metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.210D6, C12.11M4(2), C3⋊C8⋊11Q8, C3⋊5(C8⋊4Q8), (C6×Q8).9C4, C6.27(C4×Q8), C4.58(S3×Q8), C4⋊C4.9Dic3, (Q8×C12).6C2, (C4×Q8).12S3, C2.5(Q8×Dic3), C6.42(C8○D4), C12.116(C2×Q8), (C2×Q8).8Dic3, C12⋊C8.18C2, (C4×C12).95C22, C6.42(C2×M4(2)), C4.3(C4.Dic3), C12.339(C4○D4), (C2×C12).852C23, C4.59(Q8⋊3S3), C2.8(D4.Dic3), C42.S3.3C2, C22.47(C22×Dic3), (C4×C3⋊C8).8C2, (C3×C4⋊C4).14C4, (C2×C12).166(C2×C4), (C2×C3⋊C8).202C22, (C2×C4).45(C2×Dic3), C2.10(C2×C4.Dic3), (C2×C6).189(C22×C4), (C2×C4).794(C22×S3), SmallGroup(192,583)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.210D6
G = < a,b,c,d | a4=b4=1, c6=a2, d2=a2b-1, ab=ba, cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=b2c5 >
Subgroups: 152 in 94 conjugacy classes, 61 normal (33 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, C3⋊C8, C2×C12, C2×C12, C3×Q8, C4×C8, C8⋊C4, C4⋊C8, C4×Q8, C2×C3⋊C8, C2×C3⋊C8, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C6×Q8, C8⋊4Q8, C4×C3⋊C8, C42.S3, C12⋊C8, C12⋊C8, Q8×C12, C42.210D6
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, Dic3, D6, M4(2), C22×C4, C2×Q8, C4○D4, C2×Dic3, C22×S3, C4×Q8, C2×M4(2), C8○D4, C4.Dic3, S3×Q8, Q8⋊3S3, C22×Dic3, C8⋊4Q8, C2×C4.Dic3, Q8×Dic3, D4.Dic3, C42.210D6
►Regular action on 192 points
Generators in S192
(1 181 7 187)(2 188 8 182)(3 183 9 189)(4 190 10 184)(5 185 11 191)(6 192 12 186)(13 136 19 142)(14 143 20 137)(15 138 21 144)(16 133 22 139)(17 140 23 134)(18 135 24 141)(25 87 31 93)(26 94 32 88)(27 89 33 95)(28 96 34 90)(29 91 35 85)(30 86 36 92)(37 78 43 84)(38 73 44 79)(39 80 45 74)(40 75 46 81)(41 82 47 76)(42 77 48 83)(49 111 55 117)(50 118 56 112)(51 113 57 119)(52 120 58 114)(53 115 59 109)(54 110 60 116)(61 158 67 164)(62 165 68 159)(63 160 69 166)(64 167 70 161)(65 162 71 168)(66 157 72 163)(97 174 103 180)(98 169 104 175)(99 176 105 170)(100 171 106 177)(101 178 107 172)(102 173 108 179)(121 156 127 150)(122 151 128 145)(123 146 129 152)(124 153 130 147)(125 148 131 154)(126 155 132 149)
(1 153 44 163)(2 154 45 164)(3 155 46 165)(4 156 47 166)(5 145 48 167)(6 146 37 168)(7 147 38 157)(8 148 39 158)(9 149 40 159)(10 150 41 160)(11 151 42 161)(12 152 43 162)(13 93 98 116)(14 94 99 117)(15 95 100 118)(16 96 101 119)(17 85 102 120)(18 86 103 109)(19 87 104 110)(20 88 105 111)(21 89 106 112)(22 90 107 113)(23 91 108 114)(24 92 97 115)(25 169 54 136)(26 170 55 137)(27 171 56 138)(28 172 57 139)(29 173 58 140)(30 174 59 141)(31 175 60 142)(32 176 49 143)(33 177 50 144)(34 178 51 133)(35 179 52 134)(36 180 53 135)(61 188 125 74)(62 189 126 75)(63 190 127 76)(64 191 128 77)(65 192 129 78)(66 181 130 79)(67 182 131 80)(68 183 132 81)(69 184 121 82)(70 185 122 83)(71 186 123 84)(72 187 124 73)
(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)
(1 139 157 51 44 172 147 34)(2 177 158 27 45 144 148 56)(3 137 159 49 46 170 149 32)(4 175 160 25 47 142 150 54)(5 135 161 59 48 180 151 30)(6 173 162 35 37 140 152 52)(7 133 163 57 38 178 153 28)(8 171 164 33 39 138 154 50)(9 143 165 55 40 176 155 26)(10 169 166 31 41 136 156 60)(11 141 167 53 42 174 145 36)(12 179 168 29 43 134 146 58)(13 127 110 184 98 63 87 82)(14 68 111 75 99 132 88 189)(15 125 112 182 100 61 89 80)(16 66 113 73 101 130 90 187)(17 123 114 192 102 71 91 78)(18 64 115 83 103 128 92 185)(19 121 116 190 104 69 93 76)(20 62 117 81 105 126 94 183)(21 131 118 188 106 67 95 74)(22 72 119 79 107 124 96 181)(23 129 120 186 108 65 85 84)(24 70 109 77 97 122 86 191)
G:=sub<Sym(192)| (1,181,7,187)(2,188,8,182)(3,183,9,189)(4,190,10,184)(5,185,11,191)(6,192,12,186)(13,136,19,142)(14,143,20,137)(15,138,21,144)(16,133,22,139)(17,140,23,134)(18,135,24,141)(25,87,31,93)(26,94,32,88)(27,89,33,95)(28,96,34,90)(29,91,35,85)(30,86,36,92)(37,78,43,84)(38,73,44,79)(39,80,45,74)(40,75,46,81)(41,82,47,76)(42,77,48,83)(49,111,55,117)(50,118,56,112)(51,113,57,119)(52,120,58,114)(53,115,59,109)(54,110,60,116)(61,158,67,164)(62,165,68,159)(63,160,69,166)(64,167,70,161)(65,162,71,168)(66,157,72,163)(97,174,103,180)(98,169,104,175)(99,176,105,170)(100,171,106,177)(101,178,107,172)(102,173,108,179)(121,156,127,150)(122,151,128,145)(123,146,129,152)(124,153,130,147)(125,148,131,154)(126,155,132,149), (1,153,44,163)(2,154,45,164)(3,155,46,165)(4,156,47,166)(5,145,48,167)(6,146,37,168)(7,147,38,157)(8,148,39,158)(9,149,40,159)(10,150,41,160)(11,151,42,161)(12,152,43,162)(13,93,98,116)(14,94,99,117)(15,95,100,118)(16,96,101,119)(17,85,102,120)(18,86,103,109)(19,87,104,110)(20,88,105,111)(21,89,106,112)(22,90,107,113)(23,91,108,114)(24,92,97,115)(25,169,54,136)(26,170,55,137)(27,171,56,138)(28,172,57,139)(29,173,58,140)(30,174,59,141)(31,175,60,142)(32,176,49,143)(33,177,50,144)(34,178,51,133)(35,179,52,134)(36,180,53,135)(61,188,125,74)(62,189,126,75)(63,190,127,76)(64,191,128,77)(65,192,129,78)(66,181,130,79)(67,182,131,80)(68,183,132,81)(69,184,121,82)(70,185,122,83)(71,186,123,84)(72,187,124,73), (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), (1,139,157,51,44,172,147,34)(2,177,158,27,45,144,148,56)(3,137,159,49,46,170,149,32)(4,175,160,25,47,142,150,54)(5,135,161,59,48,180,151,30)(6,173,162,35,37,140,152,52)(7,133,163,57,38,178,153,28)(8,171,164,33,39,138,154,50)(9,143,165,55,40,176,155,26)(10,169,166,31,41,136,156,60)(11,141,167,53,42,174,145,36)(12,179,168,29,43,134,146,58)(13,127,110,184,98,63,87,82)(14,68,111,75,99,132,88,189)(15,125,112,182,100,61,89,80)(16,66,113,73,101,130,90,187)(17,123,114,192,102,71,91,78)(18,64,115,83,103,128,92,185)(19,121,116,190,104,69,93,76)(20,62,117,81,105,126,94,183)(21,131,118,188,106,67,95,74)(22,72,119,79,107,124,96,181)(23,129,120,186,108,65,85,84)(24,70,109,77,97,122,86,191)>;
G:=Group( (1,181,7,187)(2,188,8,182)(3,183,9,189)(4,190,10,184)(5,185,11,191)(6,192,12,186)(13,136,19,142)(14,143,20,137)(15,138,21,144)(16,133,22,139)(17,140,23,134)(18,135,24,141)(25,87,31,93)(26,94,32,88)(27,89,33,95)(28,96,34,90)(29,91,35,85)(30,86,36,92)(37,78,43,84)(38,73,44,79)(39,80,45,74)(40,75,46,81)(41,82,47,76)(42,77,48,83)(49,111,55,117)(50,118,56,112)(51,113,57,119)(52,120,58,114)(53,115,59,109)(54,110,60,116)(61,158,67,164)(62,165,68,159)(63,160,69,166)(64,167,70,161)(65,162,71,168)(66,157,72,163)(97,174,103,180)(98,169,104,175)(99,176,105,170)(100,171,106,177)(101,178,107,172)(102,173,108,179)(121,156,127,150)(122,151,128,145)(123,146,129,152)(124,153,130,147)(125,148,131,154)(126,155,132,149), (1,153,44,163)(2,154,45,164)(3,155,46,165)(4,156,47,166)(5,145,48,167)(6,146,37,168)(7,147,38,157)(8,148,39,158)(9,149,40,159)(10,150,41,160)(11,151,42,161)(12,152,43,162)(13,93,98,116)(14,94,99,117)(15,95,100,118)(16,96,101,119)(17,85,102,120)(18,86,103,109)(19,87,104,110)(20,88,105,111)(21,89,106,112)(22,90,107,113)(23,91,108,114)(24,92,97,115)(25,169,54,136)(26,170,55,137)(27,171,56,138)(28,172,57,139)(29,173,58,140)(30,174,59,141)(31,175,60,142)(32,176,49,143)(33,177,50,144)(34,178,51,133)(35,179,52,134)(36,180,53,135)(61,188,125,74)(62,189,126,75)(63,190,127,76)(64,191,128,77)(65,192,129,78)(66,181,130,79)(67,182,131,80)(68,183,132,81)(69,184,121,82)(70,185,122,83)(71,186,123,84)(72,187,124,73), (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), (1,139,157,51,44,172,147,34)(2,177,158,27,45,144,148,56)(3,137,159,49,46,170,149,32)(4,175,160,25,47,142,150,54)(5,135,161,59,48,180,151,30)(6,173,162,35,37,140,152,52)(7,133,163,57,38,178,153,28)(8,171,164,33,39,138,154,50)(9,143,165,55,40,176,155,26)(10,169,166,31,41,136,156,60)(11,141,167,53,42,174,145,36)(12,179,168,29,43,134,146,58)(13,127,110,184,98,63,87,82)(14,68,111,75,99,132,88,189)(15,125,112,182,100,61,89,80)(16,66,113,73,101,130,90,187)(17,123,114,192,102,71,91,78)(18,64,115,83,103,128,92,185)(19,121,116,190,104,69,93,76)(20,62,117,81,105,126,94,183)(21,131,118,188,106,67,95,74)(22,72,119,79,107,124,96,181)(23,129,120,186,108,65,85,84)(24,70,109,77,97,122,86,191) );
G=PermutationGroup([[(1,181,7,187),(2,188,8,182),(3,183,9,189),(4,190,10,184),(5,185,11,191),(6,192,12,186),(13,136,19,142),(14,143,20,137),(15,138,21,144),(16,133,22,139),(17,140,23,134),(18,135,24,141),(25,87,31,93),(26,94,32,88),(27,89,33,95),(28,96,34,90),(29,91,35,85),(30,86,36,92),(37,78,43,84),(38,73,44,79),(39,80,45,74),(40,75,46,81),(41,82,47,76),(42,77,48,83),(49,111,55,117),(50,118,56,112),(51,113,57,119),(52,120,58,114),(53,115,59,109),(54,110,60,116),(61,158,67,164),(62,165,68,159),(63,160,69,166),(64,167,70,161),(65,162,71,168),(66,157,72,163),(97,174,103,180),(98,169,104,175),(99,176,105,170),(100,171,106,177),(101,178,107,172),(102,173,108,179),(121,156,127,150),(122,151,128,145),(123,146,129,152),(124,153,130,147),(125,148,131,154),(126,155,132,149)], [(1,153,44,163),(2,154,45,164),(3,155,46,165),(4,156,47,166),(5,145,48,167),(6,146,37,168),(7,147,38,157),(8,148,39,158),(9,149,40,159),(10,150,41,160),(11,151,42,161),(12,152,43,162),(13,93,98,116),(14,94,99,117),(15,95,100,118),(16,96,101,119),(17,85,102,120),(18,86,103,109),(19,87,104,110),(20,88,105,111),(21,89,106,112),(22,90,107,113),(23,91,108,114),(24,92,97,115),(25,169,54,136),(26,170,55,137),(27,171,56,138),(28,172,57,139),(29,173,58,140),(30,174,59,141),(31,175,60,142),(32,176,49,143),(33,177,50,144),(34,178,51,133),(35,179,52,134),(36,180,53,135),(61,188,125,74),(62,189,126,75),(63,190,127,76),(64,191,128,77),(65,192,129,78),(66,181,130,79),(67,182,131,80),(68,183,132,81),(69,184,121,82),(70,185,122,83),(71,186,123,84),(72,187,124,73)], [(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)], [(1,139,157,51,44,172,147,34),(2,177,158,27,45,144,148,56),(3,137,159,49,46,170,149,32),(4,175,160,25,47,142,150,54),(5,135,161,59,48,180,151,30),(6,173,162,35,37,140,152,52),(7,133,163,57,38,178,153,28),(8,171,164,33,39,138,154,50),(9,143,165,55,40,176,155,26),(10,169,166,31,41,136,156,60),(11,141,167,53,42,174,145,36),(12,179,168,29,43,134,146,58),(13,127,110,184,98,63,87,82),(14,68,111,75,99,132,88,189),(15,125,112,182,100,61,89,80),(16,66,113,73,101,130,90,187),(17,123,114,192,102,71,91,78),(18,64,115,83,103,128,92,185),(19,121,116,190,104,69,93,76),(20,62,117,81,105,126,94,183),(21,131,118,188,106,67,95,74),(22,72,119,79,107,124,96,181),(23,129,120,186,108,65,85,84),(24,70,109,77,97,122,86,191)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | Q8 | D6 | Dic3 | Dic3 | M4(2) | C4○D4 | C8○D4 | C4.Dic3 | S3×Q8 | Q8⋊3S3 | D4.Dic3 |
| kernel | C42.210D6 | C4×C3⋊C8 | C42.S3 | C12⋊C8 | Q8×C12 | C3×C4⋊C4 | C6×Q8 | C4×Q8 | C3⋊C8 | C42 | C4⋊C4 | C2×Q8 | C12 | C12 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 6 | 2 | 1 | 2 | 3 | 3 | 1 | 4 | 2 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C42.210D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 62 | 0 | 0 |
| 0 | 0 | 46 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 20 |
| 0 | 0 | 0 | 0 | 40 | 41 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 66 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 45 | 46 |
| 23 | 18 | 0 | 0 | 0 | 0 |
| 68 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 70 | 0 | 0 |
| 0 | 0 | 63 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 26 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,46,0,0,0,0,62,40,0,0,0,0,0,0,32,40,0,0,0,0,20,41],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,27,0,0,0,0,0,66,46,0,0,0,0,0,0,27,45,0,0,0,0,0,46],[23,68,0,0,0,0,18,50,0,0,0,0,0,0,69,63,0,0,0,0,70,4,0,0,0,0,0,0,1,26,0,0,0,0,0,72] >;
C42.210D6 in GAP, Magma, Sage, TeX
C_4^2._{210}D_6 % in TeX
G:=Group("C4^2.210D6"); // GroupNames label
G:=SmallGroup(192,583);
// by ID
G=gap.SmallGroup(192,583);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,120,758,219,100,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=a^2*b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=b^2*c^5>;
// generators/relations