metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊1C42, C6.20(C4×D4), C6.13(C4×Q8), Dic3⋊C4⋊4C4, C2.4(S3×C42), C6.2(C2×C42), Dic3⋊4(C4⋊C4), (C4×Dic3)⋊11C4, C22.52(S3×D4), C22.11(S3×Q8), (C2×Dic3).21Q8, (C22×C4).310D6, (C2×Dic3).197D4, C6.17(C42⋊C2), C2.1(Dic3⋊4D4), C2.1(Dic6⋊C4), C2.C42.19S3, C6.C42.33C2, (C22×C6).274C23, C23.256(C22×S3), C22.28(D4⋊2S3), (C22×C12).329C22, C2.2(C23.16D6), (C22×Dic3).171C22, C3⋊1(C4×C4⋊C4), C6.2(C2×C4⋊C4), C2.1(S3×C4⋊C4), C22.30(S3×C2×C4), (C2×C6).58(C2×Q8), (C2×C4).121(C4×S3), (C2×C6).189(C2×D4), (C2×C4×Dic3).22C2, (C2×C12).138(C2×C4), (C2×C6).40(C22×C4), (C2×C6).123(C4○D4), (C2×Dic3⋊C4).25C2, (C2×Dic3).41(C2×C4), (C3×C2.C42).20C2, SmallGroup(192,208)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊C42
G = < a,b,c,d | a6=c4=d4=1, b2=a3, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a3b, cd=dc >
Subgroups: 384 in 194 conjugacy classes, 99 normal (25 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C4×Dic3, C4×Dic3, Dic3⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C4×C4⋊C4, C6.C42, C3×C2.C42, C2×C4×Dic3, C2×C4×Dic3, C2×Dic3⋊C4, Dic3⋊C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, C22×S3, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, S3×C2×C4, S3×D4, D4⋊2S3, S3×Q8, C4×C4⋊C4, S3×C42, C23.16D6, Dic3⋊4D4, Dic6⋊C4, S3×C4⋊C4, Dic3⋊C42
►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)
(1 161 4 158)(2 160 5 157)(3 159 6 162)(7 49 10 52)(8 54 11 51)(9 53 12 50)(13 151 16 154)(14 156 17 153)(15 155 18 152)(19 164 22 167)(20 163 23 166)(21 168 24 165)(25 173 28 170)(26 172 29 169)(27 171 30 174)(31 176 34 179)(32 175 35 178)(33 180 36 177)(37 185 40 182)(38 184 41 181)(39 183 42 186)(43 188 46 191)(44 187 47 190)(45 192 48 189)(55 114 58 111)(56 113 59 110)(57 112 60 109)(61 103 64 106)(62 108 65 105)(63 107 66 104)(67 126 70 123)(68 125 71 122)(69 124 72 121)(73 115 76 118)(74 120 77 117)(75 119 78 116)(79 138 82 135)(80 137 83 134)(81 136 84 133)(85 127 88 130)(86 132 89 129)(87 131 90 128)(91 150 94 147)(92 149 95 146)(93 148 96 145)(97 144 100 141)(98 143 101 140)(99 142 102 139)
(1 113 16 106)(2 112 17 105)(3 111 18 104)(4 110 13 103)(5 109 14 108)(6 114 15 107)(7 92 187 99)(8 91 188 98)(9 96 189 97)(10 95 190 102)(11 94 191 101)(12 93 192 100)(19 116 27 123)(20 115 28 122)(21 120 29 121)(22 119 30 126)(23 118 25 125)(24 117 26 124)(31 128 39 135)(32 127 40 134)(33 132 41 133)(34 131 42 138)(35 130 37 137)(36 129 38 136)(43 140 51 147)(44 139 52 146)(45 144 53 145)(46 143 54 150)(47 142 49 149)(48 141 50 148)(55 152 63 159)(56 151 64 158)(57 156 65 157)(58 155 66 162)(59 154 61 161)(60 153 62 160)(67 164 75 171)(68 163 76 170)(69 168 77 169)(70 167 78 174)(71 166 73 173)(72 165 74 172)(79 176 87 183)(80 175 88 182)(81 180 89 181)(82 179 90 186)(83 178 85 185)(84 177 86 184)
