direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3×Q16, C3⋊5(C4×Q16), (C3×Q16)⋊2C4, C6.97(C4×D4), C2.5(S3×Q16), C24.19(C2×C4), (C2×C8).241D6, (C6×Q16).4C2, (C2×Q16).7S3, C6.26(C2×Q16), C6.78(C4○D8), (C2×Q8).142D6, (Q8×Dic3).7C2, Q8.6(C2×Dic3), (C8×Dic3).4C2, C24⋊1C4.16C2, C2.14(D4×Dic3), C8.10(C2×Dic3), (C2×C24).93C22, C12.75(C22×C4), C22.118(S3×D4), C12.103(C4○D4), (C6×Q8).84C22, C4.34(D4⋊2S3), C2.5(D24⋊C2), C4.5(C22×Dic3), (C2×C12).455C23, (C2×Dic3).214D4, Q8⋊2Dic3.14C2, C4⋊Dic3.179C22, (C4×Dic3).243C22, (C3×Q8).8(C2×C4), (C2×C6).366(C2×D4), (C2×C3⋊C8).276C22, (C2×C4).543(C22×S3), SmallGroup(192,740)
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=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 232 in 110 conjugacy classes, 59 normal (27 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Q8, Q8, Dic3, 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, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4⋊Dic3, C4⋊Dic3, C2×C24, C3×Q16, C6×Q8, C4×Q16, C8×Dic3, C24⋊1C4, Q8⋊2Dic3, Q8×Dic3, C6×Q16, Dic3×Q16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, Q16, C22×C4, C2×D4, C4○D4, C2×Dic3, C22×S3, C4×D4, C2×Q16, C4○D8, S3×D4, D4⋊2S3, C22×Dic3, C4×Q16, S3×Q16, D24⋊C2, D4×Dic3, Dic3×Q16
►Regular action on 192 points
Generators in S192
(1 147 41 55 154 142)(2 148 42 56 155 143)(3 149 43 49 156 144)(4 150 44 50 157 137)(5 151 45 51 158 138)(6 152 46 52 159 139)(7 145 47 53 160 140)(8 146 48 54 153 141)(9 94 133 121 162 85)(10 95 134 122 163 86)(11 96 135 123 164 87)(12 89 136 124 165 88)(13 90 129 125 166 81)(14 91 130 126 167 82)(15 92 131 127 168 83)(16 93 132 128 161 84)(17 69 74 37 63 29)(18 70 75 38 64 30)(19 71 76 39 57 31)(20 72 77 40 58 32)(21 65 78 33 59 25)(22 66 79 34 60 26)(23 67 80 35 61 27)(24 68 73 36 62 28)(97 109 182 187 172 118)(98 110 183 188 173 119)(99 111 184 189 174 120)(100 112 177 190 175 113)(101 105 178 191 176 114)(102 106 179 192 169 115)(103 107 180 185 170 116)(104 108 181 186 171 117)
(1 129 55 81)(2 130 56 82)(3 131 49 83)(4 132 50 84)(5 133 51 85)(6 134 52 86)(7 135 53 87)(8 136 54 88)(9 138 121 45)(10 139 122 46)(11 140 123 47)(12 141 124 48)(13 142 125 41)(14 143 126 42)(15 144 127 43)(16 137 128 44)(17 99 37 189)(18 100 38 190)(19 101 39 191)(20 102 40 192)(21 103 33 185)(22 104 34 186)(23 97 35 187)(24 98 36 188)(25 107 78 170)(26 108 79 171)(27 109 80 172)(28 110 73 173)(29 111 74 174)(30 112 75 175)(31 105 76 176)(32 106 77 169)(57 178 71 114)(58 179 72 115)(59 180 65 116)(60 181 66 117)(61 182 67 118)(62 183 68 119)(63 184 69 120)(64 177 70 113)(89 153 165 146)(90 154 166 147)(91 155 167 148)(92 156 168 149)(93 157 161 150)(94 158 162 151)(95 159 163 152)(96 160 164 145)
(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 27 5 31)(2 26 6 30)(3 25 7 29)(4 32 8 28)(9 114 13 118)(10 113 14 117)(11 120 15 116)(12 119 16 115)(17 149 21 145)(18 148 22 152)(19 147 23 151)(20 146 24 150)(33 160 37 156)(34 159 38 155)(35 158 39 154)(36 157 40 153)(41 67 45 71)(42 66 46 70)(43 65 47 69)(44 72 48 68)(49 78 53 74)(50 77 54 73)(51 76 55 80)(52 75 56 79)(57 142 61 138)(58 141 62 137)(59 140 63 144)(60 139 64 143)(81 172 85 176)(82 171 86 175)(83 170 87 174)(84 169 88 173)(89 98 93 102)(90 97 94 101)(91 104 95 100)(92 103 96 99)(105 129 109 133)(106 136 110 132)(107 135 111 131)(108 134 112 130)(121 178 125 182)(122 177 126 181)(123 184 127 180)(124 183 128 179)(161 192 165 188)(162 191 166 187)(163 190 167 186)(164 189 168 185)
G:=sub<Sym(192)| (1,147,41,55,154,142)(2,148,42,56,155,143)(3,149,43,49,156,144)(4,150,44,50,157,137)(5,151,45,51,158,138)(6,152,46,52,159,139)(7,145,47,53,160,140)(8,146,48,54,153,141)(9,94,133,121,162,85)(10,95,134,122,163,86)(11,96,135,123,164,87)(12,89,136,124,165,88)(13,90,129,125,166,81)(14,91,130,126,167,82)(15,92,131,127,168,83)(16,93,132,128,161,84)(17,69,74,37,63,29)(18,70,75,38,64,30)(19,71,76,39,57,31)(20,72,77,40,58,32)(21,65,78,33,59,25)(22,66,79,34,60,26)(23,67,80,35,61,27)(24,68,73,36,62,28)(97,109,182,187,172,118)(98,110,183,188,173,119)(99,111,184,189,174,120)(100,112,177,190,175,113)(101,105,178,191,176,114)(102,106,179,192,169,115)(103,107,180,185,170,116)(104,108,181,186,171,117), (1,129,55,81)(2,130,56,82)(3,131,49,83)(4,132,50,84)(5,133,51,85)(6,134,52,86)(7,135,53,87)(8,136,54,88)(9,138,121,45)(10,139,122,46)(11,140,123,47)(12,141,124,48)(13,142,125,41)(14,143,126,42)(15,144,127,43)(16,137,128,44)(17,99,37,189)(18,100,38,190)(19,101,39,191)(20,102,40,192)(21,103,33,185)(22,104,34,186)(23,97,35,187)(24,98,36,188)(25,107,78,170)(26,108,79,171)(27,109,80,172)(28,110,73,173)(29,111,74,174)(30,112,75,175)(31,105,76,176)(32,106,77,169)(57,178,71,114)(58,179,72,115)(59,180,65,116)(60,181,66,117)(61,182,67,118)(62,183,68,119)(63,184,69,120)(64,177,70,113)(89,153,165,146)(90,154,166,147)(91,155,167,148)(92,156,168,149)(93,157,161,150)(94,158,162,151)(95,159,163,152)(96,160,164,145), (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), (1,27,5,31)(2,26,6,30)(3,25,7,29)(4,32,8,28)(9,114,13,118)(10,113,14,117)(11,120,15,116)(12,119,16,115)(17,149,21,145)(18,148,22,152)(19,147,23,151)(20,146,24,150)(33,160,37,156)(34,159,38,155)(35,158,39,154)(36,157,40,153)(41,67,45,71)(42,66,46,70)(43,65,47,69)(44,72,48,68)(49,78,53,74)(50,77,54,73)(51,76,55,80)(52,75,56,79)(57,142,61,138)(58,141,62,137)(59,140,63,144)(60,139,64,143)(81,172,85,176)(82,171,86,175)(83,170,87,174)(84,169,88,173)(89,98,93,102)(90,97,94,101)(91,104,95,100)(92,103,96,99)(105,129,109,133)(106,136,110,132)(107,135,111,131)(108,134,112,130)(121,178,125,182)(122,177,126,181)(123,184,127,180)(124,183,128,179)(161,192,165,188)(162,191,166,187)(163,190,167,186)(164,189,168,185)>;
G:=Group( (1,147,41,55,154,142)(2,148,42,56,155,143)(3,149,43,49,156,144)(4,150,44,50,157,137)(5,151,45,51,158,138)(6,152,46,52,159,139)(7,145,47,53,160,140)(8,146,48,54,153,141)(9,94,133,121,162,85)(10,95,134,122,163,86)(11,96,135,123,164,87)(12,89,136,124,165,88)(13,90,129,125,166,81)(14,91,130,126,167,82)(15,92,131,127,168,83)(16,93,132,128,161,84)(17,69,74,37,63,29)(18,70,75,38,64,30)(19,71,76,39,57,31)(20,72,77,40,58,32)(21,65,78,33,59,25)(22,66,79,34,60,26)(23,67,80,35,61,27)(24,68,73,36,62,28)(97,109,182,187,172,118)(98,110,183,188,173,119)(99,111,184,189,174,120)(100,112,177,190,175,113)(101,105,178,191,176,114)(102,106,179,192,169,115)(103,107,180,185,170,116)(104,108,181,186,171,117), (1,129,55,81)(2,130,56,82)(3,131,49,83)(4,132,50,84)(5,133,51,85)(6,134,52,86)(7,135,53,87)(8,136,54,88)(9,138,121,45)(10,139,122,46)(11,140,123,47)(12,141,124,48)(13,142,125,41)(14,143,126,42)(15,144,127,43)(16,137,128,44)(17,99,37,189)(18,100,38,190)(19,101,39,191)(20,102,40,192)(21,103,33,185)(22,104,34,186)(23,97,35,187)(24,98,36,188)(25,107,78,170)(26,108,79,171)(27,109,80,172)(28,110,73,173)(29,111,74,174)(30,112,75,175)(31,105,76,176)(32,106,77,169)(57,178,71,114)(58,179,72,115)(59,180,65,116)(60,181,66,117)(61,182,67,118)(62,183,68,119)(63,184,69,120)(64,177,70,113)(89,153,165,146)(90,154,166,147)(91,155,167,148)(92,156,168,149)(93,157,161,150)(94,158,162,151)(95,159,163,152)(96,160,164,145), (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), (1,27,5,31)(2,26,6,30)(3,25,7,29)(4,32,8,28)(9,114,13,118)(10,113,14,117)(11,120,15,116)(12,119,16,115)(17,149,21,145)(18,148,22,152)(19,147,23,151)(20,146,24,150)(33,160,37,156)(34,159,38,155)(35,158,39,154)(36,157,40,153)(41,67,45,71)(42,66,46,70)(43,65,47,69)(44,72,48,68)(49,78,53,74)(50,77,54,73)(51,76,55,80)(52,75,56,79)(57,142,61,138)(58,141,62,137)(59,140,63,144)(60,139,64,143)(81,172,85,176)(82,171,86,175)(83,170,87,174)(84,169,88,173)(89,98,93,102)(90,97,94,101)(91,104,95,100)(92,103,96,99)(105,129,109,133)(106,136,110,132)(107,135,111,131)(108,134,112,130)(121,178,125,182)(122,177,126,181)(123,184,127,180)(124,183,128,179)(161,192,165,188)(162,191,166,187)(163,190,167,186)(164,189,168,185) );
G=PermutationGroup([[(1,147,41,55,154,142),(2,148,42,56,155,143),(3,149,43,49,156,144),(4,150,44,50,157,137),(5,151,45,51,158,138),(6,152,46,52,159,139),(7,145,47,53,160,140),(8,146,48,54,153,141),(9,94,133,121,162,85),(10,95,134,122,163,86),(11,96,135,123,164,87),(12,89,136,124,165,88),(13,90,129,125,166,81),(14,91,130,126,167,82),(15,92,131,127,168,83),(16,93,132,128,161,84),(17,69,74,37,63,29),(18,70,75,38,64,30),(19,71,76,39,57,31),(20,72,77,40,58,32),(21,65,78,33,59,25),(22,66,79,34,60,26),(23,67,80,35,61,27),(24,68,73,36,62,28),(97,109,182,187,172,118),(98,110,183,188,173,119),(99,111,184,189,174,120),(100,112,177,190,175,113),(101,105,178,191,176,114),(102,106,179,192,169,115),(103,107,180,185,170,116),(104,108,181,186,171,117)], [(1,129,55,81),(2,130,56,82),(3,131,49,83),(4,132,50,84),(5,133,51,85),(6,134,52,86),(7,135,53,87),(8,136,54,88),(9,138,121,45),(10,139,122,46),(11,140,123,47),(12,141,124,48),(13,142,125,41),(14,143,126,42),(15,144,127,43),(16,137,128,44),(17,99,37,189),(18,100,38,190),(19,101,39,191),(20,102,40,192),(21,103,33,185),(22,104,34,186),(23,97,35,187),(24,98,36,188),(25,107,78,170),(26,108,79,171),(27,109,80,172),(28,110,73,173),(29,111,74,174),(30,112,75,175),(31,105,76,176),(32,106,77,169),(57,178,71,114),(58,179,72,115),(59,180,65,116),(60,181,66,117),(61,182,67,118),(62,183,68,119),(63,184,69,120),(64,177,70,113),(89,153,165,146),(90,154,166,147),(91,155,167,148),(92,156,168,149),(93,157,161,150),(94,158,162,151),(95,159,163,152),(96,160,164,145)], [(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)], [(1,27,5,31),(2,26,6,30),(3,25,7,29),(4,32,8,28),(9,114,13,118),(10,113,14,117),(11,120,15,116),(12,119,16,115),(17,149,21,145),(18,148,22,152),(19,147,23,151),(20,146,24,150),(33,160,37,156),(34,159,38,155),(35,158,39,154),(36,157,40,153),(41,67,45,71),(42,66,46,70),(43,65,47,69),(44,72,48,68),(49,78,53,74),(50,77,54,73),(51,76,55,80),(52,75,56,79),(57,142,61,138),(58,141,62,137),(59,140,63,144),(60,139,64,143),(81,172,85,176),(82,171,86,175),(83,170,87,174),(84,169,88,173),(89,98,93,102),(90,97,94,101),(91,104,95,100),(92,103,96,99),(105,129,109,133),(106,136,110,132),(107,135,111,131),(108,134,112,130),(121,178,125,182),(122,177,126,181),(123,184,127,180),(124,183,128,179),(161,192,165,188),(162,191,166,187),(163,190,167,186),(164,189,168,185)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | Dic3 | D6 | Q16 | C4○D4 | C4○D8 | D4⋊2S3 | S3×D4 | S3×Q16 | D24⋊C2 |
| kernel | Dic3×Q16 | C8×Dic3 | C24⋊1C4 | Q8⋊2Dic3 | Q8×Dic3 | C6×Q16 | C3×Q16 | C2×Q16 | C2×Dic3 | C2×C8 | Q16 | C2×Q8 | Dic3 | C12 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 8 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of Dic3×Q16 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 1 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 65 | 34 |
| 0 | 0 | 26 | 8 |
| 0 | 35 | 0 | 0 |
| 25 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 68 | 9 | 0 | 0 |
| 62 | 5 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,1,0,0,72,0],[46,0,0,0,0,46,0,0,0,0,65,26,0,0,34,8],[0,25,0,0,35,32,0,0,0,0,1,0,0,0,0,1],[68,62,0,0,9,5,0,0,0,0,72,0,0,0,0,72] >;
Dic3×Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times Q_{16} % in TeX
G:=Group("Dic3xQ16"); // GroupNames label
G:=SmallGroup(192,740);
// by ID
G=gap.SmallGroup(192,740);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,219,184,851,438,102,6278]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations