direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×Dic12, C12.57C24, C24.61C23, C23.67D12, Dic6.21C23, C6⋊1(C2×Q16), (C2×C6)⋊6Q16, C3⋊1(C22×Q16), (C2×C8).310D6, C4.47(C2×D12), (C2×C12).392D4, C12.292(C2×D4), (C2×C4).102D12, C8.52(C22×S3), C4.54(S3×C23), (C22×C8).12S3, C6.24(C22×D4), (C22×C24).16C2, (C22×C4).461D6, C2.26(C22×D12), C22.72(C2×D12), (C22×C6).147D4, (C2×C24).382C22, (C2×C12).788C23, (C22×Dic6).9C2, (C22×C12).527C22, (C2×Dic6).258C22, (C2×C6).180(C2×D4), (C2×C4).738(C22×S3), SmallGroup(192,1301)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×Dic12
G = < a,b,c,d | a2=b2=c24=1, d2=c12, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 600 in 258 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C24, Dic6, Dic6, C2×Dic3, C2×C12, C22×C6, C22×C8, C2×Q16, C22×Q8, Dic12, C2×C24, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C22×Q16, C2×Dic12, C22×C24, C22×Dic6, C22×Dic12
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C24, D12, C22×S3, C2×Q16, C22×D4, Dic12, C2×D12, S3×C23, C22×Q16, C2×Dic12, C22×D12, C22×Dic12
►Regular action on 192 points
Generators in S192
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(73 191)(74 192)(75 169)(76 170)(77 171)(78 172)(79 173)(80 174)(81 175)(82 176)(83 177)(84 178)(85 179)(86 180)(87 181)(88 182)(89 183)(90 184)(91 185)(92 186)(93 187)(94 188)(95 189)(96 190)(121 166)(122 167)(123 168)(124 145)(125 146)(126 147)(127 148)(128 149)(129 150)(130 151)(131 152)(132 153)(133 154)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 161)(141 162)(142 163)(143 164)(144 165)
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 117)(26 118)(27 119)(28 120)(29 97)(30 98)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(73 130)(74 131)(75 132)(76 133)(77 134)(78 135)(79 136)(80 137)(81 138)(82 139)(83 140)(84 141)(85 142)(86 143)(87 144)(88 121)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 129)(145 185)(146 186)(147 187)(148 188)(149 189)(150 190)(151 191)(152 192)(153 169)(154 170)(155 171)(156 172)(157 173)(158 174)(159 175)(160 176)(161 177)(162 178)(163 179)(164 180)(165 181)(166 182)(167 183)(168 184)
(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)
(1 122 13 134)(2 121 14 133)(3 144 15 132)(4 143 16 131)(5 142 17 130)(6 141 18 129)(7 140 19 128)(8 139 20 127)(9 138 21 126)(10 137 22 125)(11 136 23 124)(12 135 24 123)(25 177 37 189)(26 176 38 188)(27 175 39 187)(28 174 40 186)(29 173 41 185)(30 172 42 184)(31 171 43 183)(32 170 44 182)(33 169 45 181)(34 192 46 180)(35 191 47 179)(36 190 48 178)(49 74 61 86)(50 73 62 85)(51 96 63 84)(52 95 64 83)(53 94 65 82)(54 93 66 81)(55 92 67 80)(56 91 68 79)(57 90 69 78)(58 89 70 77)(59 88 71 76)(60 87 72 75)(97 157 109 145)(98 156 110 168)(99 155 111 167)(100 154 112 166)(101 153 113 165)(102 152 114 164)(103 151 115 163)(104 150 116 162)(105 149 117 161)(106 148 118 160)(107 147 119 159)(108 146 120 158)
G:=sub<Sym(192)| (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(73,191)(74,192)(75,169)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,177)(84,178)(85,179)(86,180)(87,181)(88,182)(89,183)(90,184)(91,185)(92,186)(93,187)(94,188)(95,189)(96,190)(121,166)(122,167)(123,168)(124,145)(125,146)(126,147)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,117)(26,118)(27,119)(28,120)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,136)(80,137)(81,138)(82,139)(83,140)(84,141)(85,142)(86,143)(87,144)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(145,185)(146,186)(147,187)(148,188)(149,189)(150,190)(151,191)(152,192)(153,169)(154,170)(155,171)(156,172)(157,173)(158,174)(159,175)(160,176)(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,183)(168,184), (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), (1,122,13,134)(2,121,14,133)(3,144,15,132)(4,143,16,131)(5,142,17,130)(6,141,18,129)(7,140,19,128)(8,139,20,127)(9,138,21,126)(10,137,22,125)(11,136,23,124)(12,135,24,123)(25,177,37,189)(26,176,38,188)(27,175,39,187)(28,174,40,186)(29,173,41,185)(30,172,42,184)(31,171,43,183)(32,170,44,182)(33,169,45,181)(34,192,46,180)(35,191,47,179)(36,190,48,178)(49,74,61,86)(50,73,62,85)(51,96,63,84)(52,95,64,83)(53,94,65,82)(54,93,66,81)(55,92,67,80)(56,91,68,79)(57,90,69,78)(58,89,70,77)(59,88,71,76)(60,87,72,75)(97,157,109,145)(98,156,110,168)(99,155,111,167)(100,154,112,166)(101,153,113,165)(102,152,114,164)(103,151,115,163)(104,150,116,162)(105,149,117,161)(106,148,118,160)(107,147,119,159)(108,146,120,158)>;
G:=Group( (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(73,191)(74,192)(75,169)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,177)(84,178)(85,179)(86,180)(87,181)(88,182)(89,183)(90,184)(91,185)(92,186)(93,187)(94,188)(95,189)(96,190)(121,166)(122,167)(123,168)(124,145)(125,146)(126,147)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,117)(26,118)(27,119)(28,120)(29,97)(30,98)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,136)(80,137)(81,138)(82,139)(83,140)(84,141)(85,142)(86,143)(87,144)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(145,185)(146,186)(147,187)(148,188)(149,189)(150,190)(151,191)(152,192)(153,169)(154,170)(155,171)(156,172)(157,173)(158,174)(159,175)(160,176)(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,183)(168,184), (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), (1,122,13,134)(2,121,14,133)(3,144,15,132)(4,143,16,131)(5,142,17,130)(6,141,18,129)(7,140,19,128)(8,139,20,127)(9,138,21,126)(10,137,22,125)(11,136,23,124)(12,135,24,123)(25,177,37,189)(26,176,38,188)(27,175,39,187)(28,174,40,186)(29,173,41,185)(30,172,42,184)(31,171,43,183)(32,170,44,182)(33,169,45,181)(34,192,46,180)(35,191,47,179)(36,190,48,178)(49,74,61,86)(50,73,62,85)(51,96,63,84)(52,95,64,83)(53,94,65,82)(54,93,66,81)(55,92,67,80)(56,91,68,79)(57,90,69,78)(58,89,70,77)(59,88,71,76)(60,87,72,75)(97,157,109,145)(98,156,110,168)(99,155,111,167)(100,154,112,166)(101,153,113,165)(102,152,114,164)(103,151,115,163)(104,150,116,162)(105,149,117,161)(106,148,118,160)(107,147,119,159)(108,146,120,158) );
G=PermutationGroup([[(1,111),(2,112),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,119),(10,120),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63),(73,191),(74,192),(75,169),(76,170),(77,171),(78,172),(79,173),(80,174),(81,175),(82,176),(83,177),(84,178),(85,179),(86,180),(87,181),(88,182),(89,183),(90,184),(91,185),(92,186),(93,187),(94,188),(95,189),(96,190),(121,166),(122,167),(123,168),(124,145),(125,146),(126,147),(127,148),(128,149),(129,150),(130,151),(131,152),(132,153),(133,154),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,161),(141,162),(142,163),(143,164),(144,165)], [(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,65),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,57),(25,117),(26,118),(27,119),(28,120),(29,97),(30,98),(31,99),(32,100),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(73,130),(74,131),(75,132),(76,133),(77,134),(78,135),(79,136),(80,137),(81,138),(82,139),(83,140),(84,141),(85,142),(86,143),(87,144),(88,121),(89,122),(90,123),(91,124),(92,125),(93,126),(94,127),(95,128),(96,129),(145,185),(146,186),(147,187),(148,188),(149,189),(150,190),(151,191),(152,192),(153,169),(154,170),(155,171),(156,172),(157,173),(158,174),(159,175),(160,176),(161,177),(162,178),(163,179),(164,180),(165,181),(166,182),(167,183),(168,184)], [(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)], [(1,122,13,134),(2,121,14,133),(3,144,15,132),(4,143,16,131),(5,142,17,130),(6,141,18,129),(7,140,19,128),(8,139,20,127),(9,138,21,126),(10,137,22,125),(11,136,23,124),(12,135,24,123),(25,177,37,189),(26,176,38,188),(27,175,39,187),(28,174,40,186),(29,173,41,185),(30,172,42,184),(31,171,43,183),(32,170,44,182),(33,169,45,181),(34,192,46,180),(35,191,47,179),(36,190,48,178),(49,74,61,86),(50,73,62,85),(51,96,63,84),(52,95,64,83),(53,94,65,82),(54,93,66,81),(55,92,67,80),(56,91,68,79),(57,90,69,78),(58,89,70,77),(59,88,71,76),(60,87,72,75),(97,157,109,145),(98,156,110,168),(99,155,111,167),(100,154,112,166),(101,153,113,165),(102,152,114,164),(103,151,115,163),(104,150,116,162),(105,149,117,161),(106,148,118,160),(107,147,119,159),(108,146,120,158)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | Q16 | D12 | D12 | Dic12 |
| kernel | C22×Dic12 | C2×Dic12 | C22×C24 | C22×Dic6 | C22×C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 2 | 1 | 3 | 1 | 6 | 1 | 8 | 6 | 2 | 16 |
Matrix representation of C22×Dic12 ►in GL5(𝔽73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 16 | 57 | 0 | 0 |
| 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 66 | 14 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 67 | 6 | 0 | 0 |
| 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 54 | 48 |
| 0 | 0 | 0 | 29 | 19 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,16,16,0,0,0,57,16,0,0,0,0,0,7,66,0,0,0,7,14],[1,0,0,0,0,0,67,6,0,0,0,6,6,0,0,0,0,0,54,29,0,0,0,48,19] >;
C22×Dic12 in GAP, Magma, Sage, TeX
C_2^2\times {\rm Dic}_{12} % in TeX
G:=Group("C2^2xDic12"); // GroupNames label
G:=SmallGroup(192,1301);
// by ID
G=gap.SmallGroup(192,1301);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,675,192,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^24=1,d^2=c^12,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations