metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊9Q8, C42.170D6, C6.812+ (1+4), C3⋊2(Q82), C4⋊Q8.15S3, C4.18(S3×Q8), C4⋊C4.215D6, C12⋊Q8.13C2, C12.53(C2×Q8), (C2×Q8).109D6, C6.45(C22×Q8), (C2×C6).266C24, (C2×C12).99C23, (C4×Dic6).26C2, Dic3.15(C2×Q8), C2.85(D4⋊6D6), (C4×C12).207C22, Dic3⋊Q8.9C2, (C6×Q8).133C22, Dic6⋊C4.13C2, Dic3⋊C4.58C22, C4⋊Dic3.383C22, C22.287(S3×C23), (C2×Dic6).186C22, (C4×Dic3).158C22, (C2×Dic3).271C23, C2.28(C2×S3×Q8), (C3×C4⋊Q8).15C2, (C2×C4).91(C22×S3), (C3×C4⋊C4).209C22, SmallGroup(192,1281)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 448 in 212 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2 [×2], C3, C4 [×4], C4 [×17], C22, C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×14], Dic3 [×8], Dic3 [×4], C12 [×4], C12 [×5], C2×C6, C42, C42 [×8], C4⋊C4 [×4], C4⋊C4 [×14], C2×Q8 [×2], C2×Q8 [×6], Dic6 [×8], Dic6 [×4], C2×Dic3 [×8], C2×C12, C2×C12 [×6], C3×Q8 [×2], C4×Q8 [×6], C4⋊Q8, C4⋊Q8 [×8], C4×Dic3 [×8], Dic3⋊C4 [×12], C4⋊Dic3 [×2], C4×C12, C3×C4⋊C4 [×4], C2×Dic6 [×6], C6×Q8 [×2], Q82, C4×Dic6 [×2], Dic6⋊C4 [×4], C12⋊Q8 [×4], Dic3⋊Q8 [×4], C3×C4⋊Q8, Dic6⋊9Q8
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×8], C23 [×15], D6 [×7], C2×Q8 [×12], C24, C22×S3 [×7], C22×Q8 [×2], 2+ (1+4), S3×Q8 [×4], S3×C23, Q82, D4⋊6D6, C2×S3×Q8 [×2], Dic6⋊9Q8
Generators and relations
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, cac-1=a7, ad=da, bc=cb, dbd-1=a6b, dcd-1=c-1 >
►Regular action on 192 points
Generators in S192
(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)
(1 157 7 163)(2 168 8 162)(3 167 9 161)(4 166 10 160)(5 165 11 159)(6 164 12 158)(13 134 19 140)(14 133 20 139)(15 144 21 138)(16 143 22 137)(17 142 23 136)(18 141 24 135)(25 85 31 91)(26 96 32 90)(27 95 33 89)(28 94 34 88)(29 93 35 87)(30 92 36 86)(37 82 43 76)(38 81 44 75)(39 80 45 74)(40 79 46 73)(41 78 47 84)(42 77 48 83)(49 118 55 112)(50 117 56 111)(51 116 57 110)(52 115 58 109)(53 114 59 120)(54 113 60 119)(61 145 67 151)(62 156 68 150)(63 155 69 149)(64 154 70 148)(65 153 71 147)(66 152 72 146)(97 173 103 179)(98 172 104 178)(99 171 105 177)(100 170 106 176)(101 169 107 175)(102 180 108 174)(121 186 127 192)(122 185 128 191)(123 184 129 190)(124 183 130 189)(125 182 131 188)(126 181 132 187)
(1 46 55 174)(2 41 56 169)(3 48 57 176)(4 43 58 171)(5 38 59 178)(6 45 60 173)(7 40 49 180)(8 47 50 175)(9 42 51 170)(10 37 52 177)(11 44 53 172)(12 39 54 179)(13 85 129 72)(14 92 130 67)(15 87 131 62)(16 94 132 69)(17 89 121 64)(18 96 122 71)(19 91 123 66)(20 86 124 61)(21 93 125 68)(22 88 126 63)(23 95 127 70)(24 90 128 65)(25 184 152 140)(26 191 153 135)(27 186 154 142)(28 181 155 137)(29 188 156 144)(30 183 145 139)(31 190 146 134)(32 185 147 141)(33 192 148 136)(34 187 149 143)(35 182 150 138)(36 189 151 133)(73 112 102 157)(74 119 103 164)(75 114 104 159)(76 109 105 166)(77 116 106 161)(78 111 107 168)(79 118 108 163)(80 113 97 158)(81 120 98 165)(82 115 99 160)(83 110 100 167)(84 117 101 162)
(1 16 55 132)(2 17 56 121)(3 18 57 122)(4 19 58 123)(5 20 59 124)(6 21 60 125)(7 22 49 126)(8 23 50 127)(9 24 51 128)(10 13 52 129)(11 14 53 130)(12 15 54 131)(25 82 152 99)(26 83 153 100)(27 84 154 101)(28 73 155 102)(29 74 156 103)(30 75 145 104)(31 76 146 105)(32 77 147 106)(33 78 148 107)(34 79 149 108)(35 80 150 97)(36 81 151 98)(37 72 177 85)(38 61 178 86)(39 62 179 87)(40 63 180 88)(41 64 169 89)(42 65 170 90)(43 66 171 91)(44 67 172 92)(45 68 173 93)(46 69 174 94)(47 70 175 95)(48 71 176 96)(109 190 166 134)(110 191 167 135)(111 192 168 136)(112 181 157 137)(113 182 158 138)(114 183 159 139)(115 184 160 140)(116 185 161 141)(117 186 162 142)(118 187 163 143)(119 188 164 144)(120 189 165 133)
G:=sub<Sym(192)| (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), (1,157,7,163)(2,168,8,162)(3,167,9,161)(4,166,10,160)(5,165,11,159)(6,164,12,158)(13,134,19,140)(14,133,20,139)(15,144,21,138)(16,143,22,137)(17,142,23,136)(18,141,24,135)(25,85,31,91)(26,96,32,90)(27,95,33,89)(28,94,34,88)(29,93,35,87)(30,92,36,86)(37,82,43,76)(38,81,44,75)(39,80,45,74)(40,79,46,73)(41,78,47,84)(42,77,48,83)(49,118,55,112)(50,117,56,111)(51,116,57,110)(52,115,58,109)(53,114,59,120)(54,113,60,119)(61,145,67,151)(62,156,68,150)(63,155,69,149)(64,154,70,148)(65,153,71,147)(66,152,72,146)(97,173,103,179)(98,172,104,178)(99,171,105,177)(100,170,106,176)(101,169,107,175)(102,180,108,174)(121,186,127,192)(122,185,128,191)(123,184,129,190)(124,183,130,189)(125,182,131,188)(126,181,132,187), (1,46,55,174)(2,41,56,169)(3,48,57,176)(4,43,58,171)(5,38,59,178)(6,45,60,173)(7,40,49,180)(8,47,50,175)(9,42,51,170)(10,37,52,177)(11,44,53,172)(12,39,54,179)(13,85,129,72)(14,92,130,67)(15,87,131,62)(16,94,132,69)(17,89,121,64)(18,96,122,71)(19,91,123,66)(20,86,124,61)(21,93,125,68)(22,88,126,63)(23,95,127,70)(24,90,128,65)(25,184,152,140)(26,191,153,135)(27,186,154,142)(28,181,155,137)(29,188,156,144)(30,183,145,139)(31,190,146,134)(32,185,147,141)(33,192,148,136)(34,187,149,143)(35,182,150,138)(36,189,151,133)(73,112,102,157)(74,119,103,164)(75,114,104,159)(76,109,105,166)(77,116,106,161)(78,111,107,168)(79,118,108,163)(80,113,97,158)(81,120,98,165)(82,115,99,160)(83,110,100,167)(84,117,101,162), (1,16,55,132)(2,17,56,121)(3,18,57,122)(4,19,58,123)(5,20,59,124)(6,21,60,125)(7,22,49,126)(8,23,50,127)(9,24,51,128)(10,13,52,129)(11,14,53,130)(12,15,54,131)(25,82,152,99)(26,83,153,100)(27,84,154,101)(28,73,155,102)(29,74,156,103)(30,75,145,104)(31,76,146,105)(32,77,147,106)(33,78,148,107)(34,79,149,108)(35,80,150,97)(36,81,151,98)(37,72,177,85)(38,61,178,86)(39,62,179,87)(40,63,180,88)(41,64,169,89)(42,65,170,90)(43,66,171,91)(44,67,172,92)(45,68,173,93)(46,69,174,94)(47,70,175,95)(48,71,176,96)(109,190,166,134)(110,191,167,135)(111,192,168,136)(112,181,157,137)(113,182,158,138)(114,183,159,139)(115,184,160,140)(116,185,161,141)(117,186,162,142)(118,187,163,143)(119,188,164,144)(120,189,165,133)>;
G:=Group( (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), (1,157,7,163)(2,168,8,162)(3,167,9,161)(4,166,10,160)(5,165,11,159)(6,164,12,158)(13,134,19,140)(14,133,20,139)(15,144,21,138)(16,143,22,137)(17,142,23,136)(18,141,24,135)(25,85,31,91)(26,96,32,90)(27,95,33,89)(28,94,34,88)(29,93,35,87)(30,92,36,86)(37,82,43,76)(38,81,44,75)(39,80,45,74)(40,79,46,73)(41,78,47,84)(42,77,48,83)(49,118,55,112)(50,117,56,111)(51,116,57,110)(52,115,58,109)(53,114,59,120)(54,113,60,119)(61,145,67,151)(62,156,68,150)(63,155,69,149)(64,154,70,148)(65,153,71,147)(66,152,72,146)(97,173,103,179)(98,172,104,178)(99,171,105,177)(100,170,106,176)(101,169,107,175)(102,180,108,174)(121,186,127,192)(122,185,128,191)(123,184,129,190)(124,183,130,189)(125,182,131,188)(126,181,132,187), (1,46,55,174)(2,41,56,169)(3,48,57,176)(4,43,58,171)(5,38,59,178)(6,45,60,173)(7,40,49,180)(8,47,50,175)(9,42,51,170)(10,37,52,177)(11,44,53,172)(12,39,54,179)(13,85,129,72)(14,92,130,67)(15,87,131,62)(16,94,132,69)(17,89,121,64)(18,96,122,71)(19,91,123,66)(20,86,124,61)(21,93,125,68)(22,88,126,63)(23,95,127,70)(24,90,128,65)(25,184,152,140)(26,191,153,135)(27,186,154,142)(28,181,155,137)(29,188,156,144)(30,183,145,139)(31,190,146,134)(32,185,147,141)(33,192,148,136)(34,187,149,143)(35,182,150,138)(36,189,151,133)(73,112,102,157)(74,119,103,164)(75,114,104,159)(76,109,105,166)(77,116,106,161)(78,111,107,168)(79,118,108,163)(80,113,97,158)(81,120,98,165)(82,115,99,160)(83,110,100,167)(84,117,101,162), (1,16,55,132)(2,17,56,121)(3,18,57,122)(4,19,58,123)(5,20,59,124)(6,21,60,125)(7,22,49,126)(8,23,50,127)(9,24,51,128)(10,13,52,129)(11,14,53,130)(12,15,54,131)(25,82,152,99)(26,83,153,100)(27,84,154,101)(28,73,155,102)(29,74,156,103)(30,75,145,104)(31,76,146,105)(32,77,147,106)(33,78,148,107)(34,79,149,108)(35,80,150,97)(36,81,151,98)(37,72,177,85)(38,61,178,86)(39,62,179,87)(40,63,180,88)(41,64,169,89)(42,65,170,90)(43,66,171,91)(44,67,172,92)(45,68,173,93)(46,69,174,94)(47,70,175,95)(48,71,176,96)(109,190,166,134)(110,191,167,135)(111,192,168,136)(112,181,157,137)(113,182,158,138)(114,183,159,139)(115,184,160,140)(116,185,161,141)(117,186,162,142)(118,187,163,143)(119,188,164,144)(120,189,165,133) );
G=PermutationGroup([(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)], [(1,157,7,163),(2,168,8,162),(3,167,9,161),(4,166,10,160),(5,165,11,159),(6,164,12,158),(13,134,19,140),(14,133,20,139),(15,144,21,138),(16,143,22,137),(17,142,23,136),(18,141,24,135),(25,85,31,91),(26,96,32,90),(27,95,33,89),(28,94,34,88),(29,93,35,87),(30,92,36,86),(37,82,43,76),(38,81,44,75),(39,80,45,74),(40,79,46,73),(41,78,47,84),(42,77,48,83),(49,118,55,112),(50,117,56,111),(51,116,57,110),(52,115,58,109),(53,114,59,120),(54,113,60,119),(61,145,67,151),(62,156,68,150),(63,155,69,149),(64,154,70,148),(65,153,71,147),(66,152,72,146),(97,173,103,179),(98,172,104,178),(99,171,105,177),(100,170,106,176),(101,169,107,175),(102,180,108,174),(121,186,127,192),(122,185,128,191),(123,184,129,190),(124,183,130,189),(125,182,131,188),(126,181,132,187)], [(1,46,55,174),(2,41,56,169),(3,48,57,176),(4,43,58,171),(5,38,59,178),(6,45,60,173),(7,40,49,180),(8,47,50,175),(9,42,51,170),(10,37,52,177),(11,44,53,172),(12,39,54,179),(13,85,129,72),(14,92,130,67),(15,87,131,62),(16,94,132,69),(17,89,121,64),(18,96,122,71),(19,91,123,66),(20,86,124,61),(21,93,125,68),(22,88,126,63),(23,95,127,70),(24,90,128,65),(25,184,152,140),(26,191,153,135),(27,186,154,142),(28,181,155,137),(29,188,156,144),(30,183,145,139),(31,190,146,134),(32,185,147,141),(33,192,148,136),(34,187,149,143),(35,182,150,138),(36,189,151,133),(73,112,102,157),(74,119,103,164),(75,114,104,159),(76,109,105,166),(77,116,106,161),(78,111,107,168),(79,118,108,163),(80,113,97,158),(81,120,98,165),(82,115,99,160),(83,110,100,167),(84,117,101,162)], [(1,16,55,132),(2,17,56,121),(3,18,57,122),(4,19,58,123),(5,20,59,124),(6,21,60,125),(7,22,49,126),(8,23,50,127),(9,24,51,128),(10,13,52,129),(11,14,53,130),(12,15,54,131),(25,82,152,99),(26,83,153,100),(27,84,154,101),(28,73,155,102),(29,74,156,103),(30,75,145,104),(31,76,146,105),(32,77,147,106),(33,78,148,107),(34,79,149,108),(35,80,150,97),(36,81,151,98),(37,72,177,85),(38,61,178,86),(39,62,179,87),(40,63,180,88),(41,64,169,89),(42,65,170,90),(43,66,171,91),(44,67,172,92),(45,68,173,93),(46,69,174,94),(47,70,175,95),(48,71,176,96),(109,190,166,134),(110,191,167,135),(111,192,168,136),(112,181,157,137),(113,182,158,138),(114,183,159,139),(115,184,160,140),(116,185,161,141),(117,186,162,142),(118,187,163,143),(119,188,164,144),(120,189,165,133)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 10 |
| 0 | 0 | 0 | 0 | 6 | 11 |
| 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 | 5 | 0 |
| 0 | 0 | 0 | 0 | 11 | 8 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 2 | 5 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 10 |
| 0 | 0 | 0 | 0 | 6 | 11 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,2,6,0,0,0,0,10,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,11,0,0,0,0,0,8],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,2,0,0,0,0,0,5],[10,4,0,0,0,0,4,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,6,0,0,0,0,10,11] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 4U | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | 2+ (1+4) | S3×Q8 | D4⋊6D6 |
| kernel | Dic6⋊9Q8 | C4×Dic6 | Dic6⋊C4 | C12⋊Q8 | Dic3⋊Q8 | C3×C4⋊Q8 | C4⋊Q8 | Dic6 | C42 | C4⋊C4 | C2×Q8 | C6 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 8 | 1 | 4 | 2 | 1 | 4 | 2 |
In GAP, Magma, Sage, TeX
Dic_6\rtimes_9Q_8
% in TeX
G:=Group("Dic6:9Q8"); // GroupNames label
G:=SmallGroup(192,1281);
// by ID
G=gap.SmallGroup(192,1281);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,100,570,185,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