metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.59D6, C3⋊Q16⋊5C4, C6.76(C4×D4), C4⋊C4.256D6, (C4×Q8).14S3, Q8.11(C4×S3), (Q8×C12).8C2, C3⋊4(Q16⋊C4), (C2×C12).260D4, (C2×Q8).187D6, C4.43(C4○D12), C12.63(C4○D4), C12.28(C22×C4), (C4×Dic6).15C2, Dic6.17(C2×C4), (C4×C12).101C22, (C2×C12).350C23, C2.4(Q8.14D6), Q8⋊2Dic3.10C2, C6.SD16.10C2, C42.S3.4C2, C12.Q8.12C2, (C6×Q8).198C22, C2.4(Q8.11D6), C6.111(C8.C22), C4⋊Dic3.333C22, (C2×Dic6).267C22, C3⋊C8.4(C2×C4), C4.28(S3×C2×C4), C2.22(C4×C3⋊D4), (C2×C6).481(C2×D4), (C3×Q8).13(C2×C4), (C2×C3⋊Q16).4C2, (C2×C3⋊C8).103C22, C22.82(C2×C3⋊D4), (C2×C4).223(C3⋊D4), (C3×C4⋊C4).287C22, (C2×C4).450(C22×S3), SmallGroup(192,589)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.59D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=b-1c5 >
Subgroups: 232 in 108 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, C3⋊C8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C4×Q8, C2×Q16, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3⋊Q16, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, Q16⋊C4, C42.S3, C12.Q8, C6.SD16, Q8⋊2Dic3, C4×Dic6, C2×C3⋊Q16, Q8×C12, C42.59D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, C8.C22, S3×C2×C4, C4○D12, C2×C3⋊D4, Q16⋊C4, C4×C3⋊D4, Q8.11D6, Q8.14D6, C42.59D6
►Regular action on 192 points
Generators in S192
(1 90 39 21)(2 85 40 16)(3 92 41 23)(4 87 42 18)(5 94 43 13)(6 89 44 20)(7 96 45 15)(8 91 46 22)(9 86 47 17)(10 93 48 24)(11 88 37 19)(12 95 38 14)(25 54 168 70)(26 49 157 65)(27 56 158 72)(28 51 159 67)(29 58 160 62)(30 53 161 69)(31 60 162 64)(32 55 163 71)(33 50 164 66)(34 57 165 61)(35 52 166 68)(36 59 167 63)(73 138 171 123)(74 133 172 130)(75 140 173 125)(76 135 174 132)(77 142 175 127)(78 137 176 122)(79 144 177 129)(80 139 178 124)(81 134 179 131)(82 141 180 126)(83 136 169 121)(84 143 170 128)(97 111 146 184)(98 118 147 191)(99 113 148 186)(100 120 149 181)(101 115 150 188)(102 110 151 183)(103 117 152 190)(104 112 153 185)(105 119 154 192)(106 114 155 187)(107 109 156 182)(108 116 145 189)
(1 182 7 188)(2 189 8 183)(3 184 9 190)(4 191 10 185)(5 186 11 192)(6 181 12 187)(13 148 19 154)(14 155 20 149)(15 150 21 156)(16 145 22 151)(17 152 23 146)(18 147 24 153)(25 177 31 171)(26 172 32 178)(27 179 33 173)(28 174 34 180)(29 169 35 175)(30 176 36 170)(37 119 43 113)(38 114 44 120)(39 109 45 115)(40 116 46 110)(41 111 47 117)(42 118 48 112)(49 130 55 124)(50 125 56 131)(51 132 57 126)(52 127 58 121)(53 122 59 128)(54 129 60 123)(61 141 67 135)(62 136 68 142)(63 143 69 137)(64 138 70 144)(65 133 71 139)(66 140 72 134)(73 168 79 162)(74 163 80 157)(75 158 81 164)(76 165 82 159)(77 160 83 166)(78 167 84 161)(85 108 91 102)(86 103 92 97)(87 98 93 104)(88 105 94 99)(89 100 95 106)(90 107 96 101)
(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)
(1 164 188 81 7 158 182 75)(2 80 183 163 8 74 189 157)(3 162 190 79 9 168 184 73)(4 78 185 161 10 84 191 167)(5 160 192 77 11 166 186 83)(6 76 187 159 12 82 181 165)(13 52 154 121 19 58 148 127)(14 132 149 51 20 126 155 57)(15 50 156 131 21 56 150 125)(16 130 151 49 22 124 145 55)(17 60 146 129 23 54 152 123)(18 128 153 59 24 122 147 53)(25 111 171 41 31 117 177 47)(26 40 178 110 32 46 172 116)(27 109 173 39 33 115 179 45)(28 38 180 120 34 44 174 114)(29 119 175 37 35 113 169 43)(30 48 170 118 36 42 176 112)(61 95 135 100 67 89 141 106)(62 99 142 94 68 105 136 88)(63 93 137 98 69 87 143 104)(64 97 144 92 70 103 138 86)(65 91 139 108 71 85 133 102)(66 107 134 90 72 101 140 96)
G:=sub<Sym(192)| (1,90,39,21)(2,85,40,16)(3,92,41,23)(4,87,42,18)(5,94,43,13)(6,89,44,20)(7,96,45,15)(8,91,46,22)(9,86,47,17)(10,93,48,24)(11,88,37,19)(12,95,38,14)(25,54,168,70)(26,49,157,65)(27,56,158,72)(28,51,159,67)(29,58,160,62)(30,53,161,69)(31,60,162,64)(32,55,163,71)(33,50,164,66)(34,57,165,61)(35,52,166,68)(36,59,167,63)(73,138,171,123)(74,133,172,130)(75,140,173,125)(76,135,174,132)(77,142,175,127)(78,137,176,122)(79,144,177,129)(80,139,178,124)(81,134,179,131)(82,141,180,126)(83,136,169,121)(84,143,170,128)(97,111,146,184)(98,118,147,191)(99,113,148,186)(100,120,149,181)(101,115,150,188)(102,110,151,183)(103,117,152,190)(104,112,153,185)(105,119,154,192)(106,114,155,187)(107,109,156,182)(108,116,145,189), (1,182,7,188)(2,189,8,183)(3,184,9,190)(4,191,10,185)(5,186,11,192)(6,181,12,187)(13,148,19,154)(14,155,20,149)(15,150,21,156)(16,145,22,151)(17,152,23,146)(18,147,24,153)(25,177,31,171)(26,172,32,178)(27,179,33,173)(28,174,34,180)(29,169,35,175)(30,176,36,170)(37,119,43,113)(38,114,44,120)(39,109,45,115)(40,116,46,110)(41,111,47,117)(42,118,48,112)(49,130,55,124)(50,125,56,131)(51,132,57,126)(52,127,58,121)(53,122,59,128)(54,129,60,123)(61,141,67,135)(62,136,68,142)(63,143,69,137)(64,138,70,144)(65,133,71,139)(66,140,72,134)(73,168,79,162)(74,163,80,157)(75,158,81,164)(76,165,82,159)(77,160,83,166)(78,167,84,161)(85,108,91,102)(86,103,92,97)(87,98,93,104)(88,105,94,99)(89,100,95,106)(90,107,96,101), (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), (1,164,188,81,7,158,182,75)(2,80,183,163,8,74,189,157)(3,162,190,79,9,168,184,73)(4,78,185,161,10,84,191,167)(5,160,192,77,11,166,186,83)(6,76,187,159,12,82,181,165)(13,52,154,121,19,58,148,127)(14,132,149,51,20,126,155,57)(15,50,156,131,21,56,150,125)(16,130,151,49,22,124,145,55)(17,60,146,129,23,54,152,123)(18,128,153,59,24,122,147,53)(25,111,171,41,31,117,177,47)(26,40,178,110,32,46,172,116)(27,109,173,39,33,115,179,45)(28,38,180,120,34,44,174,114)(29,119,175,37,35,113,169,43)(30,48,170,118,36,42,176,112)(61,95,135,100,67,89,141,106)(62,99,142,94,68,105,136,88)(63,93,137,98,69,87,143,104)(64,97,144,92,70,103,138,86)(65,91,139,108,71,85,133,102)(66,107,134,90,72,101,140,96)>;
G:=Group( (1,90,39,21)(2,85,40,16)(3,92,41,23)(4,87,42,18)(5,94,43,13)(6,89,44,20)(7,96,45,15)(8,91,46,22)(9,86,47,17)(10,93,48,24)(11,88,37,19)(12,95,38,14)(25,54,168,70)(26,49,157,65)(27,56,158,72)(28,51,159,67)(29,58,160,62)(30,53,161,69)(31,60,162,64)(32,55,163,71)(33,50,164,66)(34,57,165,61)(35,52,166,68)(36,59,167,63)(73,138,171,123)(74,133,172,130)(75,140,173,125)(76,135,174,132)(77,142,175,127)(78,137,176,122)(79,144,177,129)(80,139,178,124)(81,134,179,131)(82,141,180,126)(83,136,169,121)(84,143,170,128)(97,111,146,184)(98,118,147,191)(99,113,148,186)(100,120,149,181)(101,115,150,188)(102,110,151,183)(103,117,152,190)(104,112,153,185)(105,119,154,192)(106,114,155,187)(107,109,156,182)(108,116,145,189), (1,182,7,188)(2,189,8,183)(3,184,9,190)(4,191,10,185)(5,186,11,192)(6,181,12,187)(13,148,19,154)(14,155,20,149)(15,150,21,156)(16,145,22,151)(17,152,23,146)(18,147,24,153)(25,177,31,171)(26,172,32,178)(27,179,33,173)(28,174,34,180)(29,169,35,175)(30,176,36,170)(37,119,43,113)(38,114,44,120)(39,109,45,115)(40,116,46,110)(41,111,47,117)(42,118,48,112)(49,130,55,124)(50,125,56,131)(51,132,57,126)(52,127,58,121)(53,122,59,128)(54,129,60,123)(61,141,67,135)(62,136,68,142)(63,143,69,137)(64,138,70,144)(65,133,71,139)(66,140,72,134)(73,168,79,162)(74,163,80,157)(75,158,81,164)(76,165,82,159)(77,160,83,166)(78,167,84,161)(85,108,91,102)(86,103,92,97)(87,98,93,104)(88,105,94,99)(89,100,95,106)(90,107,96,101), (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), (1,164,188,81,7,158,182,75)(2,80,183,163,8,74,189,157)(3,162,190,79,9,168,184,73)(4,78,185,161,10,84,191,167)(5,160,192,77,11,166,186,83)(6,76,187,159,12,82,181,165)(13,52,154,121,19,58,148,127)(14,132,149,51,20,126,155,57)(15,50,156,131,21,56,150,125)(16,130,151,49,22,124,145,55)(17,60,146,129,23,54,152,123)(18,128,153,59,24,122,147,53)(25,111,171,41,31,117,177,47)(26,40,178,110,32,46,172,116)(27,109,173,39,33,115,179,45)(28,38,180,120,34,44,174,114)(29,119,175,37,35,113,169,43)(30,48,170,118,36,42,176,112)(61,95,135,100,67,89,141,106)(62,99,142,94,68,105,136,88)(63,93,137,98,69,87,143,104)(64,97,144,92,70,103,138,86)(65,91,139,108,71,85,133,102)(66,107,134,90,72,101,140,96) );
G=PermutationGroup([[(1,90,39,21),(2,85,40,16),(3,92,41,23),(4,87,42,18),(5,94,43,13),(6,89,44,20),(7,96,45,15),(8,91,46,22),(9,86,47,17),(10,93,48,24),(11,88,37,19),(12,95,38,14),(25,54,168,70),(26,49,157,65),(27,56,158,72),(28,51,159,67),(29,58,160,62),(30,53,161,69),(31,60,162,64),(32,55,163,71),(33,50,164,66),(34,57,165,61),(35,52,166,68),(36,59,167,63),(73,138,171,123),(74,133,172,130),(75,140,173,125),(76,135,174,132),(77,142,175,127),(78,137,176,122),(79,144,177,129),(80,139,178,124),(81,134,179,131),(82,141,180,126),(83,136,169,121),(84,143,170,128),(97,111,146,184),(98,118,147,191),(99,113,148,186),(100,120,149,181),(101,115,150,188),(102,110,151,183),(103,117,152,190),(104,112,153,185),(105,119,154,192),(106,114,155,187),(107,109,156,182),(108,116,145,189)], [(1,182,7,188),(2,189,8,183),(3,184,9,190),(4,191,10,185),(5,186,11,192),(6,181,12,187),(13,148,19,154),(14,155,20,149),(15,150,21,156),(16,145,22,151),(17,152,23,146),(18,147,24,153),(25,177,31,171),(26,172,32,178),(27,179,33,173),(28,174,34,180),(29,169,35,175),(30,176,36,170),(37,119,43,113),(38,114,44,120),(39,109,45,115),(40,116,46,110),(41,111,47,117),(42,118,48,112),(49,130,55,124),(50,125,56,131),(51,132,57,126),(52,127,58,121),(53,122,59,128),(54,129,60,123),(61,141,67,135),(62,136,68,142),(63,143,69,137),(64,138,70,144),(65,133,71,139),(66,140,72,134),(73,168,79,162),(74,163,80,157),(75,158,81,164),(76,165,82,159),(77,160,83,166),(78,167,84,161),(85,108,91,102),(86,103,92,97),(87,98,93,104),(88,105,94,99),(89,100,95,106),(90,107,96,101)], [(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)], [(1,164,188,81,7,158,182,75),(2,80,183,163,8,74,189,157),(3,162,190,79,9,168,184,73),(4,78,185,161,10,84,191,167),(5,160,192,77,11,166,186,83),(6,76,187,159,12,82,181,165),(13,52,154,121,19,58,148,127),(14,132,149,51,20,126,155,57),(15,50,156,131,21,56,150,125),(16,130,151,49,22,124,145,55),(17,60,146,129,23,54,152,123),(18,128,153,59,24,122,147,53),(25,111,171,41,31,117,177,47),(26,40,178,110,32,46,172,116),(27,109,173,39,33,115,179,45),(28,38,180,120,34,44,174,114),(29,119,175,37,35,113,169,43),(30,48,170,118,36,42,176,112),(61,95,135,100,67,89,141,106),(62,99,142,94,68,105,136,88),(63,93,137,98,69,87,143,104),(64,97,144,92,70,103,138,86),(65,91,139,108,71,85,133,102),(66,107,134,90,72,101,140,96)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C4×S3 | C4○D12 | C8.C22 | Q8.11D6 | Q8.14D6 |
| kernel | C42.59D6 | C42.S3 | C12.Q8 | C6.SD16 | Q8⋊2Dic3 | C4×Dic6 | C2×C3⋊Q16 | Q8×C12 | C3⋊Q16 | C4×Q8 | C2×C12 | C42 | C4⋊C4 | C2×Q8 | C12 | C2×C4 | Q8 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 2 |
Matrix representation of C42.59D6 ►in GL6(𝔽73)
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 13 | 60 | 13 |
| 0 | 0 | 60 | 30 | 0 | 60 |
| 0 | 0 | 17 | 17 | 43 | 60 |
| 0 | 0 | 0 | 17 | 13 | 30 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 25 | 25 |
| 0 | 0 | 0 | 1 | 23 | 25 |
| 0 | 0 | 72 | 1 | 72 | 0 |
| 0 | 0 | 71 | 72 | 0 | 72 |
| 30 | 43 | 0 | 0 | 0 | 0 |
| 30 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 13 | 1 | 0 |
| 0 | 0 | 60 | 1 | 72 | 1 |
| 0 | 0 | 46 | 23 | 72 | 60 |
| 0 | 0 | 23 | 46 | 13 | 59 |
| 63 | 51 | 0 | 0 | 0 | 0 |
| 61 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 46 | 56 | 36 |
| 0 | 0 | 68 | 22 | 36 | 54 |
| 0 | 0 | 8 | 19 | 7 | 69 |
| 0 | 0 | 19 | 11 | 3 | 66 |
G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,43,60,17,0,0,0,13,30,17,17,0,0,60,0,43,13,0,0,13,60,60,30],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,72,71,0,0,0,1,1,72,0,0,25,23,72,0,0,0,25,25,0,72],[30,30,0,0,0,0,43,60,0,0,0,0,0,0,14,60,46,23,0,0,13,1,23,46,0,0,1,72,72,13,0,0,0,1,60,59],[63,61,0,0,0,0,51,10,0,0,0,0,0,0,51,68,8,19,0,0,46,22,19,11,0,0,56,36,7,3,0,0,36,54,69,66] >;
C42.59D6 in GAP, Magma, Sage, TeX
C_4^2._{59}D_6 % in TeX
G:=Group("C4^2.59D6"); // GroupNames label
G:=SmallGroup(192,589);
// by ID
G=gap.SmallGroup(192,589);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,232,387,58,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations