | C2×C3⋊Q16 | Q8×Dic3 | C3⋊Q16 | Q8⋊C4 | C2×Dic3 | C4⋊C4 | C2×C8 | C2×Q8 | Dic3 | C12 | Q8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of Dic3⋊4Q16 ►in GL4(𝔽73) generated by
| 1 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 5 | 54 | 0 | 0 |
| 59 | 68 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 11 | 2 | 0 | 0 |
| 13 | 62 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 57 | 32 |
| 11 | 2 | 0 | 0 |
| 13 | 62 | 0 | 0 |
| 0 | 0 | 51 | 27 |
| 0 | 0 | 28 | 22 |
G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[5,59,0,0,54,68,0,0,0,0,46,0,0,0,0,46],[11,13,0,0,2,62,0,0,0,0,0,57,0,0,32,32],[11,13,0,0,2,62,0,0,0,0,51,28,0,0,27,22] >;
Dic3⋊4Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_4Q_{16} % in TeX
G:=Group("Dic3:4Q16"); // GroupNames label
G:=SmallGroup(192,349);
// by ID
G=gap.SmallGroup(192,349);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,135,268,570,297,136,6278]);
// 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=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations