metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊6Q8, C42.85D6, C12.21SD16, C4⋊C4.88D6, C4⋊Q8.13S3, C4.14(S3×Q8), C3⋊5(Q8⋊Q8), C12.41(C2×Q8), (C2×C12).163D4, C4.8(D4.S3), C12⋊C8.23C2, C6.62(C2×SD16), C12.88(C4○D4), (C4×Dic6).19C2, C6.78(C22⋊Q8), (C4×C12).141C22, (C2×C12).412C23, C2.15(D6⋊3Q8), C4.37(Q8⋊3S3), C6.SD16.16C2, C12.Q8.18C2, C6.100(C8.C22), C4⋊Dic3.351C22, C2.21(Q8.11D6), (C2×Dic6).277C22, (C3×C4⋊Q8).13C2, (C2×C6).543(C2×D4), C2.16(C2×D4.S3), (C2×C3⋊C8).142C22, (C2×C4).192(C3⋊D4), (C3×C4⋊C4).135C22, (C2×C4).509(C22×S3), C22.215(C2×C3⋊D4), SmallGroup(192,653)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊6Q8
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, cac-1=a7, ad=da, cbc-1=a9b, bd=db, dcd-1=c-1 >
Subgroups: 224 in 96 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, Q8⋊Q8, C12⋊C8, C12.Q8, C6.SD16, C4×Dic6, C3×C4⋊Q8, Dic6⋊6Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, C4○D4, C3⋊D4, C22×S3, C22⋊Q8, C2×SD16, C8.C22, D4.S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, Q8⋊Q8, C2×D4.S3, Q8.11D6, D6⋊3Q8, Dic6⋊6Q8
►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)
(1 127 7 121)(2 126 8 132)(3 125 9 131)(4 124 10 130)(5 123 11 129)(6 122 12 128)(13 176 19 170)(14 175 20 169)(15 174 21 180)(16 173 22 179)(17 172 23 178)(18 171 24 177)(25 98 31 104)(26 97 32 103)(27 108 33 102)(28 107 34 101)(29 106 35 100)(30 105 36 99)(37 60 43 54)(38 59 44 53)(39 58 45 52)(40 57 46 51)(41 56 47 50)(42 55 48 49)(61 185 67 191)(62 184 68 190)(63 183 69 189)(64 182 70 188)(65 181 71 187)(66 192 72 186)(73 155 79 149)(74 154 80 148)(75 153 81 147)(76 152 82 146)(77 151 83 145)(78 150 84 156)(85 163 91 157)(86 162 92 168)(87 161 93 167)(88 160 94 166)(89 159 95 165)(90 158 96 164)(109 141 115 135)(110 140 116 134)(111 139 117 133)(112 138 118 144)(113 137 119 143)(114 136 120 142)
(1 65 107 13)(2 72 108 20)(3 67 97 15)(4 62 98 22)(5 69 99 17)(6 64 100 24)(7 71 101 19)(8 66 102 14)(9 61 103 21)(10 68 104 16)(11 63 105 23)(12 70 106 18)(25 176 130 181)(26 171 131 188)(27 178 132 183)(28 173 121 190)(29 180 122 185)(30 175 123 192)(31 170 124 187)(32 177 125 182)(33 172 126 189)(34 179 127 184)(35 174 128 191)(36 169 129 186)(37 159 140 74)(38 166 141 81)(39 161 142 76)(40 168 143 83)(41 163 144 78)(42 158 133 73)(43 165 134 80)(44 160 135 75)(45 167 136 82)(46 162 137 77)(47 157 138 84)(48 164 139 79)(49 93 117 152)(50 88 118 147)(51 95 119 154)(52 90 120 149)(53 85 109 156)(54 92 110 151)(55 87 111 146)(56 94 112 153)(57 89 113 148)(58 96 114 155)(59 91 115 150)(60 86 116 145)
(1 82 107 167)(2 83 108 168)(3 84 97 157)(4 73 98 158)(5 74 99 159)(6 75 100 160)(7 76 101 161)(8 77 102 162)(9 78 103 163)(10 79 104 164)(11 80 105 165)(12 81 106 166)(13 45 65 136)(14 46 66 137)(15 47 67 138)(16 48 68 139)(17 37 69 140)(18 38 70 141)(19 39 71 142)(20 40 72 143)(21 41 61 144)(22 42 62 133)(23 43 63 134)(24 44 64 135)(25 90 130 149)(26 91 131 150)(27 92 132 151)(28 93 121 152)(29 94 122 153)(30 95 123 154)(31 96 124 155)(32 85 125 156)(33 86 126 145)(34 87 127 146)(35 88 128 147)(36 89 129 148)(49 190 117 173)(50 191 118 174)(51 192 119 175)(52 181 120 176)(53 182 109 177)(54 183 110 178)(55 184 111 179)(56 185 112 180)(57 186 113 169)(58 187 114 170)(59 188 115 171)(60 189 116 172)
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), (1,127,7,121)(2,126,8,132)(3,125,9,131)(4,124,10,130)(5,123,11,129)(6,122,12,128)(13,176,19,170)(14,175,20,169)(15,174,21,180)(16,173,22,179)(17,172,23,178)(18,171,24,177)(25,98,31,104)(26,97,32,103)(27,108,33,102)(28,107,34,101)(29,106,35,100)(30,105,36,99)(37,60,43,54)(38,59,44,53)(39,58,45,52)(40,57,46,51)(41,56,47,50)(42,55,48,49)(61,185,67,191)(62,184,68,190)(63,183,69,189)(64,182,70,188)(65,181,71,187)(66,192,72,186)(73,155,79,149)(74,154,80,148)(75,153,81,147)(76,152,82,146)(77,151,83,145)(78,150,84,156)(85,163,91,157)(86,162,92,168)(87,161,93,167)(88,160,94,166)(89,159,95,165)(90,158,96,164)(109,141,115,135)(110,140,116,134)(111,139,117,133)(112,138,118,144)(113,137,119,143)(114,136,120,142), (1,65,107,13)(2,72,108,20)(3,67,97,15)(4,62,98,22)(5,69,99,17)(6,64,100,24)(7,71,101,19)(8,66,102,14)(9,61,103,21)(10,68,104,16)(11,63,105,23)(12,70,106,18)(25,176,130,181)(26,171,131,188)(27,178,132,183)(28,173,121,190)(29,180,122,185)(30,175,123,192)(31,170,124,187)(32,177,125,182)(33,172,126,189)(34,179,127,184)(35,174,128,191)(36,169,129,186)(37,159,140,74)(38,166,141,81)(39,161,142,76)(40,168,143,83)(41,163,144,78)(42,158,133,73)(43,165,134,80)(44,160,135,75)(45,167,136,82)(46,162,137,77)(47,157,138,84)(48,164,139,79)(49,93,117,152)(50,88,118,147)(51,95,119,154)(52,90,120,149)(53,85,109,156)(54,92,110,151)(55,87,111,146)(56,94,112,153)(57,89,113,148)(58,96,114,155)(59,91,115,150)(60,86,116,145), (1,82,107,167)(2,83,108,168)(3,84,97,157)(4,73,98,158)(5,74,99,159)(6,75,100,160)(7,76,101,161)(8,77,102,162)(9,78,103,163)(10,79,104,164)(11,80,105,165)(12,81,106,166)(13,45,65,136)(14,46,66,137)(15,47,67,138)(16,48,68,139)(17,37,69,140)(18,38,70,141)(19,39,71,142)(20,40,72,143)(21,41,61,144)(22,42,62,133)(23,43,63,134)(24,44,64,135)(25,90,130,149)(26,91,131,150)(27,92,132,151)(28,93,121,152)(29,94,122,153)(30,95,123,154)(31,96,124,155)(32,85,125,156)(33,86,126,145)(34,87,127,146)(35,88,128,147)(36,89,129,148)(49,190,117,173)(50,191,118,174)(51,192,119,175)(52,181,120,176)(53,182,109,177)(54,183,110,178)(55,184,111,179)(56,185,112,180)(57,186,113,169)(58,187,114,170)(59,188,115,171)(60,189,116,172)>;
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), (1,127,7,121)(2,126,8,132)(3,125,9,131)(4,124,10,130)(5,123,11,129)(6,122,12,128)(13,176,19,170)(14,175,20,169)(15,174,21,180)(16,173,22,179)(17,172,23,178)(18,171,24,177)(25,98,31,104)(26,97,32,103)(27,108,33,102)(28,107,34,101)(29,106,35,100)(30,105,36,99)(37,60,43,54)(38,59,44,53)(39,58,45,52)(40,57,46,51)(41,56,47,50)(42,55,48,49)(61,185,67,191)(62,184,68,190)(63,183,69,189)(64,182,70,188)(65,181,71,187)(66,192,72,186)(73,155,79,149)(74,154,80,148)(75,153,81,147)(76,152,82,146)(77,151,83,145)(78,150,84,156)(85,163,91,157)(86,162,92,168)(87,161,93,167)(88,160,94,166)(89,159,95,165)(90,158,96,164)(109,141,115,135)(110,140,116,134)(111,139,117,133)(112,138,118,144)(113,137,119,143)(114,136,120,142), (1,65,107,13)(2,72,108,20)(3,67,97,15)(4,62,98,22)(5,69,99,17)(6,64,100,24)(7,71,101,19)(8,66,102,14)(9,61,103,21)(10,68,104,16)(11,63,105,23)(12,70,106,18)(25,176,130,181)(26,171,131,188)(27,178,132,183)(28,173,121,190)(29,180,122,185)(30,175,123,192)(31,170,124,187)(32,177,125,182)(33,172,126,189)(34,179,127,184)(35,174,128,191)(36,169,129,186)(37,159,140,74)(38,166,141,81)(39,161,142,76)(40,168,143,83)(41,163,144,78)(42,158,133,73)(43,165,134,80)(44,160,135,75)(45,167,136,82)(46,162,137,77)(47,157,138,84)(48,164,139,79)(49,93,117,152)(50,88,118,147)(51,95,119,154)(52,90,120,149)(53,85,109,156)(54,92,110,151)(55,87,111,146)(56,94,112,153)(57,89,113,148)(58,96,114,155)(59,91,115,150)(60,86,116,145), (1,82,107,167)(2,83,108,168)(3,84,97,157)(4,73,98,158)(5,74,99,159)(6,75,100,160)(7,76,101,161)(8,77,102,162)(9,78,103,163)(10,79,104,164)(11,80,105,165)(12,81,106,166)(13,45,65,136)(14,46,66,137)(15,47,67,138)(16,48,68,139)(17,37,69,140)(18,38,70,141)(19,39,71,142)(20,40,72,143)(21,41,61,144)(22,42,62,133)(23,43,63,134)(24,44,64,135)(25,90,130,149)(26,91,131,150)(27,92,132,151)(28,93,121,152)(29,94,122,153)(30,95,123,154)(31,96,124,155)(32,85,125,156)(33,86,126,145)(34,87,127,146)(35,88,128,147)(36,89,129,148)(49,190,117,173)(50,191,118,174)(51,192,119,175)(52,181,120,176)(53,182,109,177)(54,183,110,178)(55,184,111,179)(56,185,112,180)(57,186,113,169)(58,187,114,170)(59,188,115,171)(60,189,116,172) );
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)], [(1,127,7,121),(2,126,8,132),(3,125,9,131),(4,124,10,130),(5,123,11,129),(6,122,12,128),(13,176,19,170),(14,175,20,169),(15,174,21,180),(16,173,22,179),(17,172,23,178),(18,171,24,177),(25,98,31,104),(26,97,32,103),(27,108,33,102),(28,107,34,101),(29,106,35,100),(30,105,36,99),(37,60,43,54),(38,59,44,53),(39,58,45,52),(40,57,46,51),(41,56,47,50),(42,55,48,49),(61,185,67,191),(62,184,68,190),(63,183,69,189),(64,182,70,188),(65,181,71,187),(66,192,72,186),(73,155,79,149),(74,154,80,148),(75,153,81,147),(76,152,82,146),(77,151,83,145),(78,150,84,156),(85,163,91,157),(86,162,92,168),(87,161,93,167),(88,160,94,166),(89,159,95,165),(90,158,96,164),(109,141,115,135),(110,140,116,134),(111,139,117,133),(112,138,118,144),(113,137,119,143),(114,136,120,142)], [(1,65,107,13),(2,72,108,20),(3,67,97,15),(4,62,98,22),(5,69,99,17),(6,64,100,24),(7,71,101,19),(8,66,102,14),(9,61,103,21),(10,68,104,16),(11,63,105,23),(12,70,106,18),(25,176,130,181),(26,171,131,188),(27,178,132,183),(28,173,121,190),(29,180,122,185),(30,175,123,192),(31,170,124,187),(32,177,125,182),(33,172,126,189),(34,179,127,184),(35,174,128,191),(36,169,129,186),(37,159,140,74),(38,166,141,81),(39,161,142,76),(40,168,143,83),(41,163,144,78),(42,158,133,73),(43,165,134,80),(44,160,135,75),(45,167,136,82),(46,162,137,77),(47,157,138,84),(48,164,139,79),(49,93,117,152),(50,88,118,147),(51,95,119,154),(52,90,120,149),(53,85,109,156),(54,92,110,151),(55,87,111,146),(56,94,112,153),(57,89,113,148),(58,96,114,155),(59,91,115,150),(60,86,116,145)], [(1,82,107,167),(2,83,108,168),(3,84,97,157),(4,73,98,158),(5,74,99,159),(6,75,100,160),(7,76,101,161),(8,77,102,162),(9,78,103,163),(10,79,104,164),(11,80,105,165),(12,81,106,166),(13,45,65,136),(14,46,66,137),(15,47,67,138),(16,48,68,139),(17,37,69,140),(18,38,70,141),(19,39,71,142),(20,40,72,143),(21,41,61,144),(22,42,62,133),(23,43,63,134),(24,44,64,135),(25,90,130,149),(26,91,131,150),(27,92,132,151),(28,93,121,152),(29,94,122,153),(30,95,123,154),(31,96,124,155),(32,85,125,156),(33,86,126,145),(34,87,127,146),(35,88,128,147),(36,89,129,148),(49,190,117,173),(50,191,118,174),(51,192,119,175),(52,181,120,176),(53,182,109,177),(54,183,110,178),(55,184,111,179),(56,185,112,180),(57,186,113,169),(58,187,114,170),(59,188,115,171),(60,189,116,172)]])
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 | ··· | 12F | 12G | 12H | 12I | 12J |
| 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 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 | S3 | Q8 | D4 | D6 | D6 | SD16 | C4○D4 | C3⋊D4 | C8.C22 | D4.S3 | S3×Q8 | Q8⋊3S3 | Q8.11D6 |
| kernel | Dic6⋊6Q8 | C12⋊C8 | C12.Q8 | C6.SD16 | C4×Dic6 | C3×C4⋊Q8 | C4⋊Q8 | Dic6 | C2×C12 | C42 | C4⋊C4 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 1 | 2 | 1 | 1 | 2 |
Matrix representation of Dic6⋊6Q8 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 42 | 0 | 0 |
| 0 | 0 | 45 | 70 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 50 |
| 0 | 0 | 0 | 0 | 50 | 45 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 43 |
| 0 | 0 | 0 | 0 | 43 | 14 |
| 43 | 62 | 0 | 0 | 0 | 0 |
| 62 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,45,0,0,0,0,42,70,0,0,0,0,0,0,28,50,0,0,0,0,50,45],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,59,43,0,0,0,0,43,14],[43,62,0,0,0,0,62,30,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic6⋊6Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_6Q_8 % in TeX
G:=Group("Dic6:6Q8"); // GroupNames label
G:=SmallGroup(192,653);
// by ID
G=gap.SmallGroup(192,653);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,100,1123,297,136,6278]);
// 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,c*b*c^-1=a^9*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations