metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊7Q8, C42.146D6, C6.942- (1+4), C4.15(S3×Q8), C4⋊C4.202D6, C3⋊4(Q8⋊3Q8), C12⋊Q8.12C2, C12.47(C2×Q8), C42.C2.6S3, C6.39(C22×Q8), (C2×C12).84C23, (C2×C6).229C24, C2.55(Q8○D12), Dic3.13(C2×Q8), (C4×Dic6).24C2, Dic3.Q8.2C2, (C4×C12).190C22, C12.3Q8.12C2, Dic3.40(C4○D4), Dic6⋊C4.11C2, C4⋊Dic3.378C22, C22.250(S3×C23), Dic3⋊C4.143C22, (C4×Dic3).137C22, (C2×Dic3).119C23, (C2×Dic6).298C22, C2.22(C2×S3×Q8), C2.82(S3×C4○D4), C6.193(C2×C4○D4), (C2×C4).75(C22×S3), (C3×C42.C2).5C2, (C3×C4⋊C4).184C22, SmallGroup(192,1244)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 400 in 200 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×17], C22, C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], Dic3 [×6], Dic3 [×5], C12 [×2], C12 [×6], C2×C6, C42, C42 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×16], C2×Q8 [×4], Dic6 [×4], Dic6 [×6], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×4], C4×Q8 [×6], C42.C2, C42.C2 [×5], C4⋊Q8 [×3], C4×Dic3 [×2], C4×Dic3 [×6], Dic3⋊C4 [×2], Dic3⋊C4 [×10], C4⋊Dic3 [×2], C4⋊Dic3 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×4], C2×Dic6 [×2], C2×Dic6 [×2], Q8⋊3Q8, C4×Dic6 [×2], Dic6⋊C4 [×2], Dic6⋊C4 [×2], C12⋊Q8, C12⋊Q8 [×2], Dic3.Q8 [×4], C12.3Q8, C3×C42.C2, Dic6⋊7Q8
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×Q8, C2×C4○D4, 2- (1+4), S3×Q8 [×2], S3×C23, Q8⋊3Q8, C2×S3×Q8, S3×C4○D4, Q8○D12, Dic6⋊7Q8
Generators and relations
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, ac=ca, dad-1=a5, cbc-1=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)
(1 166 7 160)(2 165 8 159)(3 164 9 158)(4 163 10 157)(5 162 11 168)(6 161 12 167)(13 81 19 75)(14 80 20 74)(15 79 21 73)(16 78 22 84)(17 77 23 83)(18 76 24 82)(25 189 31 183)(26 188 32 182)(27 187 33 181)(28 186 34 192)(29 185 35 191)(30 184 36 190)(37 70 43 64)(38 69 44 63)(39 68 45 62)(40 67 46 61)(41 66 47 72)(42 65 48 71)(49 96 55 90)(50 95 56 89)(51 94 57 88)(52 93 58 87)(53 92 59 86)(54 91 60 85)(97 155 103 149)(98 154 104 148)(99 153 105 147)(100 152 106 146)(101 151 107 145)(102 150 108 156)(109 177 115 171)(110 176 116 170)(111 175 117 169)(112 174 118 180)(113 173 119 179)(114 172 120 178)(121 136 127 142)(122 135 128 141)(123 134 129 140)(124 133 130 139)(125 144 131 138)(126 143 132 137)
(1 64 109 138)(2 65 110 139)(3 66 111 140)(4 67 112 141)(5 68 113 142)(6 69 114 143)(7 70 115 144)(8 71 116 133)(9 72 117 134)(10 61 118 135)(11 62 119 136)(12 63 120 137)(13 51 100 182)(14 52 101 183)(15 53 102 184)(16 54 103 185)(17 55 104 186)(18 56 105 187)(19 57 106 188)(20 58 107 189)(21 59 108 190)(22 60 97 191)(23 49 98 192)(24 50 99 181)(25 74 93 145)(26 75 94 146)(27 76 95 147)(28 77 96 148)(29 78 85 149)(30 79 86 150)(31 80 87 151)(32 81 88 152)(33 82 89 153)(34 83 90 154)(35 84 91 155)(36 73 92 156)(37 171 125 160)(38 172 126 161)(39 173 127 162)(40 174 128 163)(41 175 129 164)(42 176 130 165)(43 177 131 166)(44 178 132 167)(45 179 121 168)(46 180 122 157)(47 169 123 158)(48 170 124 159)
(1 60 109 191)(2 53 110 184)(3 58 111 189)(4 51 112 182)(5 56 113 187)(6 49 114 192)(7 54 115 185)(8 59 116 190)(9 52 117 183)(10 57 118 188)(11 50 119 181)(12 55 120 186)(13 141 100 67)(14 134 101 72)(15 139 102 65)(16 144 103 70)(17 137 104 63)(18 142 105 68)(19 135 106 61)(20 140 107 66)(21 133 108 71)(22 138 97 64)(23 143 98 69)(24 136 99 62)(25 164 93 175)(26 157 94 180)(27 162 95 173)(28 167 96 178)(29 160 85 171)(30 165 86 176)(31 158 87 169)(32 163 88 174)(33 168 89 179)(34 161 90 172)(35 166 91 177)(36 159 92 170)(37 78 125 149)(38 83 126 154)(39 76 127 147)(40 81 128 152)(41 74 129 145)(42 79 130 150)(43 84 131 155)(44 77 132 148)(45 82 121 153)(46 75 122 146)(47 80 123 151)(48 73 124 156)
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), (1,166,7,160)(2,165,8,159)(3,164,9,158)(4,163,10,157)(5,162,11,168)(6,161,12,167)(13,81,19,75)(14,80,20,74)(15,79,21,73)(16,78,22,84)(17,77,23,83)(18,76,24,82)(25,189,31,183)(26,188,32,182)(27,187,33,181)(28,186,34,192)(29,185,35,191)(30,184,36,190)(37,70,43,64)(38,69,44,63)(39,68,45,62)(40,67,46,61)(41,66,47,72)(42,65,48,71)(49,96,55,90)(50,95,56,89)(51,94,57,88)(52,93,58,87)(53,92,59,86)(54,91,60,85)(97,155,103,149)(98,154,104,148)(99,153,105,147)(100,152,106,146)(101,151,107,145)(102,150,108,156)(109,177,115,171)(110,176,116,170)(111,175,117,169)(112,174,118,180)(113,173,119,179)(114,172,120,178)(121,136,127,142)(122,135,128,141)(123,134,129,140)(124,133,130,139)(125,144,131,138)(126,143,132,137), (1,64,109,138)(2,65,110,139)(3,66,111,140)(4,67,112,141)(5,68,113,142)(6,69,114,143)(7,70,115,144)(8,71,116,133)(9,72,117,134)(10,61,118,135)(11,62,119,136)(12,63,120,137)(13,51,100,182)(14,52,101,183)(15,53,102,184)(16,54,103,185)(17,55,104,186)(18,56,105,187)(19,57,106,188)(20,58,107,189)(21,59,108,190)(22,60,97,191)(23,49,98,192)(24,50,99,181)(25,74,93,145)(26,75,94,146)(27,76,95,147)(28,77,96,148)(29,78,85,149)(30,79,86,150)(31,80,87,151)(32,81,88,152)(33,82,89,153)(34,83,90,154)(35,84,91,155)(36,73,92,156)(37,171,125,160)(38,172,126,161)(39,173,127,162)(40,174,128,163)(41,175,129,164)(42,176,130,165)(43,177,131,166)(44,178,132,167)(45,179,121,168)(46,180,122,157)(47,169,123,158)(48,170,124,159), (1,60,109,191)(2,53,110,184)(3,58,111,189)(4,51,112,182)(5,56,113,187)(6,49,114,192)(7,54,115,185)(8,59,116,190)(9,52,117,183)(10,57,118,188)(11,50,119,181)(12,55,120,186)(13,141,100,67)(14,134,101,72)(15,139,102,65)(16,144,103,70)(17,137,104,63)(18,142,105,68)(19,135,106,61)(20,140,107,66)(21,133,108,71)(22,138,97,64)(23,143,98,69)(24,136,99,62)(25,164,93,175)(26,157,94,180)(27,162,95,173)(28,167,96,178)(29,160,85,171)(30,165,86,176)(31,158,87,169)(32,163,88,174)(33,168,89,179)(34,161,90,172)(35,166,91,177)(36,159,92,170)(37,78,125,149)(38,83,126,154)(39,76,127,147)(40,81,128,152)(41,74,129,145)(42,79,130,150)(43,84,131,155)(44,77,132,148)(45,82,121,153)(46,75,122,146)(47,80,123,151)(48,73,124,156)>;
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), (1,166,7,160)(2,165,8,159)(3,164,9,158)(4,163,10,157)(5,162,11,168)(6,161,12,167)(13,81,19,75)(14,80,20,74)(15,79,21,73)(16,78,22,84)(17,77,23,83)(18,76,24,82)(25,189,31,183)(26,188,32,182)(27,187,33,181)(28,186,34,192)(29,185,35,191)(30,184,36,190)(37,70,43,64)(38,69,44,63)(39,68,45,62)(40,67,46,61)(41,66,47,72)(42,65,48,71)(49,96,55,90)(50,95,56,89)(51,94,57,88)(52,93,58,87)(53,92,59,86)(54,91,60,85)(97,155,103,149)(98,154,104,148)(99,153,105,147)(100,152,106,146)(101,151,107,145)(102,150,108,156)(109,177,115,171)(110,176,116,170)(111,175,117,169)(112,174,118,180)(113,173,119,179)(114,172,120,178)(121,136,127,142)(122,135,128,141)(123,134,129,140)(124,133,130,139)(125,144,131,138)(126,143,132,137), (1,64,109,138)(2,65,110,139)(3,66,111,140)(4,67,112,141)(5,68,113,142)(6,69,114,143)(7,70,115,144)(8,71,116,133)(9,72,117,134)(10,61,118,135)(11,62,119,136)(12,63,120,137)(13,51,100,182)(14,52,101,183)(15,53,102,184)(16,54,103,185)(17,55,104,186)(18,56,105,187)(19,57,106,188)(20,58,107,189)(21,59,108,190)(22,60,97,191)(23,49,98,192)(24,50,99,181)(25,74,93,145)(26,75,94,146)(27,76,95,147)(28,77,96,148)(29,78,85,149)(30,79,86,150)(31,80,87,151)(32,81,88,152)(33,82,89,153)(34,83,90,154)(35,84,91,155)(36,73,92,156)(37,171,125,160)(38,172,126,161)(39,173,127,162)(40,174,128,163)(41,175,129,164)(42,176,130,165)(43,177,131,166)(44,178,132,167)(45,179,121,168)(46,180,122,157)(47,169,123,158)(48,170,124,159), (1,60,109,191)(2,53,110,184)(3,58,111,189)(4,51,112,182)(5,56,113,187)(6,49,114,192)(7,54,115,185)(8,59,116,190)(9,52,117,183)(10,57,118,188)(11,50,119,181)(12,55,120,186)(13,141,100,67)(14,134,101,72)(15,139,102,65)(16,144,103,70)(17,137,104,63)(18,142,105,68)(19,135,106,61)(20,140,107,66)(21,133,108,71)(22,138,97,64)(23,143,98,69)(24,136,99,62)(25,164,93,175)(26,157,94,180)(27,162,95,173)(28,167,96,178)(29,160,85,171)(30,165,86,176)(31,158,87,169)(32,163,88,174)(33,168,89,179)(34,161,90,172)(35,166,91,177)(36,159,92,170)(37,78,125,149)(38,83,126,154)(39,76,127,147)(40,81,128,152)(41,74,129,145)(42,79,130,150)(43,84,131,155)(44,77,132,148)(45,82,121,153)(46,75,122,146)(47,80,123,151)(48,73,124,156) );
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)], [(1,166,7,160),(2,165,8,159),(3,164,9,158),(4,163,10,157),(5,162,11,168),(6,161,12,167),(13,81,19,75),(14,80,20,74),(15,79,21,73),(16,78,22,84),(17,77,23,83),(18,76,24,82),(25,189,31,183),(26,188,32,182),(27,187,33,181),(28,186,34,192),(29,185,35,191),(30,184,36,190),(37,70,43,64),(38,69,44,63),(39,68,45,62),(40,67,46,61),(41,66,47,72),(42,65,48,71),(49,96,55,90),(50,95,56,89),(51,94,57,88),(52,93,58,87),(53,92,59,86),(54,91,60,85),(97,155,103,149),(98,154,104,148),(99,153,105,147),(100,152,106,146),(101,151,107,145),(102,150,108,156),(109,177,115,171),(110,176,116,170),(111,175,117,169),(112,174,118,180),(113,173,119,179),(114,172,120,178),(121,136,127,142),(122,135,128,141),(123,134,129,140),(124,133,130,139),(125,144,131,138),(126,143,132,137)], [(1,64,109,138),(2,65,110,139),(3,66,111,140),(4,67,112,141),(5,68,113,142),(6,69,114,143),(7,70,115,144),(8,71,116,133),(9,72,117,134),(10,61,118,135),(11,62,119,136),(12,63,120,137),(13,51,100,182),(14,52,101,183),(15,53,102,184),(16,54,103,185),(17,55,104,186),(18,56,105,187),(19,57,106,188),(20,58,107,189),(21,59,108,190),(22,60,97,191),(23,49,98,192),(24,50,99,181),(25,74,93,145),(26,75,94,146),(27,76,95,147),(28,77,96,148),(29,78,85,149),(30,79,86,150),(31,80,87,151),(32,81,88,152),(33,82,89,153),(34,83,90,154),(35,84,91,155),(36,73,92,156),(37,171,125,160),(38,172,126,161),(39,173,127,162),(40,174,128,163),(41,175,129,164),(42,176,130,165),(43,177,131,166),(44,178,132,167),(45,179,121,168),(46,180,122,157),(47,169,123,158),(48,170,124,159)], [(1,60,109,191),(2,53,110,184),(3,58,111,189),(4,51,112,182),(5,56,113,187),(6,49,114,192),(7,54,115,185),(8,59,116,190),(9,52,117,183),(10,57,118,188),(11,50,119,181),(12,55,120,186),(13,141,100,67),(14,134,101,72),(15,139,102,65),(16,144,103,70),(17,137,104,63),(18,142,105,68),(19,135,106,61),(20,140,107,66),(21,133,108,71),(22,138,97,64),(23,143,98,69),(24,136,99,62),(25,164,93,175),(26,157,94,180),(27,162,95,173),(28,167,96,178),(29,160,85,171),(30,165,86,176),(31,158,87,169),(32,163,88,174),(33,168,89,179),(34,161,90,172),(35,166,91,177),(36,159,92,170),(37,78,125,149),(38,83,126,154),(39,76,127,147),(40,81,128,152),(41,74,129,145),(42,79,130,150),(43,84,131,155),(44,77,132,148),(45,82,121,153),(46,75,122,146),(47,80,123,151),(48,73,124,156)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,2,0,0,0,0,11,4,0,0,0,0,0,0,12,0,0,0,0,0,1,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
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 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4○D4 | 2- (1+4) | S3×Q8 | S3×C4○D4 | Q8○D12 |
| kernel | Dic6⋊7Q8 | C4×Dic6 | Dic6⋊C4 | C12⋊Q8 | Dic3.Q8 | C12.3Q8 | C3×C42.C2 | C42.C2 | Dic6 | C42 | C4⋊C4 | Dic3 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 4 | 3 | 4 | 1 | 1 | 1 | 4 | 1 | 6 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
Dic_6\rtimes_7Q_8
% in TeX
G:=Group("Dic6:7Q8"); // GroupNames label
G:=SmallGroup(192,1244);
// by ID
G=gap.SmallGroup(192,1244);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,219,268,1571,297,192,6278]);
// 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,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
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