metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.2D6, C6.6C4≀C2, C8⋊C4.3S3, C4⋊Dic3.1C4, (C2×C4).106D12, (C2×C12).224D4, (C4×C12).11C22, C12.6Q8.4C2, C2.7(C42⋊4S3), C2.5(D12⋊C4), C22.57(D6⋊C4), C6.1(C4.10D4), C42.S3.1C2, C2.3(C12.47D4), C3⋊1(C42.2C22), (C2×C4).11(C4×S3), (C3×C8⋊C4).7C2, (C2×C12).23(C2×C4), (C2×C4).207(C3⋊D4), (C2×C6).38(C22⋊C4), SmallGroup(192,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.2D6
G = < a,b,c,d | a4=b4=1, c6=a-1, d2=ba=ab, ac=ca, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=bc5 >
Subgroups: 152 in 60 conjugacy classes, 25 normal (all characteristic)
C1, C2, C3, C4, C22, C6, C8, C2×C4, C2×C4, Dic3, C12, C2×C6, C42, C4⋊C4, C2×C8, C3⋊C8, C24, C2×Dic3, C2×C12, C8⋊C4, C8⋊C4, C42.C2, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C24, C42.2C22, C42.S3, C3×C8⋊C4, C12.6Q8, C42.2D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C4.10D4, C4≀C2, D6⋊C4, C42.2C22, C42⋊4S3, C12.47D4, D12⋊C4, C42.2D6
►Regular action on 192 points
Generators in S192
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 43 37 31)(26 44 38 32)(27 45 39 33)(28 46 40 34)(29 47 41 35)(30 48 42 36)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)(73 91 85 79)(74 92 86 80)(75 93 87 81)(76 94 88 82)(77 95 89 83)(78 96 90 84)(97 115 109 103)(98 116 110 104)(99 117 111 105)(100 118 112 106)(101 119 113 107)(102 120 114 108)(121 139 133 127)(122 140 134 128)(123 141 135 129)(124 142 136 130)(125 143 137 131)(126 144 138 132)(145 163 157 151)(146 164 158 152)(147 165 159 153)(148 166 160 154)(149 167 161 155)(150 168 162 156)(169 187 181 175)(170 188 182 176)(171 189 183 177)(172 190 184 178)(173 191 185 179)(174 192 186 180)
(1 127 107 173)(2 140 108 186)(3 129 109 175)(4 142 110 188)(5 131 111 177)(6 144 112 190)(7 133 113 179)(8 122 114 192)(9 135 115 181)(10 124 116 170)(11 137 117 183)(12 126 118 172)(13 139 119 185)(14 128 120 174)(15 141 97 187)(16 130 98 176)(17 143 99 189)(18 132 100 178)(19 121 101 191)(20 134 102 180)(21 123 103 169)(22 136 104 182)(23 125 105 171)(24 138 106 184)(25 56 155 85)(26 69 156 74)(27 58 157 87)(28 71 158 76)(29 60 159 89)(30 49 160 78)(31 62 161 91)(32 51 162 80)(33 64 163 93)(34 53 164 82)(35 66 165 95)(36 55 166 84)(37 68 167 73)(38 57 168 86)(39 70 145 75)(40 59 146 88)(41 72 147 77)(42 61 148 90)(43 50 149 79)(44 63 150 92)(45 52 151 81)(46 65 152 94)(47 54 153 83)(48 67 154 96)
(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 71 121 152 119 88 179 34)(2 151 134 75 120 33 192 58)(3 74 123 44 97 57 181 162)(4 43 136 68 98 161 170 85)(5 67 125 148 99 84 183 30)(6 147 138 95 100 29 172 54)(7 94 127 40 101 53 185 158)(8 39 140 64 102 157 174 81)(9 63 129 168 103 80 187 26)(10 167 142 91 104 25 176 50)(11 90 131 36 105 49 189 154)(12 35 144 60 106 153 178 77)(13 59 133 164 107 76 191 46)(14 163 122 87 108 45 180 70)(15 86 135 32 109 69 169 150)(16 31 124 56 110 149 182 73)(17 55 137 160 111 96 171 42)(18 159 126 83 112 41 184 66)(19 82 139 28 113 65 173 146)(20 27 128 52 114 145 186 93)(21 51 141 156 115 92 175 38)(22 155 130 79 116 37 188 62)(23 78 143 48 117 61 177 166)(24 47 132 72 118 165 190 89)
G:=sub<Sym(192)| (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84)(97,115,109,103)(98,116,110,104)(99,117,111,105)(100,118,112,106)(101,119,113,107)(102,120,114,108)(121,139,133,127)(122,140,134,128)(123,141,135,129)(124,142,136,130)(125,143,137,131)(126,144,138,132)(145,163,157,151)(146,164,158,152)(147,165,159,153)(148,166,160,154)(149,167,161,155)(150,168,162,156)(169,187,181,175)(170,188,182,176)(171,189,183,177)(172,190,184,178)(173,191,185,179)(174,192,186,180), (1,127,107,173)(2,140,108,186)(3,129,109,175)(4,142,110,188)(5,131,111,177)(6,144,112,190)(7,133,113,179)(8,122,114,192)(9,135,115,181)(10,124,116,170)(11,137,117,183)(12,126,118,172)(13,139,119,185)(14,128,120,174)(15,141,97,187)(16,130,98,176)(17,143,99,189)(18,132,100,178)(19,121,101,191)(20,134,102,180)(21,123,103,169)(22,136,104,182)(23,125,105,171)(24,138,106,184)(25,56,155,85)(26,69,156,74)(27,58,157,87)(28,71,158,76)(29,60,159,89)(30,49,160,78)(31,62,161,91)(32,51,162,80)(33,64,163,93)(34,53,164,82)(35,66,165,95)(36,55,166,84)(37,68,167,73)(38,57,168,86)(39,70,145,75)(40,59,146,88)(41,72,147,77)(42,61,148,90)(43,50,149,79)(44,63,150,92)(45,52,151,81)(46,65,152,94)(47,54,153,83)(48,67,154,96), (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), (1,71,121,152,119,88,179,34)(2,151,134,75,120,33,192,58)(3,74,123,44,97,57,181,162)(4,43,136,68,98,161,170,85)(5,67,125,148,99,84,183,30)(6,147,138,95,100,29,172,54)(7,94,127,40,101,53,185,158)(8,39,140,64,102,157,174,81)(9,63,129,168,103,80,187,26)(10,167,142,91,104,25,176,50)(11,90,131,36,105,49,189,154)(12,35,144,60,106,153,178,77)(13,59,133,164,107,76,191,46)(14,163,122,87,108,45,180,70)(15,86,135,32,109,69,169,150)(16,31,124,56,110,149,182,73)(17,55,137,160,111,96,171,42)(18,159,126,83,112,41,184,66)(19,82,139,28,113,65,173,146)(20,27,128,52,114,145,186,93)(21,51,141,156,115,92,175,38)(22,155,130,79,116,37,188,62)(23,78,143,48,117,61,177,166)(24,47,132,72,118,165,190,89)>;
G:=Group( (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36)(49,67,61,55)(50,68,62,56)(51,69,63,57)(52,70,64,58)(53,71,65,59)(54,72,66,60)(73,91,85,79)(74,92,86,80)(75,93,87,81)(76,94,88,82)(77,95,89,83)(78,96,90,84)(97,115,109,103)(98,116,110,104)(99,117,111,105)(100,118,112,106)(101,119,113,107)(102,120,114,108)(121,139,133,127)(122,140,134,128)(123,141,135,129)(124,142,136,130)(125,143,137,131)(126,144,138,132)(145,163,157,151)(146,164,158,152)(147,165,159,153)(148,166,160,154)(149,167,161,155)(150,168,162,156)(169,187,181,175)(170,188,182,176)(171,189,183,177)(172,190,184,178)(173,191,185,179)(174,192,186,180), (1,127,107,173)(2,140,108,186)(3,129,109,175)(4,142,110,188)(5,131,111,177)(6,144,112,190)(7,133,113,179)(8,122,114,192)(9,135,115,181)(10,124,116,170)(11,137,117,183)(12,126,118,172)(13,139,119,185)(14,128,120,174)(15,141,97,187)(16,130,98,176)(17,143,99,189)(18,132,100,178)(19,121,101,191)(20,134,102,180)(21,123,103,169)(22,136,104,182)(23,125,105,171)(24,138,106,184)(25,56,155,85)(26,69,156,74)(27,58,157,87)(28,71,158,76)(29,60,159,89)(30,49,160,78)(31,62,161,91)(32,51,162,80)(33,64,163,93)(34,53,164,82)(35,66,165,95)(36,55,166,84)(37,68,167,73)(38,57,168,86)(39,70,145,75)(40,59,146,88)(41,72,147,77)(42,61,148,90)(43,50,149,79)(44,63,150,92)(45,52,151,81)(46,65,152,94)(47,54,153,83)(48,67,154,96), (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), (1,71,121,152,119,88,179,34)(2,151,134,75,120,33,192,58)(3,74,123,44,97,57,181,162)(4,43,136,68,98,161,170,85)(5,67,125,148,99,84,183,30)(6,147,138,95,100,29,172,54)(7,94,127,40,101,53,185,158)(8,39,140,64,102,157,174,81)(9,63,129,168,103,80,187,26)(10,167,142,91,104,25,176,50)(11,90,131,36,105,49,189,154)(12,35,144,60,106,153,178,77)(13,59,133,164,107,76,191,46)(14,163,122,87,108,45,180,70)(15,86,135,32,109,69,169,150)(16,31,124,56,110,149,182,73)(17,55,137,160,111,96,171,42)(18,159,126,83,112,41,184,66)(19,82,139,28,113,65,173,146)(20,27,128,52,114,145,186,93)(21,51,141,156,115,92,175,38)(22,155,130,79,116,37,188,62)(23,78,143,48,117,61,177,166)(24,47,132,72,118,165,190,89) );
G=PermutationGroup([[(1,19,13,7),(2,20,14,8),(3,21,15,9),(4,22,16,10),(5,23,17,11),(6,24,18,12),(25,43,37,31),(26,44,38,32),(27,45,39,33),(28,46,40,34),(29,47,41,35),(30,48,42,36),(49,67,61,55),(50,68,62,56),(51,69,63,57),(52,70,64,58),(53,71,65,59),(54,72,66,60),(73,91,85,79),(74,92,86,80),(75,93,87,81),(76,94,88,82),(77,95,89,83),(78,96,90,84),(97,115,109,103),(98,116,110,104),(99,117,111,105),(100,118,112,106),(101,119,113,107),(102,120,114,108),(121,139,133,127),(122,140,134,128),(123,141,135,129),(124,142,136,130),(125,143,137,131),(126,144,138,132),(145,163,157,151),(146,164,158,152),(147,165,159,153),(148,166,160,154),(149,167,161,155),(150,168,162,156),(169,187,181,175),(170,188,182,176),(171,189,183,177),(172,190,184,178),(173,191,185,179),(174,192,186,180)], [(1,127,107,173),(2,140,108,186),(3,129,109,175),(4,142,110,188),(5,131,111,177),(6,144,112,190),(7,133,113,179),(8,122,114,192),(9,135,115,181),(10,124,116,170),(11,137,117,183),(12,126,118,172),(13,139,119,185),(14,128,120,174),(15,141,97,187),(16,130,98,176),(17,143,99,189),(18,132,100,178),(19,121,101,191),(20,134,102,180),(21,123,103,169),(22,136,104,182),(23,125,105,171),(24,138,106,184),(25,56,155,85),(26,69,156,74),(27,58,157,87),(28,71,158,76),(29,60,159,89),(30,49,160,78),(31,62,161,91),(32,51,162,80),(33,64,163,93),(34,53,164,82),(35,66,165,95),(36,55,166,84),(37,68,167,73),(38,57,168,86),(39,70,145,75),(40,59,146,88),(41,72,147,77),(42,61,148,90),(43,50,149,79),(44,63,150,92),(45,52,151,81),(46,65,152,94),(47,54,153,83),(48,67,154,96)], [(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)], [(1,71,121,152,119,88,179,34),(2,151,134,75,120,33,192,58),(3,74,123,44,97,57,181,162),(4,43,136,68,98,161,170,85),(5,67,125,148,99,84,183,30),(6,147,138,95,100,29,172,54),(7,94,127,40,101,53,185,158),(8,39,140,64,102,157,174,81),(9,63,129,168,103,80,187,26),(10,167,142,91,104,25,176,50),(11,90,131,36,105,49,189,154),(12,35,144,60,106,153,178,77),(13,59,133,164,107,76,191,46),(14,163,122,87,108,45,180,70),(15,86,135,32,109,69,169,150),(16,31,124,56,110,149,182,73),(17,55,137,160,111,96,171,42),(18,159,126,83,112,41,184,66),(19,82,139,28,113,65,173,146),(20,27,128,52,114,145,186,93),(21,51,141,156,115,92,175,38),(22,155,130,79,116,37,188,62),(23,78,143,48,117,61,177,166),(24,47,132,72,118,165,190,89)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | C4×S3 | D12 | C3⋊D4 | C4≀C2 | C42⋊4S3 | C4.10D4 | C12.47D4 | D12⋊C4 |
| kernel | C42.2D6 | C42.S3 | C3×C8⋊C4 | C12.6Q8 | C4⋊Dic3 | C8⋊C4 | C2×C12 | C42 | C2×C4 | C2×C4 | C2×C4 | C6 | C2 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 2 | 2 | 2 | 8 | 8 | 1 | 2 | 2 |
Matrix representation of C42.2D6 ►in GL4(𝔽73) generated by
| 30 | 60 | 0 | 0 |
| 13 | 43 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 7 | 14 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 21 | 27 |
| 39 | 62 | 0 | 0 |
| 11 | 50 | 0 | 0 |
| 0 | 0 | 62 | 3 |
| 0 | 0 | 48 | 11 |
| 1 | 28 | 0 | 0 |
| 27 | 72 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 71 | 1 |
G:=sub<GL(4,GF(73))| [30,13,0,0,60,43,0,0,0,0,46,0,0,0,0,46],[7,59,0,0,14,66,0,0,0,0,46,21,0,0,0,27],[39,11,0,0,62,50,0,0,0,0,62,48,0,0,3,11],[1,27,0,0,28,72,0,0,0,0,27,71,0,0,0,1] >;
C42.2D6 in GAP, Magma, Sage, TeX
C_4^2._2D_6
% in TeX
G:=Group("C4^2.2D6"); // GroupNames label
G:=SmallGroup(192,24);
// by ID
G=gap.SmallGroup(192,24);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,422,184,1571,570,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^-1,d^2=b*a=a*b,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=b*c^5>;
// generators/relations