metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊3Q16, (C2×C8).36D6, (C3×Q8).7D4, C3⋊5(C4⋊2Q16), (C2×Q16).3S3, (C6×Q16).8C2, C2.16(S3×Q16), C6.27(C2×Q16), C12.184(C2×D4), Dic3⋊C8.9C2, (C2×Q8).143D6, (Q8×Dic3).8C2, Q8.9(C3⋊D4), C22.274(S3×D4), C2.Dic12.8C2, C12.104(C4○D4), (C6×Q8).85C22, C4.13(D4⋊2S3), C6.119(C4⋊D4), (C2×C12).456C23, (C2×C24).178C22, (C2×Dic3).186D4, Dic3⋊Q8.5C2, Q8⋊2Dic3.15C2, C2.26(Q16⋊S3), C6.75(C8.C22), C4⋊Dic3.180C22, (C4×Dic3).53C22, C2.28(C23.14D6), (C2×Dic6).130C22, C4.46(C2×C3⋊D4), (C2×C6).367(C2×D4), (C2×C3⋊Q16).8C2, (C2×C3⋊C8).162C22, (C2×C4).544(C22×S3), SmallGroup(192,741)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊3Q16
G = < a,b,c,d | a6=c8=1, b2=a3, d2=c4, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >
Subgroups: 264 in 108 conjugacy classes, 43 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, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C2×Q16, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3⋊Q16, C2×C24, C3×Q16, C2×Dic6, C6×Q8, C4⋊2Q16, Dic3⋊C8, C2.Dic12, Q8⋊2Dic3, C2×C3⋊Q16, Dic3⋊Q8, Q8×Dic3, C6×Q16, Dic3⋊3Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×Q16, C8.C22, S3×D4, D4⋊2S3, C2×C3⋊D4, C4⋊2Q16, S3×Q16, Q16⋊S3, C23.14D6, Dic3⋊3Q16
►Regular action on 192 points
Generators in S192
(1 126 149 27 51 137)(2 127 150 28 52 138)(3 128 151 29 53 139)(4 121 152 30 54 140)(5 122 145 31 55 141)(6 123 146 32 56 142)(7 124 147 25 49 143)(8 125 148 26 50 144)(9 78 163 130 157 81)(10 79 164 131 158 82)(11 80 165 132 159 83)(12 73 166 133 160 84)(13 74 167 134 153 85)(14 75 168 135 154 86)(15 76 161 136 155 87)(16 77 162 129 156 88)(17 89 60 33 42 67)(18 90 61 34 43 68)(19 91 62 35 44 69)(20 92 63 36 45 70)(21 93 64 37 46 71)(22 94 57 38 47 72)(23 95 58 39 48 65)(24 96 59 40 41 66)(97 109 169 187 117 182)(98 110 170 188 118 183)(99 111 171 189 119 184)(100 112 172 190 120 177)(101 105 173 191 113 178)(102 106 174 192 114 179)(103 107 175 185 115 180)(104 108 176 186 116 181)
(1 16 27 129)(2 130 28 9)(3 10 29 131)(4 132 30 11)(5 12 31 133)(6 134 32 13)(7 14 25 135)(8 136 26 15)(17 115 33 107)(18 108 34 116)(19 117 35 109)(20 110 36 118)(21 119 37 111)(22 112 38 120)(23 113 39 105)(24 106 40 114)(41 192 96 102)(42 103 89 185)(43 186 90 104)(44 97 91 187)(45 188 92 98)(46 99 93 189)(47 190 94 100)(48 101 95 191)(49 168 124 86)(50 87 125 161)(51 162 126 88)(52 81 127 163)(53 164 128 82)(54 83 121 165)(55 166 122 84)(56 85 123 167)(57 177 72 172)(58 173 65 178)(59 179 66 174)(60 175 67 180)(61 181 68 176)(62 169 69 182)(63 183 70 170)(64 171 71 184)(73 145 160 141)(74 142 153 146)(75 147 154 143)(76 144 155 148)(77 149 156 137)(78 138 157 150)(79 151 158 139)(80 140 159 152)
(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 112 13 108)(10 111 14 107)(11 110 15 106)(12 109 16 105)(25 33 29 37)(26 40 30 36)(27 39 31 35)(28 38 32 34)(41 54 45 50)(42 53 46 49)(43 52 47 56)(44 51 48 55)(57 146 61 150)(58 145 62 149)(59 152 63 148)(60 151 64 147)(65 141 69 137)(66 140 70 144)(67 139 71 143)(68 138 72 142)(73 169 77 173)(74 176 78 172)(75 175 79 171)(76 174 80 170)(81 100 85 104)(82 99 86 103)(83 98 87 102)(84 97 88 101)(89 128 93 124)(90 127 94 123)(91 126 95 122)(92 125 96 121)(113 133 117 129)(114 132 118 136)(115 131 119 135)(116 130 120 134)(153 181 157 177)(154 180 158 184)(155 179 159 183)(156 178 160 182)(161 192 165 188)(162 191 166 187)(163 190 167 186)(164 189 168 185)
G:=sub<Sym(192)| (1,126,149,27,51,137)(2,127,150,28,52,138)(3,128,151,29,53,139)(4,121,152,30,54,140)(5,122,145,31,55,141)(6,123,146,32,56,142)(7,124,147,25,49,143)(8,125,148,26,50,144)(9,78,163,130,157,81)(10,79,164,131,158,82)(11,80,165,132,159,83)(12,73,166,133,160,84)(13,74,167,134,153,85)(14,75,168,135,154,86)(15,76,161,136,155,87)(16,77,162,129,156,88)(17,89,60,33,42,67)(18,90,61,34,43,68)(19,91,62,35,44,69)(20,92,63,36,45,70)(21,93,64,37,46,71)(22,94,57,38,47,72)(23,95,58,39,48,65)(24,96,59,40,41,66)(97,109,169,187,117,182)(98,110,170,188,118,183)(99,111,171,189,119,184)(100,112,172,190,120,177)(101,105,173,191,113,178)(102,106,174,192,114,179)(103,107,175,185,115,180)(104,108,176,186,116,181), (1,16,27,129)(2,130,28,9)(3,10,29,131)(4,132,30,11)(5,12,31,133)(6,134,32,13)(7,14,25,135)(8,136,26,15)(17,115,33,107)(18,108,34,116)(19,117,35,109)(20,110,36,118)(21,119,37,111)(22,112,38,120)(23,113,39,105)(24,106,40,114)(41,192,96,102)(42,103,89,185)(43,186,90,104)(44,97,91,187)(45,188,92,98)(46,99,93,189)(47,190,94,100)(48,101,95,191)(49,168,124,86)(50,87,125,161)(51,162,126,88)(52,81,127,163)(53,164,128,82)(54,83,121,165)(55,166,122,84)(56,85,123,167)(57,177,72,172)(58,173,65,178)(59,179,66,174)(60,175,67,180)(61,181,68,176)(62,169,69,182)(63,183,70,170)(64,171,71,184)(73,145,160,141)(74,142,153,146)(75,147,154,143)(76,144,155,148)(77,149,156,137)(78,138,157,150)(79,151,158,139)(80,140,159,152), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,112,13,108)(10,111,14,107)(11,110,15,106)(12,109,16,105)(25,33,29,37)(26,40,30,36)(27,39,31,35)(28,38,32,34)(41,54,45,50)(42,53,46,49)(43,52,47,56)(44,51,48,55)(57,146,61,150)(58,145,62,149)(59,152,63,148)(60,151,64,147)(65,141,69,137)(66,140,70,144)(67,139,71,143)(68,138,72,142)(73,169,77,173)(74,176,78,172)(75,175,79,171)(76,174,80,170)(81,100,85,104)(82,99,86,103)(83,98,87,102)(84,97,88,101)(89,128,93,124)(90,127,94,123)(91,126,95,122)(92,125,96,121)(113,133,117,129)(114,132,118,136)(115,131,119,135)(116,130,120,134)(153,181,157,177)(154,180,158,184)(155,179,159,183)(156,178,160,182)(161,192,165,188)(162,191,166,187)(163,190,167,186)(164,189,168,185)>;
G:=Group( (1,126,149,27,51,137)(2,127,150,28,52,138)(3,128,151,29,53,139)(4,121,152,30,54,140)(5,122,145,31,55,141)(6,123,146,32,56,142)(7,124,147,25,49,143)(8,125,148,26,50,144)(9,78,163,130,157,81)(10,79,164,131,158,82)(11,80,165,132,159,83)(12,73,166,133,160,84)(13,74,167,134,153,85)(14,75,168,135,154,86)(15,76,161,136,155,87)(16,77,162,129,156,88)(17,89,60,33,42,67)(18,90,61,34,43,68)(19,91,62,35,44,69)(20,92,63,36,45,70)(21,93,64,37,46,71)(22,94,57,38,47,72)(23,95,58,39,48,65)(24,96,59,40,41,66)(97,109,169,187,117,182)(98,110,170,188,118,183)(99,111,171,189,119,184)(100,112,172,190,120,177)(101,105,173,191,113,178)(102,106,174,192,114,179)(103,107,175,185,115,180)(104,108,176,186,116,181), (1,16,27,129)(2,130,28,9)(3,10,29,131)(4,132,30,11)(5,12,31,133)(6,134,32,13)(7,14,25,135)(8,136,26,15)(17,115,33,107)(18,108,34,116)(19,117,35,109)(20,110,36,118)(21,119,37,111)(22,112,38,120)(23,113,39,105)(24,106,40,114)(41,192,96,102)(42,103,89,185)(43,186,90,104)(44,97,91,187)(45,188,92,98)(46,99,93,189)(47,190,94,100)(48,101,95,191)(49,168,124,86)(50,87,125,161)(51,162,126,88)(52,81,127,163)(53,164,128,82)(54,83,121,165)(55,166,122,84)(56,85,123,167)(57,177,72,172)(58,173,65,178)(59,179,66,174)(60,175,67,180)(61,181,68,176)(62,169,69,182)(63,183,70,170)(64,171,71,184)(73,145,160,141)(74,142,153,146)(75,147,154,143)(76,144,155,148)(77,149,156,137)(78,138,157,150)(79,151,158,139)(80,140,159,152), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,112,13,108)(10,111,14,107)(11,110,15,106)(12,109,16,105)(25,33,29,37)(26,40,30,36)(27,39,31,35)(28,38,32,34)(41,54,45,50)(42,53,46,49)(43,52,47,56)(44,51,48,55)(57,146,61,150)(58,145,62,149)(59,152,63,148)(60,151,64,147)(65,141,69,137)(66,140,70,144)(67,139,71,143)(68,138,72,142)(73,169,77,173)(74,176,78,172)(75,175,79,171)(76,174,80,170)(81,100,85,104)(82,99,86,103)(83,98,87,102)(84,97,88,101)(89,128,93,124)(90,127,94,123)(91,126,95,122)(92,125,96,121)(113,133,117,129)(114,132,118,136)(115,131,119,135)(116,130,120,134)(153,181,157,177)(154,180,158,184)(155,179,159,183)(156,178,160,182)(161,192,165,188)(162,191,166,187)(163,190,167,186)(164,189,168,185) );
G=PermutationGroup([[(1,126,149,27,51,137),(2,127,150,28,52,138),(3,128,151,29,53,139),(4,121,152,30,54,140),(5,122,145,31,55,141),(6,123,146,32,56,142),(7,124,147,25,49,143),(8,125,148,26,50,144),(9,78,163,130,157,81),(10,79,164,131,158,82),(11,80,165,132,159,83),(12,73,166,133,160,84),(13,74,167,134,153,85),(14,75,168,135,154,86),(15,76,161,136,155,87),(16,77,162,129,156,88),(17,89,60,33,42,67),(18,90,61,34,43,68),(19,91,62,35,44,69),(20,92,63,36,45,70),(21,93,64,37,46,71),(22,94,57,38,47,72),(23,95,58,39,48,65),(24,96,59,40,41,66),(97,109,169,187,117,182),(98,110,170,188,118,183),(99,111,171,189,119,184),(100,112,172,190,120,177),(101,105,173,191,113,178),(102,106,174,192,114,179),(103,107,175,185,115,180),(104,108,176,186,116,181)], [(1,16,27,129),(2,130,28,9),(3,10,29,131),(4,132,30,11),(5,12,31,133),(6,134,32,13),(7,14,25,135),(8,136,26,15),(17,115,33,107),(18,108,34,116),(19,117,35,109),(20,110,36,118),(21,119,37,111),(22,112,38,120),(23,113,39,105),(24,106,40,114),(41,192,96,102),(42,103,89,185),(43,186,90,104),(44,97,91,187),(45,188,92,98),(46,99,93,189),(47,190,94,100),(48,101,95,191),(49,168,124,86),(50,87,125,161),(51,162,126,88),(52,81,127,163),(53,164,128,82),(54,83,121,165),(55,166,122,84),(56,85,123,167),(57,177,72,172),(58,173,65,178),(59,179,66,174),(60,175,67,180),(61,181,68,176),(62,169,69,182),(63,183,70,170),(64,171,71,184),(73,145,160,141),(74,142,153,146),(75,147,154,143),(76,144,155,148),(77,149,156,137),(78,138,157,150),(79,151,158,139),(80,140,159,152)], [(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)], [(1,23,5,19),(2,22,6,18),(3,21,7,17),(4,20,8,24),(9,112,13,108),(10,111,14,107),(11,110,15,106),(12,109,16,105),(25,33,29,37),(26,40,30,36),(27,39,31,35),(28,38,32,34),(41,54,45,50),(42,53,46,49),(43,52,47,56),(44,51,48,55),(57,146,61,150),(58,145,62,149),(59,152,63,148),(60,151,64,147),(65,141,69,137),(66,140,70,144),(67,139,71,143),(68,138,72,142),(73,169,77,173),(74,176,78,172),(75,175,79,171),(76,174,80,170),(81,100,85,104),(82,99,86,103),(83,98,87,102),(84,97,88,101),(89,128,93,124),(90,127,94,123),(91,126,95,122),(92,125,96,121),(113,133,117,129),(114,132,118,136),(115,131,119,135),(116,130,120,134),(153,181,157,177),(154,180,158,184),(155,179,159,183),(156,178,160,182),(161,192,165,188),(162,191,166,187),(163,190,167,186),(164,189,168,185)]])
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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | Q16 | C4○D4 | C3⋊D4 | C8.C22 | D4⋊2S3 | S3×D4 | S3×Q16 | Q16⋊S3 |
| kernel | Dic3⋊3Q16 | Dic3⋊C8 | C2.Dic12 | Q8⋊2Dic3 | C2×C3⋊Q16 | Dic3⋊Q8 | Q8×Dic3 | C6×Q16 | C2×Q16 | C2×Dic3 | C3×Q8 | C2×C8 | C2×Q8 | Dic3 | C12 | Q8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Dic3⋊3Q16 ►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 |
| 6 | 65 | 0 | 0 | 0 | 0 |
| 32 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 29 |
| 0 | 0 | 0 | 0 | 48 | 54 |
| 0 | 23 | 0 | 0 | 0 | 0 |
| 19 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 48 | 0 | 0 |
| 0 | 0 | 38 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 57 | 3 | 0 | 0 | 0 | 0 |
| 61 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 69 | 0 | 0 |
| 0 | 0 | 30 | 51 | 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],[6,32,0,0,0,0,65,67,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,48,0,0,0,0,29,54],[0,19,0,0,0,0,23,0,0,0,0,0,0,0,41,38,0,0,0,0,48,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[57,61,0,0,0,0,3,16,0,0,0,0,0,0,22,30,0,0,0,0,69,51,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
Dic3⋊3Q16 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_3Q_{16} % in TeX
G:=Group("Dic3:3Q16"); // GroupNames label
G:=SmallGroup(192,741);
// by ID
G=gap.SmallGroup(192,741);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,422,135,184,570,297,136,6278]);
// 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,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations