metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊5Q16, Dic12⋊6C4, C3⋊3(C4×Q16), C8.15(C4×S3), C6.53(C4×D4), C2.3(S3×Q16), C2.D8.9S3, C24.16(C2×C4), C4⋊C4.167D6, (C2×C8).226D6, C6.20(C2×Q16), C6.26(C4○D8), Dic6.9(C2×C4), (C8×Dic3).1C2, C22.88(S3×D4), C12.36(C4○D4), C2.3(D8⋊3S3), C12.47(C22×C4), (C2×C24).78C22, C4.8(Q8⋊3S3), (C2×Dic12).9C2, C6.SD16.7C2, (C2×C12).289C23, Dic6⋊C4.7C2, (C2×Dic3).208D4, C2.13(Dic3⋊5D4), (C2×Dic6).86C22, (C4×Dic3).232C22, C4.44(S3×C2×C4), (C3×C2.D8).4C2, (C2×C6).294(C2×D4), (C3×C4⋊C4).82C22, (C2×C3⋊C8).230C22, (C2×C4).392(C22×S3), SmallGroup(192,432)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊5Q16
G = < a,b,c,d | a6=c8=1, b2=a3, d2=c4, bab-1=dad-1=a-1, ac=ca, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 256 in 110 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, Dic12, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, C4×Q16, C6.SD16, C8×Dic3, C3×C2.D8, Dic6⋊C4, C2×Dic12, Dic3⋊5Q16
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, Q8⋊3S3, C4×Q16, Dic3⋊5D4, D8⋊3S3, S3×Q16, Dic3⋊5Q16
►Regular action on 192 points
Generators in S192
(1 126 51 43 145 141)(2 127 52 44 146 142)(3 128 53 45 147 143)(4 121 54 46 148 144)(5 122 55 47 149 137)(6 123 56 48 150 138)(7 124 49 41 151 139)(8 125 50 42 152 140)(9 78 168 129 159 81)(10 79 161 130 160 82)(11 80 162 131 153 83)(12 73 163 132 154 84)(13 74 164 133 155 85)(14 75 165 134 156 86)(15 76 166 135 157 87)(16 77 167 136 158 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)(97 187 115 182 111 175)(98 188 116 183 112 176)(99 189 117 184 105 169)(100 190 118 177 106 170)(101 191 119 178 107 171)(102 192 120 179 108 172)(103 185 113 180 109 173)(104 186 114 181 110 174)
(1 132 43 12)(2 133 44 13)(3 134 45 14)(4 135 46 15)(5 136 47 16)(6 129 48 9)(7 130 41 10)(8 131 42 11)(17 169 64 117)(18 170 57 118)(19 171 58 119)(20 172 59 120)(21 173 60 113)(22 174 61 114)(23 175 62 115)(24 176 63 116)(25 184 67 99)(26 177 68 100)(27 178 69 101)(28 179 70 102)(29 180 71 103)(30 181 72 104)(31 182 65 97)(32 183 66 98)(33 187 95 111)(34 188 96 112)(35 189 89 105)(36 190 90 106)(37 191 91 107)(38 192 92 108)(39 185 93 109)(40 186 94 110)(49 79 139 160)(50 80 140 153)(51 73 141 154)(52 74 142 155)(53 75 143 156)(54 76 144 157)(55 77 137 158)(56 78 138 159)(81 123 168 150)(82 124 161 151)(83 125 162 152)(84 126 163 145)(85 127 164 146)(86 128 165 147)(87 121 166 148)(88 122 167 149)
(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)
(1 115 5 119)(2 114 6 118)(3 113 7 117)(4 120 8 116)(9 57 13 61)(10 64 14 60)(11 63 15 59)(12 62 16 58)(17 134 21 130)(18 133 22 129)(19 132 23 136)(20 131 24 135)(25 75 29 79)(26 74 30 78)(27 73 31 77)(28 80 32 76)(33 88 37 84)(34 87 38 83)(35 86 39 82)(36 85 40 81)(41 169 45 173)(42 176 46 172)(43 175 47 171)(44 174 48 170)(49 99 53 103)(50 98 54 102)(51 97 55 101)(52 104 56 100)(65 158 69 154)(66 157 70 153)(67 156 71 160)(68 155 72 159)(89 165 93 161)(90 164 94 168)(91 163 95 167)(92 162 96 166)(105 147 109 151)(106 146 110 150)(107 145 111 149)(108 152 112 148)(121 192 125 188)(122 191 126 187)(123 190 127 186)(124 189 128 185)(137 178 141 182)(138 177 142 181)(139 184 143 180)(140 183 144 179)
G:=sub<Sym(192)| (1,126,51,43,145,141)(2,127,52,44,146,142)(3,128,53,45,147,143)(4,121,54,46,148,144)(5,122,55,47,149,137)(6,123,56,48,150,138)(7,124,49,41,151,139)(8,125,50,42,152,140)(9,78,168,129,159,81)(10,79,161,130,160,82)(11,80,162,131,153,83)(12,73,163,132,154,84)(13,74,164,133,155,85)(14,75,165,134,156,86)(15,76,166,135,157,87)(16,77,167,136,158,88)(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)(97,187,115,182,111,175)(98,188,116,183,112,176)(99,189,117,184,105,169)(100,190,118,177,106,170)(101,191,119,178,107,171)(102,192,120,179,108,172)(103,185,113,180,109,173)(104,186,114,181,110,174), (1,132,43,12)(2,133,44,13)(3,134,45,14)(4,135,46,15)(5,136,47,16)(6,129,48,9)(7,130,41,10)(8,131,42,11)(17,169,64,117)(18,170,57,118)(19,171,58,119)(20,172,59,120)(21,173,60,113)(22,174,61,114)(23,175,62,115)(24,176,63,116)(25,184,67,99)(26,177,68,100)(27,178,69,101)(28,179,70,102)(29,180,71,103)(30,181,72,104)(31,182,65,97)(32,183,66,98)(33,187,95,111)(34,188,96,112)(35,189,89,105)(36,190,90,106)(37,191,91,107)(38,192,92,108)(39,185,93,109)(40,186,94,110)(49,79,139,160)(50,80,140,153)(51,73,141,154)(52,74,142,155)(53,75,143,156)(54,76,144,157)(55,77,137,158)(56,78,138,159)(81,123,168,150)(82,124,161,151)(83,125,162,152)(84,126,163,145)(85,127,164,146)(86,128,165,147)(87,121,166,148)(88,122,167,149), (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), (1,115,5,119)(2,114,6,118)(3,113,7,117)(4,120,8,116)(9,57,13,61)(10,64,14,60)(11,63,15,59)(12,62,16,58)(17,134,21,130)(18,133,22,129)(19,132,23,136)(20,131,24,135)(25,75,29,79)(26,74,30,78)(27,73,31,77)(28,80,32,76)(33,88,37,84)(34,87,38,83)(35,86,39,82)(36,85,40,81)(41,169,45,173)(42,176,46,172)(43,175,47,171)(44,174,48,170)(49,99,53,103)(50,98,54,102)(51,97,55,101)(52,104,56,100)(65,158,69,154)(66,157,70,153)(67,156,71,160)(68,155,72,159)(89,165,93,161)(90,164,94,168)(91,163,95,167)(92,162,96,166)(105,147,109,151)(106,146,110,150)(107,145,111,149)(108,152,112,148)(121,192,125,188)(122,191,126,187)(123,190,127,186)(124,189,128,185)(137,178,141,182)(138,177,142,181)(139,184,143,180)(140,183,144,179)>;
G:=Group( (1,126,51,43,145,141)(2,127,52,44,146,142)(3,128,53,45,147,143)(4,121,54,46,148,144)(5,122,55,47,149,137)(6,123,56,48,150,138)(7,124,49,41,151,139)(8,125,50,42,152,140)(9,78,168,129,159,81)(10,79,161,130,160,82)(11,80,162,131,153,83)(12,73,163,132,154,84)(13,74,164,133,155,85)(14,75,165,134,156,86)(15,76,166,135,157,87)(16,77,167,136,158,88)(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)(97,187,115,182,111,175)(98,188,116,183,112,176)(99,189,117,184,105,169)(100,190,118,177,106,170)(101,191,119,178,107,171)(102,192,120,179,108,172)(103,185,113,180,109,173)(104,186,114,181,110,174), (1,132,43,12)(2,133,44,13)(3,134,45,14)(4,135,46,15)(5,136,47,16)(6,129,48,9)(7,130,41,10)(8,131,42,11)(17,169,64,117)(18,170,57,118)(19,171,58,119)(20,172,59,120)(21,173,60,113)(22,174,61,114)(23,175,62,115)(24,176,63,116)(25,184,67,99)(26,177,68,100)(27,178,69,101)(28,179,70,102)(29,180,71,103)(30,181,72,104)(31,182,65,97)(32,183,66,98)(33,187,95,111)(34,188,96,112)(35,189,89,105)(36,190,90,106)(37,191,91,107)(38,192,92,108)(39,185,93,109)(40,186,94,110)(49,79,139,160)(50,80,140,153)(51,73,141,154)(52,74,142,155)(53,75,143,156)(54,76,144,157)(55,77,137,158)(56,78,138,159)(81,123,168,150)(82,124,161,151)(83,125,162,152)(84,126,163,145)(85,127,164,146)(86,128,165,147)(87,121,166,148)(88,122,167,149), (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), (1,115,5,119)(2,114,6,118)(3,113,7,117)(4,120,8,116)(9,57,13,61)(10,64,14,60)(11,63,15,59)(12,62,16,58)(17,134,21,130)(18,133,22,129)(19,132,23,136)(20,131,24,135)(25,75,29,79)(26,74,30,78)(27,73,31,77)(28,80,32,76)(33,88,37,84)(34,87,38,83)(35,86,39,82)(36,85,40,81)(41,169,45,173)(42,176,46,172)(43,175,47,171)(44,174,48,170)(49,99,53,103)(50,98,54,102)(51,97,55,101)(52,104,56,100)(65,158,69,154)(66,157,70,153)(67,156,71,160)(68,155,72,159)(89,165,93,161)(90,164,94,168)(91,163,95,167)(92,162,96,166)(105,147,109,151)(106,146,110,150)(107,145,111,149)(108,152,112,148)(121,192,125,188)(122,191,126,187)(123,190,127,186)(124,189,128,185)(137,178,141,182)(138,177,142,181)(139,184,143,180)(140,183,144,179) );
G=PermutationGroup([[(1,126,51,43,145,141),(2,127,52,44,146,142),(3,128,53,45,147,143),(4,121,54,46,148,144),(5,122,55,47,149,137),(6,123,56,48,150,138),(7,124,49,41,151,139),(8,125,50,42,152,140),(9,78,168,129,159,81),(10,79,161,130,160,82),(11,80,162,131,153,83),(12,73,163,132,154,84),(13,74,164,133,155,85),(14,75,165,134,156,86),(15,76,166,135,157,87),(16,77,167,136,158,88),(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),(97,187,115,182,111,175),(98,188,116,183,112,176),(99,189,117,184,105,169),(100,190,118,177,106,170),(101,191,119,178,107,171),(102,192,120,179,108,172),(103,185,113,180,109,173),(104,186,114,181,110,174)], [(1,132,43,12),(2,133,44,13),(3,134,45,14),(4,135,46,15),(5,136,47,16),(6,129,48,9),(7,130,41,10),(8,131,42,11),(17,169,64,117),(18,170,57,118),(19,171,58,119),(20,172,59,120),(21,173,60,113),(22,174,61,114),(23,175,62,115),(24,176,63,116),(25,184,67,99),(26,177,68,100),(27,178,69,101),(28,179,70,102),(29,180,71,103),(30,181,72,104),(31,182,65,97),(32,183,66,98),(33,187,95,111),(34,188,96,112),(35,189,89,105),(36,190,90,106),(37,191,91,107),(38,192,92,108),(39,185,93,109),(40,186,94,110),(49,79,139,160),(50,80,140,153),(51,73,141,154),(52,74,142,155),(53,75,143,156),(54,76,144,157),(55,77,137,158),(56,78,138,159),(81,123,168,150),(82,124,161,151),(83,125,162,152),(84,126,163,145),(85,127,164,146),(86,128,165,147),(87,121,166,148),(88,122,167,149)], [(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)], [(1,115,5,119),(2,114,6,118),(3,113,7,117),(4,120,8,116),(9,57,13,61),(10,64,14,60),(11,63,15,59),(12,62,16,58),(17,134,21,130),(18,133,22,129),(19,132,23,136),(20,131,24,135),(25,75,29,79),(26,74,30,78),(27,73,31,77),(28,80,32,76),(33,88,37,84),(34,87,38,83),(35,86,39,82),(36,85,40,81),(41,169,45,173),(42,176,46,172),(43,175,47,171),(44,174,48,170),(49,99,53,103),(50,98,54,102),(51,97,55,101),(52,104,56,100),(65,158,69,154),(66,157,70,153),(67,156,71,160),(68,155,72,159),(89,165,93,161),(90,164,94,168),(91,163,95,167),(92,162,96,166),(105,147,109,151),(106,146,110,150),(107,145,111,149),(108,152,112,148),(121,192,125,188),(122,191,126,187),(123,190,127,186),(124,189,128,185),(137,178,141,182),(138,177,142,181),(139,184,143,180),(140,183,144,179)]])
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 | D6 | Q16 | C4○D4 | C4×S3 | C4○D8 | Q8⋊3S3 | S3×D4 | D8⋊3S3 | S3×Q16 |
| kernel | Dic3⋊5Q16 | C6.SD16 | C8×Dic3 | C3×C2.D8 | Dic6⋊C4 | C2×Dic12 | Dic12 | C2.D8 | C2×Dic3 | C4⋊C4 | C2×C8 | Dic3 | C12 | C8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of Dic3⋊5Q16 ►in GL4(𝔽73) generated by
| 1 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 37 | 11 | 0 | 0 |
| 48 | 36 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 41 |
| 0 | 0 | 16 | 41 |
| 50 | 5 | 0 | 0 |
| 55 | 23 | 0 | 0 |
| 0 | 0 | 17 | 55 |
| 0 | 0 | 8 | 56 |
G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[37,48,0,0,11,36,0,0,0,0,46,0,0,0,0,46],[72,0,0,0,0,72,0,0,0,0,0,16,0,0,41,41],[50,55,0,0,5,23,0,0,0,0,17,8,0,0,55,56] >;
Dic3⋊5Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_5Q_{16} % in TeX
G:=Group("Dic3:5Q16"); // GroupNames label
G:=SmallGroup(192,432);
// by ID
G=gap.SmallGroup(192,432);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,135,268,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^8=1,b^2=a^3,d^2=c^4,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations