metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.8D6, C6.19C4≀C2, C4⋊C4.1Dic3, (C2×C12).232D4, C42.C2.1S3, (C4×C12).236C22, C6.8(C4.10D4), C42.S3.9C2, C2.7(Q8⋊3Dic3), C2.3(C12.10D4), C3⋊2(C42.2C22), C22.40(C6.D4), (C3×C4⋊C4).1C4, (C2×C12).170(C2×C4), (C2×C4).10(C2×Dic3), (C3×C42.C2).7C2, (C2×C4).166(C3⋊D4), (C2×C6).101(C22⋊C4), SmallGroup(192,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.8D6
G = < a,b,c,d | a4=b4=1, c6=a2b2, d2=a-1b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=a2bc5 >
Subgroups: 128 in 60 conjugacy classes, 27 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C3⋊C8, C2×C12, C2×C12, C2×C12, C8⋊C4, C42.C2, C2×C3⋊C8, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C42.2C22, C42.S3, C3×C42.C2, C42.8D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C2×Dic3, C3⋊D4, C4.10D4, C4≀C2, C6.D4, C42.2C22, C12.10D4, Q8⋊3Dic3, C42.8D6
►Regular action on 192 points
Generators in S192
(1 118 184 168)(2 113 185 163)(3 120 186 158)(4 115 187 165)(5 110 188 160)(6 117 189 167)(7 112 190 162)(8 119 191 157)(9 114 192 164)(10 109 181 159)(11 116 182 166)(12 111 183 161)(13 67 48 105)(14 62 37 100)(15 69 38 107)(16 64 39 102)(17 71 40 97)(18 66 41 104)(19 61 42 99)(20 68 43 106)(21 63 44 101)(22 70 45 108)(23 65 46 103)(24 72 47 98)(25 141 51 91)(26 136 52 86)(27 143 53 93)(28 138 54 88)(29 133 55 95)(30 140 56 90)(31 135 57 85)(32 142 58 92)(33 137 59 87)(34 144 60 94)(35 139 49 89)(36 134 50 96)(73 179 122 156)(74 174 123 151)(75 169 124 146)(76 176 125 153)(77 171 126 148)(78 178 127 155)(79 173 128 150)(80 180 129 145)(81 175 130 152)(82 170 131 147)(83 177 132 154)(84 172 121 149)
(1 61 190 105)(2 106 191 62)(3 63 192 107)(4 108 181 64)(5 65 182 97)(6 98 183 66)(7 67 184 99)(8 100 185 68)(9 69 186 101)(10 102 187 70)(11 71 188 103)(12 104 189 72)(13 118 42 162)(14 163 43 119)(15 120 44 164)(16 165 45 109)(17 110 46 166)(18 167 47 111)(19 112 48 168)(20 157 37 113)(21 114 38 158)(22 159 39 115)(23 116 40 160)(24 161 41 117)(25 75 57 130)(26 131 58 76)(27 77 59 132)(28 121 60 78)(29 79 49 122)(30 123 50 80)(31 81 51 124)(32 125 52 82)(33 83 53 126)(34 127 54 84)(35 73 55 128)(36 129 56 74)(85 152 141 169)(86 170 142 153)(87 154 143 171)(88 172 144 155)(89 156 133 173)(90 174 134 145)(91 146 135 175)(92 176 136 147)(93 148 137 177)(94 178 138 149)(95 150 139 179)(96 180 140 151)
(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)
(1 180 19 30 7 174 13 36)(2 133 43 79 8 139 37 73)(3 178 21 28 9 172 15 34)(4 143 45 77 10 137 39 83)(5 176 23 26 11 170 17 32)(6 141 47 75 12 135 41 81)(14 122 185 95 20 128 191 89)(16 132 187 93 22 126 181 87)(18 130 189 91 24 124 183 85)(25 66 146 167 31 72 152 161)(27 64 148 165 33 70 154 159)(29 62 150 163 35 68 156 157)(38 60 186 155 44 54 192 149)(40 58 188 153 46 52 182 147)(42 56 190 151 48 50 184 145)(49 106 179 119 55 100 173 113)(51 104 169 117 57 98 175 111)(53 102 171 115 59 108 177 109)(61 134 112 129 67 140 118 123)(63 144 114 127 69 138 120 121)(65 142 116 125 71 136 110 131)(74 99 96 162 80 105 90 168)(76 97 86 160 82 103 92 166)(78 107 88 158 84 101 94 164)
G:=sub<Sym(192)| (1,118,184,168)(2,113,185,163)(3,120,186,158)(4,115,187,165)(5,110,188,160)(6,117,189,167)(7,112,190,162)(8,119,191,157)(9,114,192,164)(10,109,181,159)(11,116,182,166)(12,111,183,161)(13,67,48,105)(14,62,37,100)(15,69,38,107)(16,64,39,102)(17,71,40,97)(18,66,41,104)(19,61,42,99)(20,68,43,106)(21,63,44,101)(22,70,45,108)(23,65,46,103)(24,72,47,98)(25,141,51,91)(26,136,52,86)(27,143,53,93)(28,138,54,88)(29,133,55,95)(30,140,56,90)(31,135,57,85)(32,142,58,92)(33,137,59,87)(34,144,60,94)(35,139,49,89)(36,134,50,96)(73,179,122,156)(74,174,123,151)(75,169,124,146)(76,176,125,153)(77,171,126,148)(78,178,127,155)(79,173,128,150)(80,180,129,145)(81,175,130,152)(82,170,131,147)(83,177,132,154)(84,172,121,149), (1,61,190,105)(2,106,191,62)(3,63,192,107)(4,108,181,64)(5,65,182,97)(6,98,183,66)(7,67,184,99)(8,100,185,68)(9,69,186,101)(10,102,187,70)(11,71,188,103)(12,104,189,72)(13,118,42,162)(14,163,43,119)(15,120,44,164)(16,165,45,109)(17,110,46,166)(18,167,47,111)(19,112,48,168)(20,157,37,113)(21,114,38,158)(22,159,39,115)(23,116,40,160)(24,161,41,117)(25,75,57,130)(26,131,58,76)(27,77,59,132)(28,121,60,78)(29,79,49,122)(30,123,50,80)(31,81,51,124)(32,125,52,82)(33,83,53,126)(34,127,54,84)(35,73,55,128)(36,129,56,74)(85,152,141,169)(86,170,142,153)(87,154,143,171)(88,172,144,155)(89,156,133,173)(90,174,134,145)(91,146,135,175)(92,176,136,147)(93,148,137,177)(94,178,138,149)(95,150,139,179)(96,180,140,151), (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), (1,180,19,30,7,174,13,36)(2,133,43,79,8,139,37,73)(3,178,21,28,9,172,15,34)(4,143,45,77,10,137,39,83)(5,176,23,26,11,170,17,32)(6,141,47,75,12,135,41,81)(14,122,185,95,20,128,191,89)(16,132,187,93,22,126,181,87)(18,130,189,91,24,124,183,85)(25,66,146,167,31,72,152,161)(27,64,148,165,33,70,154,159)(29,62,150,163,35,68,156,157)(38,60,186,155,44,54,192,149)(40,58,188,153,46,52,182,147)(42,56,190,151,48,50,184,145)(49,106,179,119,55,100,173,113)(51,104,169,117,57,98,175,111)(53,102,171,115,59,108,177,109)(61,134,112,129,67,140,118,123)(63,144,114,127,69,138,120,121)(65,142,116,125,71,136,110,131)(74,99,96,162,80,105,90,168)(76,97,86,160,82,103,92,166)(78,107,88,158,84,101,94,164)>;
G:=Group( (1,118,184,168)(2,113,185,163)(3,120,186,158)(4,115,187,165)(5,110,188,160)(6,117,189,167)(7,112,190,162)(8,119,191,157)(9,114,192,164)(10,109,181,159)(11,116,182,166)(12,111,183,161)(13,67,48,105)(14,62,37,100)(15,69,38,107)(16,64,39,102)(17,71,40,97)(18,66,41,104)(19,61,42,99)(20,68,43,106)(21,63,44,101)(22,70,45,108)(23,65,46,103)(24,72,47,98)(25,141,51,91)(26,136,52,86)(27,143,53,93)(28,138,54,88)(29,133,55,95)(30,140,56,90)(31,135,57,85)(32,142,58,92)(33,137,59,87)(34,144,60,94)(35,139,49,89)(36,134,50,96)(73,179,122,156)(74,174,123,151)(75,169,124,146)(76,176,125,153)(77,171,126,148)(78,178,127,155)(79,173,128,150)(80,180,129,145)(81,175,130,152)(82,170,131,147)(83,177,132,154)(84,172,121,149), (1,61,190,105)(2,106,191,62)(3,63,192,107)(4,108,181,64)(5,65,182,97)(6,98,183,66)(7,67,184,99)(8,100,185,68)(9,69,186,101)(10,102,187,70)(11,71,188,103)(12,104,189,72)(13,118,42,162)(14,163,43,119)(15,120,44,164)(16,165,45,109)(17,110,46,166)(18,167,47,111)(19,112,48,168)(20,157,37,113)(21,114,38,158)(22,159,39,115)(23,116,40,160)(24,161,41,117)(25,75,57,130)(26,131,58,76)(27,77,59,132)(28,121,60,78)(29,79,49,122)(30,123,50,80)(31,81,51,124)(32,125,52,82)(33,83,53,126)(34,127,54,84)(35,73,55,128)(36,129,56,74)(85,152,141,169)(86,170,142,153)(87,154,143,171)(88,172,144,155)(89,156,133,173)(90,174,134,145)(91,146,135,175)(92,176,136,147)(93,148,137,177)(94,178,138,149)(95,150,139,179)(96,180,140,151), (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), (1,180,19,30,7,174,13,36)(2,133,43,79,8,139,37,73)(3,178,21,28,9,172,15,34)(4,143,45,77,10,137,39,83)(5,176,23,26,11,170,17,32)(6,141,47,75,12,135,41,81)(14,122,185,95,20,128,191,89)(16,132,187,93,22,126,181,87)(18,130,189,91,24,124,183,85)(25,66,146,167,31,72,152,161)(27,64,148,165,33,70,154,159)(29,62,150,163,35,68,156,157)(38,60,186,155,44,54,192,149)(40,58,188,153,46,52,182,147)(42,56,190,151,48,50,184,145)(49,106,179,119,55,100,173,113)(51,104,169,117,57,98,175,111)(53,102,171,115,59,108,177,109)(61,134,112,129,67,140,118,123)(63,144,114,127,69,138,120,121)(65,142,116,125,71,136,110,131)(74,99,96,162,80,105,90,168)(76,97,86,160,82,103,92,166)(78,107,88,158,84,101,94,164) );
G=PermutationGroup([[(1,118,184,168),(2,113,185,163),(3,120,186,158),(4,115,187,165),(5,110,188,160),(6,117,189,167),(7,112,190,162),(8,119,191,157),(9,114,192,164),(10,109,181,159),(11,116,182,166),(12,111,183,161),(13,67,48,105),(14,62,37,100),(15,69,38,107),(16,64,39,102),(17,71,40,97),(18,66,41,104),(19,61,42,99),(20,68,43,106),(21,63,44,101),(22,70,45,108),(23,65,46,103),(24,72,47,98),(25,141,51,91),(26,136,52,86),(27,143,53,93),(28,138,54,88),(29,133,55,95),(30,140,56,90),(31,135,57,85),(32,142,58,92),(33,137,59,87),(34,144,60,94),(35,139,49,89),(36,134,50,96),(73,179,122,156),(74,174,123,151),(75,169,124,146),(76,176,125,153),(77,171,126,148),(78,178,127,155),(79,173,128,150),(80,180,129,145),(81,175,130,152),(82,170,131,147),(83,177,132,154),(84,172,121,149)], [(1,61,190,105),(2,106,191,62),(3,63,192,107),(4,108,181,64),(5,65,182,97),(6,98,183,66),(7,67,184,99),(8,100,185,68),(9,69,186,101),(10,102,187,70),(11,71,188,103),(12,104,189,72),(13,118,42,162),(14,163,43,119),(15,120,44,164),(16,165,45,109),(17,110,46,166),(18,167,47,111),(19,112,48,168),(20,157,37,113),(21,114,38,158),(22,159,39,115),(23,116,40,160),(24,161,41,117),(25,75,57,130),(26,131,58,76),(27,77,59,132),(28,121,60,78),(29,79,49,122),(30,123,50,80),(31,81,51,124),(32,125,52,82),(33,83,53,126),(34,127,54,84),(35,73,55,128),(36,129,56,74),(85,152,141,169),(86,170,142,153),(87,154,143,171),(88,172,144,155),(89,156,133,173),(90,174,134,145),(91,146,135,175),(92,176,136,147),(93,148,137,177),(94,178,138,149),(95,150,139,179),(96,180,140,151)], [(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)], [(1,180,19,30,7,174,13,36),(2,133,43,79,8,139,37,73),(3,178,21,28,9,172,15,34),(4,143,45,77,10,137,39,83),(5,176,23,26,11,170,17,32),(6,141,47,75,12,135,41,81),(14,122,185,95,20,128,191,89),(16,132,187,93,22,126,181,87),(18,130,189,91,24,124,183,85),(25,66,146,167,31,72,152,161),(27,64,148,165,33,70,154,159),(29,62,150,163,35,68,156,157),(38,60,186,155,44,54,192,149),(40,58,188,153,46,52,182,147),(42,56,190,151,48,50,184,145),(49,106,179,119,55,100,173,113),(51,104,169,117,57,98,175,111),(53,102,171,115,59,108,177,109),(61,134,112,129,67,140,118,123),(63,144,114,127,69,138,120,121),(65,142,116,125,71,136,110,131),(74,99,96,162,80,105,90,168),(76,97,86,160,82,103,92,166),(78,107,88,158,84,101,94,164)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 12 | ··· | 12 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | D6 | Dic3 | C3⋊D4 | C4≀C2 | C4.10D4 | C12.10D4 | Q8⋊3Dic3 |
| kernel | C42.8D6 | C42.S3 | C3×C42.C2 | C3×C4⋊C4 | C42.C2 | C2×C12 | C42 | C4⋊C4 | C2×C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 2 | 4 | 8 | 1 | 2 | 4 |
Matrix representation of C42.8D6 ►in GL6(𝔽73)
| 0 | 27 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 11 | 43 | 0 | 0 | 0 | 0 |
| 43 | 62 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 67 |
| 0 | 0 | 0 | 0 | 67 | 29 |
| 10 | 47 | 0 | 0 | 0 | 0 |
| 47 | 63 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 32 | 0 | 0 |
| 0 | 0 | 10 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 60 |
| 0 | 0 | 0 | 0 | 13 | 13 |
G:=sub<GL(6,GF(73))| [0,46,0,0,0,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[11,43,0,0,0,0,43,62,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,44,67,0,0,0,0,67,29],[10,47,0,0,0,0,47,63,0,0,0,0,0,0,51,10,0,0,0,0,32,22,0,0,0,0,0,0,13,13,0,0,0,0,60,13] >;
C42.8D6 in GAP, Magma, Sage, TeX
C_4^2._8D_6
% in TeX
G:=Group("C4^2.8D6"); // GroupNames label
G:=SmallGroup(192,102);
// by ID
G=gap.SmallGroup(192,102);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,120,219,268,1571,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b^2,d^2=a^-1*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*b*c^5>;
// generators/relations