metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3.5C42, C6.1(C2×C42), C2.3(S3×C42), C3⋊1(C42⋊4C4), (C4×Dic3)⋊10C4, (C22×C4).309D6, C6.16(C42⋊C2), C2.C42.18S3, C6.C42.32C2, (C22×C6).273C23, C23.255(C22×S3), C22.27(D4⋊2S3), (C22×C12).328C22, C22.12(Q8⋊3S3), C2.1(C23.16D6), (C22×Dic3).170C22, C22.29(S3×C2×C4), (C2×C4).120(C4×S3), C2.1(C4⋊C4⋊7S3), (C2×C4×Dic3).21C2, (C2×C12).137(C2×C4), (C2×C6).39(C22×C4), (C2×C6).122(C4○D4), (C2×Dic3).74(C2×C4), (C3×C2.C42).19C2, SmallGroup(192,207)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3.5C42
G = < a,b,c,d | a6=c4=d4=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a3c >
Subgroups: 352 in 178 conjugacy classes, 91 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C42, C4×Dic3, C22×Dic3, C22×Dic3, C22×C12, C42⋊4C4, C6.C42, C3×C2.C42, C2×C4×Dic3, Dic3.5C42
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, C22×C4, C4○D4, C4×S3, C22×S3, C2×C42, C42⋊C2, S3×C2×C4, D4⋊2S3, Q8⋊3S3, C42⋊4C4, S3×C42, C23.16D6, C4⋊C4⋊7S3, Dic3.5C42
►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)
(1 100 4 97)(2 99 5 102)(3 98 6 101)(7 104 10 107)(8 103 11 106)(9 108 12 105)(13 110 16 113)(14 109 17 112)(15 114 18 111)(19 116 22 119)(20 115 23 118)(21 120 24 117)(25 122 28 125)(26 121 29 124)(27 126 30 123)(31 128 34 131)(32 127 35 130)(33 132 36 129)(37 134 40 137)(38 133 41 136)(39 138 42 135)(43 140 46 143)(44 139 47 142)(45 144 48 141)(49 146 52 149)(50 145 53 148)(51 150 54 147)(55 152 58 155)(56 151 59 154)(57 156 60 153)(61 158 64 161)(62 157 65 160)(63 162 66 159)(67 164 70 167)(68 163 71 166)(69 168 72 165)(73 170 76 173)(74 169 77 172)(75 174 78 171)(79 176 82 179)(80 175 83 178)(81 180 84 177)(85 182 88 185)(86 181 89 184)(87 186 90 183)(91 188 94 191)(92 187 95 190)(93 192 96 189)
(1 59 11 53)(2 60 12 54)(3 55 7 49)(4 56 8 50)(5 57 9 51)(6 58 10 52)(13 67 19 61)(14 68 20 62)(15 69 21 63)(16 70 22 64)(17 71 23 65)(18 72 24 66)(25 79 31 73)(26 80 32 74)(27 81 33 75)(28 82 34 76)(29 83 35 77)(30 84 36 78)(37 91 43 85)(38 92 44 86)(39 93 45 87)(40 94 46 88)(41 95 47 89)(42 96 48 90)(97 151 103 145)(98 152 104 146)(99 153 105 147)(100 154 106 148)(101 155 107 149)(102 156 108 150)(109 163 115 157)(110 164 116 158)(111 165 117 159)(112 166 118 160)(113 167 119 161)(114 168 120 162)(121 175 127 169)(122 176 128 170)(123 177 129 171)(124 178 130 172)(125 179 131 173)(126 180 132 174)(133 187 139 181)(134 188 140 182)(135 189 141 183)(136 190 142 184)(137 191 143 185)(138 192 144 186)
(1 41 17 29)(2 42 18 30)(3 37 13 25)(4 38 14 26)(5 39 15 27)(6 40 16 28)(7 43 19 31)(8 44 20 32)(9 45 21 33)(10 46 22 34)(11 47 23 35)(12 48 24 36)(49 88 61 76)(50 89 62 77)(51 90 63 78)(52 85 64 73)(53 86 65 74)(54 87 66 75)(55 94 67 82)(56 95 68 83)(57 96 69 84)(58 91 70 79)(59 92 71 80)(60 93 72 81)(97 133 109 121)(98 134 110 122)(99 135 111 123)(100 136 112 124)(101 137 113 125)(102 138 114 126)(103 139 115 127)(104 140 116 128)(105 141 117 129)(106 142 118 130)(107 143 119 131)(108 144 120 132)(145 184 157 172)(146 185 158 173)(147 186 159 174)(148 181 160 169)(149 182 161 170)(150 183 162 171)(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), (1,100,4,97)(2,99,5,102)(3,98,6,101)(7,104,10,107)(8,103,11,106)(9,108,12,105)(13,110,16,113)(14,109,17,112)(15,114,18,111)(19,116,22,119)(20,115,23,118)(21,120,24,117)(25,122,28,125)(26,121,29,124)(27,126,30,123)(31,128,34,131)(32,127,35,130)(33,132,36,129)(37,134,40,137)(38,133,41,136)(39,138,42,135)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,146,52,149)(50,145,53,148)(51,150,54,147)(55,152,58,155)(56,151,59,154)(57,156,60,153)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,170,76,173)(74,169,77,172)(75,174,78,171)(79,176,82,179)(80,175,83,178)(81,180,84,177)(85,182,88,185)(86,181,89,184)(87,186,90,183)(91,188,94,191)(92,187,95,190)(93,192,96,189), (1,59,11,53)(2,60,12,54)(3,55,7,49)(4,56,8,50)(5,57,9,51)(6,58,10,52)(13,67,19,61)(14,68,20,62)(15,69,21,63)(16,70,22,64)(17,71,23,65)(18,72,24,66)(25,79,31,73)(26,80,32,74)(27,81,33,75)(28,82,34,76)(29,83,35,77)(30,84,36,78)(37,91,43,85)(38,92,44,86)(39,93,45,87)(40,94,46,88)(41,95,47,89)(42,96,48,90)(97,151,103,145)(98,152,104,146)(99,153,105,147)(100,154,106,148)(101,155,107,149)(102,156,108,150)(109,163,115,157)(110,164,116,158)(111,165,117,159)(112,166,118,160)(113,167,119,161)(114,168,120,162)(121,175,127,169)(122,176,128,170)(123,177,129,171)(124,178,130,172)(125,179,131,173)(126,180,132,174)(133,187,139,181)(134,188,140,182)(135,189,141,183)(136,190,142,184)(137,191,143,185)(138,192,144,186), (1,41,17,29)(2,42,18,30)(3,37,13,25)(4,38,14,26)(5,39,15,27)(6,40,16,28)(7,43,19,31)(8,44,20,32)(9,45,21,33)(10,46,22,34)(11,47,23,35)(12,48,24,36)(49,88,61,76)(50,89,62,77)(51,90,63,78)(52,85,64,73)(53,86,65,74)(54,87,66,75)(55,94,67,82)(56,95,68,83)(57,96,69,84)(58,91,70,79)(59,92,71,80)(60,93,72,81)(97,133,109,121)(98,134,110,122)(99,135,111,123)(100,136,112,124)(101,137,113,125)(102,138,114,126)(103,139,115,127)(104,140,116,128)(105,141,117,129)(106,142,118,130)(107,143,119,131)(108,144,120,132)(145,184,157,172)(146,185,158,173)(147,186,159,174)(148,181,160,169)(149,182,161,170)(150,183,162,171)(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), (1,100,4,97)(2,99,5,102)(3,98,6,101)(7,104,10,107)(8,103,11,106)(9,108,12,105)(13,110,16,113)(14,109,17,112)(15,114,18,111)(19,116,22,119)(20,115,23,118)(21,120,24,117)(25,122,28,125)(26,121,29,124)(27,126,30,123)(31,128,34,131)(32,127,35,130)(33,132,36,129)(37,134,40,137)(38,133,41,136)(39,138,42,135)(43,140,46,143)(44,139,47,142)(45,144,48,141)(49,146,52,149)(50,145,53,148)(51,150,54,147)(55,152,58,155)(56,151,59,154)(57,156,60,153)(61,158,64,161)(62,157,65,160)(63,162,66,159)(67,164,70,167)(68,163,71,166)(69,168,72,165)(73,170,76,173)(74,169,77,172)(75,174,78,171)(79,176,82,179)(80,175,83,178)(81,180,84,177)(85,182,88,185)(86,181,89,184)(87,186,90,183)(91,188,94,191)(92,187,95,190)(93,192,96,189), (1,59,11,53)(2,60,12,54)(3,55,7,49)(4,56,8,50)(5,57,9,51)(6,58,10,52)(13,67,19,61)(14,68,20,62)(15,69,21,63)(16,70,22,64)(17,71,23,65)(18,72,24,66)(25,79,31,73)(26,80,32,74)(27,81,33,75)(28,82,34,76)(29,83,35,77)(30,84,36,78)(37,91,43,85)(38,92,44,86)(39,93,45,87)(40,94,46,88)(41,95,47,89)(42,96,48,90)(97,151,103,145)(98,152,104,146)(99,153,105,147)(100,154,106,148)(101,155,107,149)(102,156,108,150)(109,163,115,157)(110,164,116,158)(111,165,117,159)(112,166,118,160)(113,167,119,161)(114,168,120,162)(121,175,127,169)(122,176,128,170)(123,177,129,171)(124,178,130,172)(125,179,131,173)(126,180,132,174)(133,187,139,181)(134,188,140,182)(135,189,141,183)(136,190,142,184)(137,191,143,185)(138,192,144,186), (1,41,17,29)(2,42,18,30)(3,37,13,25)(4,38,14,26)(5,39,15,27)(6,40,16,28)(7,43,19,31)(8,44,20,32)(9,45,21,33)(10,46,22,34)(11,47,23,35)(12,48,24,36)(49,88,61,76)(50,89,62,77)(51,90,63,78)(52,85,64,73)(53,86,65,74)(54,87,66,75)(55,94,67,82)(56,95,68,83)(57,96,69,84)(58,91,70,79)(59,92,71,80)(60,93,72,81)(97,133,109,121)(98,134,110,122)(99,135,111,123)(100,136,112,124)(101,137,113,125)(102,138,114,126)(103,139,115,127)(104,140,116,128)(105,141,117,129)(106,142,118,130)(107,143,119,131)(108,144,120,132)(145,184,157,172)(146,185,158,173)(147,186,159,174)(148,181,160,169)(149,182,161,170)(150,183,162,171)(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)], [(1,100,4,97),(2,99,5,102),(3,98,6,101),(7,104,10,107),(8,103,11,106),(9,108,12,105),(13,110,16,113),(14,109,17,112),(15,114,18,111),(19,116,22,119),(20,115,23,118),(21,120,24,117),(25,122,28,125),(26,121,29,124),(27,126,30,123),(31,128,34,131),(32,127,35,130),(33,132,36,129),(37,134,40,137),(38,133,41,136),(39,138,42,135),(43,140,46,143),(44,139,47,142),(45,144,48,141),(49,146,52,149),(50,145,53,148),(51,150,54,147),(55,152,58,155),(56,151,59,154),(57,156,60,153),(61,158,64,161),(62,157,65,160),(63,162,66,159),(67,164,70,167),(68,163,71,166),(69,168,72,165),(73,170,76,173),(74,169,77,172),(75,174,78,171),(79,176,82,179),(80,175,83,178),(81,180,84,177),(85,182,88,185),(86,181,89,184),(87,186,90,183),(91,188,94,191),(92,187,95,190),(93,192,96,189)], [(1,59,11,53),(2,60,12,54),(3,55,7,49),(4,56,8,50),(5,57,9,51),(6,58,10,52),(13,67,19,61),(14,68,20,62),(15,69,21,63),(16,70,22,64),(17,71,23,65),(18,72,24,66),(25,79,31,73),(26,80,32,74),(27,81,33,75),(28,82,34,76),(29,83,35,77),(30,84,36,78),(37,91,43,85),(38,92,44,86),(39,93,45,87),(40,94,46,88),(41,95,47,89),(42,96,48,90),(97,151,103,145),(98,152,104,146),(99,153,105,147),(100,154,106,148),(101,155,107,149),(102,156,108,150),(109,163,115,157),(110,164,116,158),(111,165,117,159),(112,166,118,160),(113,167,119,161),(114,168,120,162),(121,175,127,169),(122,176,128,170),(123,177,129,171),(124,178,130,172),(125,179,131,173),(126,180,132,174),(133,187,139,181),(134,188,140,182),(135,189,141,183),(136,190,142,184),(137,191,143,185),(138,192,144,186)], [(1,41,17,29),(2,42,18,30),(3,37,13,25),(4,38,14,26),(5,39,15,27),(6,40,16,28),(7,43,19,31),(8,44,20,32),(9,45,21,33),(10,46,22,34),(11,47,23,35),(12,48,24,36),(49,88,61,76),(50,89,62,77),(51,90,63,78),(52,85,64,73),(53,86,65,74),(54,87,66,75),(55,94,67,82),(56,95,68,83),(57,96,69,84),(58,91,70,79),(59,92,71,80),(60,93,72,81),(97,133,109,121),(98,134,110,122),(99,135,111,123),(100,136,112,124),(101,137,113,125),(102,138,114,126),(103,139,115,127),(104,140,116,128),(105,141,117,129),(106,142,118,130),(107,143,119,131),(108,144,120,132),(145,184,157,172),(146,185,158,173),(147,186,159,174),(148,181,160,169),(149,182,161,170),(150,183,162,171),(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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4○D4 | C4×S3 | D4⋊2S3 | Q8⋊3S3 |
| kernel | Dic3.5C42 | C6.C42 | C3×C2.C42 | C2×C4×Dic3 | C4×Dic3 | C2.C42 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 |
| # reps | 1 | 3 | 1 | 3 | 24 | 1 | 3 | 8 | 12 | 3 | 1 |
Matrix representation of Dic3.5C42 ►in GL6(𝔽13)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 12 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 12 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 8 | 0 | 0 |
| 0 | 0 | 7 | 6 | 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 | 2 | 11 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [9,12,0,0,0,0,0,3,0,0,0,0,0,0,3,2,0,0,0,0,0,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[9,12,0,0,0,0,2,4,0,0,0,0,0,0,7,7,0,0,0,0,8,6,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,2,9,0,0,0,0,11,11],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,1,12] >;
Dic3.5C42 in GAP, Magma, Sage, TeX
{\rm Dic}_3._5C_4^2 % in TeX
G:=Group("Dic3.5C4^2"); // GroupNames label
G:=SmallGroup(192,207);
// by ID
G=gap.SmallGroup(192,207);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^4=d^4=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^3*c>;
// generators/relations