(1 49 25 37)(2 50 26 38)(3 51 27 39)(4 52 28 40)(5 53 29 41)(6 54 30 42)(7 173 182 161)(8 174 183 162)(9 169 184 157)(10 170 185 158)(11 171 186 159)(12 172 181 160)(13 44 20 32)(14 45 21 33)(15 46 22 34)(16 47 23 35)(17 48 24 36)(18 43 19 31)(55 94 67 82)(56 95 68 83)(57 96 69 84)(58 91 70 79)(59 92 71 80)(60 93 72 81)(61 99 73 88)(62 100 74 89)(63 101 75 90)(64 102 76 85)(65 97 77 86)(66 98 78 87)(103 139 115 127)(104 140 116 128)(105 141 117 129)(106 142 118 130)(107 143 119 131)(108 144 120 132)(109 145 121 133)(110 146 122 134)(111 147 123 135)(112 148 124 136)(113 149 125 137)(114 150 126 138)(151 190 163 178)(152 191 164 179)(153 192 165 180)(154 187 166 175)(155 188 167 176)(156 189 168 177)
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), (1,161,4,158)(2,160,5,157)(3,159,6,162)(7,49,10,52)(8,54,11,51)(9,53,12,50)(13,151,16,154)(14,156,17,153)(15,155,18,152)(19,164,22,167)(20,163,23,166)(21,168,24,165)(25,173,28,170)(26,172,29,169)(27,171,30,174)(31,176,34,179)(32,175,35,178)(33,180,36,177)(37,185,40,182)(38,184,41,181)(39,183,42,186)(43,188,46,191)(44,187,47,190)(45,192,48,189)(55,114,58,111)(56,113,59,110)(57,112,60,109)(61,103,64,106)(62,108,65,105)(63,107,66,104)(67,126,70,123)(68,125,71,122)(69,124,72,121)(73,115,76,118)(74,120,77,117)(75,119,78,116)(79,138,82,135)(80,137,83,134)(81,136,84,133)(85,127,88,130)(86,132,89,129)(87,131,90,128)(91,150,94,147)(92,149,95,146)(93,148,96,145)(97,144,100,141)(98,143,101,140)(99,142,102,139), (1,113,16,106)(2,112,17,105)(3,111,18,104)(4,110,13,103)(5,109,14,108)(6,114,15,107)(7,92,187,99)(8,91,188,98)(9,96,189,97)(10,95,190,102)(11,94,191,101)(12,93,192,100)(19,116,27,123)(20,115,28,122)(21,120,29,121)(22,119,30,126)(23,118,25,125)(24,117,26,124)(31,128,39,135)(32,127,40,134)(33,132,41,133)(34,131,42,138)(35,130,37,137)(36,129,38,136)(43,140,51,147)(44,139,52,146)(45,144,53,145)(46,143,54,150)(47,142,49,149)(48,141,50,148)(55,152,63,159)(56,151,64,158)(57,156,65,157)(58,155,66,162)(59,154,61,161)(60,153,62,160)(67,164,75,171)(68,163,76,170)(69,168,77,169)(70,167,78,174)(71,166,73,173)(72,165,74,172)(79,176,87,183)(80,175,88,182)(81,180,89,181)(82,179,90,186)(83,178,85,185)(84,177,86,184), (1,49,25,37)(2,50,26,38)(3,51,27,39)(4,52,28,40)(5,53,29,41)(6,54,30,42)(7,173,182,161)(8,174,183,162)(9,169,184,157)(10,170,185,158)(11,171,186,159)(12,172,181,160)(13,44,20,32)(14,45,21,33)(15,46,22,34)(16,47,23,35)(17,48,24,36)(18,43,19,31)(55,94,67,82)(56,95,68,83)(57,96,69,84)(58,91,70,79)(59,92,71,80)(60,93,72,81)(61,99,73,88)(62,100,74,89)(63,101,75,90)(64,102,76,85)(65,97,77,86)(66,98,78,87)(103,139,115,127)(104,140,116,128)(105,141,117,129)(106,142,118,130)(107,143,119,131)(108,144,120,132)(109,145,121,133)(110,146,122,134)(111,147,123,135)(112,148,124,136)(113,149,125,137)(114,150,126,138)(151,190,163,178)(152,191,164,179)(153,192,165,180)(154,187,166,175)(155,188,167,176)(156,189,168,177)>;
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), (1,161,4,158)(2,160,5,157)(3,159,6,162)(7,49,10,52)(8,54,11,51)(9,53,12,50)(13,151,16,154)(14,156,17,153)(15,155,18,152)(19,164,22,167)(20,163,23,166)(21,168,24,165)(25,173,28,170)(26,172,29,169)(27,171,30,174)(31,176,34,179)(32,175,35,178)(33,180,36,177)(37,185,40,182)(38,184,41,181)(39,183,42,186)(43,188,46,191)(44,187,47,190)(45,192,48,189)(55,114,58,111)(56,113,59,110)(57,112,60,109)(61,103,64,106)(62,108,65,105)(63,107,66,104)(67,126,70,123)(68,125,71,122)(69,124,72,121)(73,115,76,118)(74,120,77,117)(75,119,78,116)(79,138,82,135)(80,137,83,134)(81,136,84,133)(85,127,88,130)(86,132,89,129)(87,131,90,128)(91,150,94,147)(92,149,95,146)(93,148,96,145)(97,144,100,141)(98,143,101,140)(99,142,102,139), (1,113,16,106)(2,112,17,105)(3,111,18,104)(4,110,13,103)(5,109,14,108)(6,114,15,107)(7,92,187,99)(8,91,188,98)(9,96,189,97)(10,95,190,102)(11,94,191,101)(12,93,192,100)(19,116,27,123)(20,115,28,122)(21,120,29,121)(22,119,30,126)(23,118,25,125)(24,117,26,124)(31,128,39,135)(32,127,40,134)(33,132,41,133)(34,131,42,138)(35,130,37,137)(36,129,38,136)(43,140,51,147)(44,139,52,146)(45,144,53,145)(46,143,54,150)(47,142,49,149)(48,141,50,148)(55,152,63,159)(56,151,64,158)(57,156,65,157)(58,155,66,162)(59,154,61,161)(60,153,62,160)(67,164,75,171)(68,163,76,170)(69,168,77,169)(70,167,78,174)(71,166,73,173)(72,165,74,172)(79,176,87,183)(80,175,88,182)(81,180,89,181)(82,179,90,186)(83,178,85,185)(84,177,86,184), (1,49,25,37)(2,50,26,38)(3,51,27,39)(4,52,28,40)(5,53,29,41)(6,54,30,42)(7,173,182,161)(8,174,183,162)(9,169,184,157)(10,170,185,158)(11,171,186,159)(12,172,181,160)(13,44,20,32)(14,45,21,33)(15,46,22,34)(16,47,23,35)(17,48,24,36)(18,43,19,31)(55,94,67,82)(56,95,68,83)(57,96,69,84)(58,91,70,79)(59,92,71,80)(60,93,72,81)(61,99,73,88)(62,100,74,89)(63,101,75,90)(64,102,76,85)(65,97,77,86)(66,98,78,87)(103,139,115,127)(104,140,116,128)(105,141,117,129)(106,142,118,130)(107,143,119,131)(108,144,120,132)(109,145,121,133)(110,146,122,134)(111,147,123,135)(112,148,124,136)(113,149,125,137)(114,150,126,138)(151,190,163,178)(152,191,164,179)(153,192,165,180)(154,187,166,175)(155,188,167,176)(156,189,168,177) );
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)], [(1,161,4,158),(2,160,5,157),(3,159,6,162),(7,49,10,52),(8,54,11,51),(9,53,12,50),(13,151,16,154),(14,156,17,153),(15,155,18,152),(19,164,22,167),(20,163,23,166),(21,168,24,165),(25,173,28,170),(26,172,29,169),(27,171,30,174),(31,176,34,179),(32,175,35,178),(33,180,36,177),(37,185,40,182),(38,184,41,181),(39,183,42,186),(43,188,46,191),(44,187,47,190),(45,192,48,189),(55,114,58,111),(56,113,59,110),(57,112,60,109),(61,103,64,106),(62,108,65,105),(63,107,66,104),(67,126,70,123),(68,125,71,122),(69,124,72,121),(73,115,76,118),(74,120,77,117),(75,119,78,116),(79,138,82,135),(80,137,83,134),(81,136,84,133),(85,127,88,130),(86,132,89,129),(87,131,90,128),(91,150,94,147),(92,149,95,146),(93,148,96,145),(97,144,100,141),(98,143,101,140),(99,142,102,139)], [(1,113,16,106),(2,112,17,105),(3,111,18,104),(4,110,13,103),(5,109,14,108),(6,114,15,107),(7,92,187,99),(8,91,188,98),(9,96,189,97),(10,95,190,102),(11,94,191,101),(12,93,192,100),(19,116,27,123),(20,115,28,122),(21,120,29,121),(22,119,30,126),(23,118,25,125),(24,117,26,124),(31,128,39,135),(32,127,40,134),(33,132,41,133),(34,131,42,138),(35,130,37,137),(36,129,38,136),(43,140,51,147),(44,139,52,146),(45,144,53,145),(46,143,54,150),(47,142,49,149),(48,141,50,148),(55,152,63,159),(56,151,64,158),(57,156,65,157),(58,155,66,162),(59,154,61,161),(60,153,62,160),(67,164,75,171),(68,163,76,170),(69,168,77,169),(70,167,78,174),(71,166,73,173),(72,165,74,172),(79,176,87,183),(80,175,88,182),(81,180,89,181),(82,179,90,186),(83,178,85,185),(84,177,86,184)], [(1,49,25,37),(2,50,26,38),(3,51,27,39),(4,52,28,40),(5,53,29,41),(6,54,30,42),(7,173,182,161),(8,174,183,162),(9,169,184,157),(10,170,185,158),(11,171,186,159),(12,172,181,160),(13,44,20,32),(14,45,21,33),(15,46,22,34),(16,47,23,35),(17,48,24,36),(18,43,19,31),(55,94,67,82),(56,95,68,83),(57,96,69,84),(58,91,70,79),(59,92,71,80),(60,93,72,81),(61,99,73,88),(62,100,74,89),(63,101,75,90),(64,102,76,85),(65,97,77,86),(66,98,78,87),(103,139,115,127),(104,140,116,128),(105,141,117,129),(106,142,118,130),(107,143,119,131),(108,144,120,132),(109,145,121,133),(110,146,122,134),(111,147,123,135),(112,148,124,136),(113,149,125,137),(114,150,126,138),(151,190,163,178),(152,191,164,179),(153,192,165,180),(154,187,166,175),(155,188,167,176),(156,189,168,177)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4L | 4M | ··· | 4T | 4U | ··· | 4AF | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | D6 | C4○D4 | C4×S3 | S3×D4 | D4⋊2S3 | S3×Q8 |
| kernel | Dic3⋊C42 | C6.C42 | C3×C2.C42 | C2×C4×Dic3 | C2×Dic3⋊C4 | C4×Dic3 | Dic3⋊C4 | C2.C42 | C2×Dic3 | C2×Dic3 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 3 | 2 | 8 | 16 | 1 | 2 | 2 | 3 | 4 | 12 | 1 | 2 | 1 |
Matrix representation of Dic3⋊C42 ►in GL6(𝔽13)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 3 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 10 | 7 | 0 | 0 | 0 | 0 |
| 10 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 10 | 5 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 2 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [3,3,0,0,0,0,0,9,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[10,10,0,0,0,0,7,3,0,0,0,0,0,0,3,10,0,0,0,0,7,10,0,0,0,0,0,0,8,10,0,0,0,0,0,5],[2,2,0,0,0,0,4,11,0,0,0,0,0,0,3,10,0,0,0,0,7,10,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,12,1] >;
Dic3⋊C42 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes C_4^2 % in TeX
G:=Group("Dic3:C4^2"); // GroupNames label
G:=SmallGroup(192,208);
// by ID
G=gap.SmallGroup(192,208);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,555,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^4=d^4=1,b^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,c*d=d*c>;
// generators/relations