direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×Dic12, C24.61C23, C12.57C24, C23.67D12, Dic6.21C23, C6⋊1(C2×Q16), (C2×C6)⋊6Q16, C3⋊1(C22×Q16), (C2×C8).310D6, C4.47(C2×D12), C12.292(C2×D4), (C2×C12).392D4, (C2×C4).102D12, C8.52(C22×S3), C4.54(S3×C23), (C22×C8).12S3, C6.24(C22×D4), (C22×C24).16C2, C22.72(C2×D12), (C22×C6).147D4, C2.26(C22×D12), (C22×C4).461D6, (C2×C12).788C23, (C2×C24).382C22, (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
Subgroups: 600 in 258 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×3], C4 [×8], C22 [×7], C6, C6 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×12], Q8 [×20], C23, Dic3 [×8], C12, C12 [×3], C2×C6 [×7], C2×C8 [×6], Q16 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×18], C24 [×4], Dic6 [×8], Dic6 [×12], C2×Dic3 [×12], C2×C12 [×6], C22×C6, C22×C8, C2×Q16 [×12], C22×Q8 [×2], Dic12 [×16], C2×C24 [×6], C2×Dic6 [×12], C2×Dic6 [×6], C22×Dic3 [×2], C22×C12, C22×Q16, C2×Dic12 [×12], C22×C24, C22×Dic6 [×2], C22×Dic12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], Q16 [×4], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C2×Q16 [×6], C22×D4, Dic12 [×4], C2×D12 [×6], S3×C23, C22×Q16, C2×Dic12 [×6], C22×D12, C22×Dic12
Generators and relations
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 >
►Regular action on 192 points
Generators in S192
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 97)(62 98)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 144)(82 121)(83 122)(84 123)(85 124)(86 125)(87 126)(88 127)(89 128)(90 129)(91 130)(92 131)(93 132)(94 133)(95 134)(96 135)(145 173)(146 174)(147 175)(148 176)(149 177)(150 178)(151 179)(152 180)(153 181)(154 182)(155 183)(156 184)(157 185)(158 186)(159 187)(160 188)(161 189)(162 190)(163 191)(164 192)(165 169)(166 170)(167 171)(168 172)
(1 170)(2 171)(3 172)(4 173)(5 174)(6 175)(7 176)(8 177)(9 178)(10 179)(11 180)(12 181)(13 182)(14 183)(15 184)(16 185)(17 186)(18 187)(19 188)(20 189)(21 190)(22 191)(23 192)(24 169)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 161)(32 162)(33 163)(34 164)(35 165)(36 166)(37 167)(38 168)(39 145)(40 146)(41 147)(42 148)(43 149)(44 150)(45 151)(46 152)(47 153)(48 154)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)(97 136)(98 137)(99 138)(100 139)(101 140)(102 141)(103 142)(104 143)(105 144)(106 121)(107 122)(108 123)(109 124)(110 125)(111 126)(112 127)(113 128)(114 129)(115 130)(116 131)(117 132)(118 133)(119 134)(120 135)
(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 13 59)(2 70 14 58)(3 69 15 57)(4 68 16 56)(5 67 17 55)(6 66 18 54)(7 65 19 53)(8 64 20 52)(9 63 21 51)(10 62 22 50)(11 61 23 49)(12 60 24 72)(25 118 37 106)(26 117 38 105)(27 116 39 104)(28 115 40 103)(29 114 41 102)(30 113 42 101)(31 112 43 100)(32 111 44 99)(33 110 45 98)(34 109 46 97)(35 108 47 120)(36 107 48 119)(73 192 85 180)(74 191 86 179)(75 190 87 178)(76 189 88 177)(77 188 89 176)(78 187 90 175)(79 186 91 174)(80 185 92 173)(81 184 93 172)(82 183 94 171)(83 182 95 170)(84 181 96 169)(121 155 133 167)(122 154 134 166)(123 153 135 165)(124 152 136 164)(125 151 137 163)(126 150 138 162)(127 149 139 161)(128 148 140 160)(129 147 141 159)(130 146 142 158)(131 145 143 157)(132 168 144 156)
G:=sub<Sym(192)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,121)(83,122)(84,123)(85,124)(86,125)(87,126)(88,127)(89,128)(90,129)(91,130)(92,131)(93,132)(94,133)(95,134)(96,135)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(151,179)(152,180)(153,181)(154,182)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(161,189)(162,190)(163,191)(164,192)(165,169)(166,170)(167,171)(168,172), (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,181)(13,182)(14,183)(15,184)(16,185)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,169)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,145)(40,146)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135), (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), (1,71,13,59)(2,70,14,58)(3,69,15,57)(4,68,16,56)(5,67,17,55)(6,66,18,54)(7,65,19,53)(8,64,20,52)(9,63,21,51)(10,62,22,50)(11,61,23,49)(12,60,24,72)(25,118,37,106)(26,117,38,105)(27,116,39,104)(28,115,40,103)(29,114,41,102)(30,113,42,101)(31,112,43,100)(32,111,44,99)(33,110,45,98)(34,109,46,97)(35,108,47,120)(36,107,48,119)(73,192,85,180)(74,191,86,179)(75,190,87,178)(76,189,88,177)(77,188,89,176)(78,187,90,175)(79,186,91,174)(80,185,92,173)(81,184,93,172)(82,183,94,171)(83,182,95,170)(84,181,96,169)(121,155,133,167)(122,154,134,166)(123,153,135,165)(124,152,136,164)(125,151,137,163)(126,150,138,162)(127,149,139,161)(128,148,140,160)(129,147,141,159)(130,146,142,158)(131,145,143,157)(132,168,144,156)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,35)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,97)(62,98)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,121)(83,122)(84,123)(85,124)(86,125)(87,126)(88,127)(89,128)(90,129)(91,130)(92,131)(93,132)(94,133)(95,134)(96,135)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(151,179)(152,180)(153,181)(154,182)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(161,189)(162,190)(163,191)(164,192)(165,169)(166,170)(167,171)(168,172), (1,170)(2,171)(3,172)(4,173)(5,174)(6,175)(7,176)(8,177)(9,178)(10,179)(11,180)(12,181)(13,182)(14,183)(15,184)(16,185)(17,186)(18,187)(19,188)(20,189)(21,190)(22,191)(23,192)(24,169)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,145)(40,146)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,81)(70,82)(71,83)(72,84)(97,136)(98,137)(99,138)(100,139)(101,140)(102,141)(103,142)(104,143)(105,144)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135), (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), (1,71,13,59)(2,70,14,58)(3,69,15,57)(4,68,16,56)(5,67,17,55)(6,66,18,54)(7,65,19,53)(8,64,20,52)(9,63,21,51)(10,62,22,50)(11,61,23,49)(12,60,24,72)(25,118,37,106)(26,117,38,105)(27,116,39,104)(28,115,40,103)(29,114,41,102)(30,113,42,101)(31,112,43,100)(32,111,44,99)(33,110,45,98)(34,109,46,97)(35,108,47,120)(36,107,48,119)(73,192,85,180)(74,191,86,179)(75,190,87,178)(76,189,88,177)(77,188,89,176)(78,187,90,175)(79,186,91,174)(80,185,92,173)(81,184,93,172)(82,183,94,171)(83,182,95,170)(84,181,96,169)(121,155,133,167)(122,154,134,166)(123,153,135,165)(124,152,136,164)(125,151,137,163)(126,150,138,162)(127,149,139,161)(128,148,140,160)(129,147,141,159)(130,146,142,158)(131,145,143,157)(132,168,144,156) );
G=PermutationGroup([(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(23,34),(24,35),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,97),(62,98),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,136),(74,137),(75,138),(76,139),(77,140),(78,141),(79,142),(80,143),(81,144),(82,121),(83,122),(84,123),(85,124),(86,125),(87,126),(88,127),(89,128),(90,129),(91,130),(92,131),(93,132),(94,133),(95,134),(96,135),(145,173),(146,174),(147,175),(148,176),(149,177),(150,178),(151,179),(152,180),(153,181),(154,182),(155,183),(156,184),(157,185),(158,186),(159,187),(160,188),(161,189),(162,190),(163,191),(164,192),(165,169),(166,170),(167,171),(168,172)], [(1,170),(2,171),(3,172),(4,173),(5,174),(6,175),(7,176),(8,177),(9,178),(10,179),(11,180),(12,181),(13,182),(14,183),(15,184),(16,185),(17,186),(18,187),(19,188),(20,189),(21,190),(22,191),(23,192),(24,169),(25,155),(26,156),(27,157),(28,158),(29,159),(30,160),(31,161),(32,162),(33,163),(34,164),(35,165),(36,166),(37,167),(38,168),(39,145),(40,146),(41,147),(42,148),(43,149),(44,150),(45,151),(46,152),(47,153),(48,154),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(67,79),(68,80),(69,81),(70,82),(71,83),(72,84),(97,136),(98,137),(99,138),(100,139),(101,140),(102,141),(103,142),(104,143),(105,144),(106,121),(107,122),(108,123),(109,124),(110,125),(111,126),(112,127),(113,128),(114,129),(115,130),(116,131),(117,132),(118,133),(119,134),(120,135)], [(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)], [(1,71,13,59),(2,70,14,58),(3,69,15,57),(4,68,16,56),(5,67,17,55),(6,66,18,54),(7,65,19,53),(8,64,20,52),(9,63,21,51),(10,62,22,50),(11,61,23,49),(12,60,24,72),(25,118,37,106),(26,117,38,105),(27,116,39,104),(28,115,40,103),(29,114,41,102),(30,113,42,101),(31,112,43,100),(32,111,44,99),(33,110,45,98),(34,109,46,97),(35,108,47,120),(36,107,48,119),(73,192,85,180),(74,191,86,179),(75,190,87,178),(76,189,88,177),(77,188,89,176),(78,187,90,175),(79,186,91,174),(80,185,92,173),(81,184,93,172),(82,183,94,171),(83,182,95,170),(84,181,96,169),(121,155,133,167),(122,154,134,166),(123,153,135,165),(124,152,136,164),(125,151,137,163),(126,150,138,162),(127,149,139,161),(128,148,140,160),(129,147,141,159),(130,146,142,158),(131,145,143,157),(132,168,144,156)])
Matrix representation ►G ⊆ 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] >;
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 |
In GAP, Magma, Sage, TeX
C_2^2\times 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
×
⋊
ℤ
𝔽
○
≀