G = C3⋊(C42⋊5C4) order 192 = 26·3
The semidirect product of C3 and C42⋊5C4 acting via C42⋊5C4/C2.C42=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊1(C42⋊5C4), (C4×Dic3)⋊13C4, (C22×C4).24D6, C6.2(C42⋊C2), C6.6(C42⋊2C2), C2.6(C42⋊2S3), C2.C42.3S3, C6.C42.2C2, (C22×C12).4C22, C22.28(C4○D12), (C22×C6).276C23, C23.258(C22×S3), C2.1(C23.8D6), C22.30(D4⋊2S3), C22.13(Q8⋊3S3), C2.6(C23.16D6), (C22×Dic3).172C22, C22.83(S3×C2×C4), (C2×C4).123(C4×S3), C2.1(C4⋊C4⋊S3), C2.5(C4⋊C4⋊7S3), (C2×C4×Dic3).24C2, (C2×C12).140(C2×C4), (C2×C6).42(C22×C4), (C2×C6).124(C4○D4), (C2×Dic3).76(C2×C4), (C3×C2.C42).22C2, SmallGroup(192,210)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊(C42⋊5C4)
G = < a,b,c,d | a3=b4=c4=d4=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c-1 >
Subgroups: 320 in 138 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C42, C4×Dic3, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C42⋊5C4, C6.C42, C6.C42, C3×C2.C42, C2×C4×Dic3, C3⋊(C42⋊5C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, C42⋊2C2, S3×C2×C4, C4○D12, D4⋊2S3, Q8⋊3S3, C42⋊5C4, C42⋊2S3, C23.16D6, C23.8D6, C4⋊C4⋊7S3, C4⋊C4⋊S3, C3⋊(C42⋊5C4)
►Regular action on 192 points
Generators in S192
(1 182 110)(2 111 183)(3 184 112)(4 109 181)(5 155 80)(6 77 156)(7 153 78)(8 79 154)(9 61 158)(10 159 62)(11 63 160)(12 157 64)(13 127 147)(14 148 128)(15 125 145)(16 146 126)(17 143 92)(18 89 144)(19 141 90)(20 91 142)(21 88 120)(22 117 85)(23 86 118)(24 119 87)(25 36 116)(26 113 33)(27 34 114)(28 115 35)(29 53 187)(30 188 54)(31 55 185)(32 186 56)(37 48 104)(38 101 45)(39 46 102)(40 103 47)(41 140 175)(42 176 137)(43 138 173)(44 174 139)(49 167 65)(50 66 168)(51 165 67)(52 68 166)(57 106 190)(58 191 107)(59 108 192)(60 189 105)(69 98 178)(70 179 99)(71 100 180)(72 177 97)(73 169 129)(74 130 170)(75 171 131)(76 132 172)(81 133 163)(82 164 134)(83 135 161)(84 162 136)(93 121 151)(94 152 122)(95 123 149)(96 150 124)
(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)
(1 121 31 143)(2 122 32 144)(3 123 29 141)(4 124 30 142)(5 67 81 158)(6 68 82 159)(7 65 83 160)(8 66 84 157)(9 155 51 133)(10 156 52 134)(11 153 49 135)(12 154 50 136)(13 100 87 46)(14 97 88 47)(15 98 85 48)(16 99 86 45)(17 110 93 185)(18 111 94 186)(19 112 95 187)(20 109 96 188)(21 103 128 177)(22 104 125 178)(23 101 126 179)(24 102 127 180)(25 42 57 129)(26 43 58 130)(27 44 59 131)(28 41 60 132)(33 173 107 74)(34 174 108 75)(35 175 105 76)(36 176 106 73)(37 145 69 117)(38 146 70 118)(39 147 71 119)(40 148 72 120)(53 90 184 149)(54 91 181 150)(55 92 182 151)(56 89 183 152)(61 80 165 163)(62 77 166 164)(63 78 167 161)(64 79 168 162)(113 138 191 170)(114 139 192 171)(115 140 189 172)(116 137 190 169)
(1 58 70 51)(2 27 71 10)(3 60 72 49)(4 25 69 12)(5 95 173 88)(6 20 174 15)(7 93 175 86)(8 18 176 13)(9 31 26 38)(11 29 28 40)(14 81 19 74)(16 83 17 76)(21 80 149 138)(22 164 150 171)(23 78 151 140)(24 162 152 169)(30 57 37 50)(32 59 39 52)(33 45 158 185)(34 100 159 111)(35 47 160 187)(36 98 157 109)(41 118 153 121)(42 147 154 144)(43 120 155 123)(44 145 156 142)(46 68 186 108)(48 66 188 106)(53 115 103 63)(54 190 104 168)(55 113 101 61)(56 192 102 166)(62 183 114 180)(64 181 116 178)(65 112 105 97)(67 110 107 99)(73 87 84 94)(75 85 82 96)(77 91 139 125)(79 89 137 127)(90 170 128 163)(92 172 126 161)(117 134 124 131)(119 136 122 129)(130 148 133 141)(132 146 135 143)(165 182 191 179)(167 184 189 177)
G:=sub<Sym(192)| (1,182,110)(2,111,183)(3,184,112)(4,109,181)(5,155,80)(6,77,156)(7,153,78)(8,79,154)(9,61,158)(10,159,62)(11,63,160)(12,157,64)(13,127,147)(14,148,128)(15,125,145)(16,146,126)(17,143,92)(18,89,144)(19,141,90)(20,91,142)(21,88,120)(22,117,85)(23,86,118)(24,119,87)(25,36,116)(26,113,33)(27,34,114)(28,115,35)(29,53,187)(30,188,54)(31,55,185)(32,186,56)(37,48,104)(38,101,45)(39,46,102)(40,103,47)(41,140,175)(42,176,137)(43,138,173)(44,174,139)(49,167,65)(50,66,168)(51,165,67)(52,68,166)(57,106,190)(58,191,107)(59,108,192)(60,189,105)(69,98,178)(70,179,99)(71,100,180)(72,177,97)(73,169,129)(74,130,170)(75,171,131)(76,132,172)(81,133,163)(82,164,134)(83,135,161)(84,162,136)(93,121,151)(94,152,122)(95,123,149)(96,150,124), (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), (1,121,31,143)(2,122,32,144)(3,123,29,141)(4,124,30,142)(5,67,81,158)(6,68,82,159)(7,65,83,160)(8,66,84,157)(9,155,51,133)(10,156,52,134)(11,153,49,135)(12,154,50,136)(13,100,87,46)(14,97,88,47)(15,98,85,48)(16,99,86,45)(17,110,93,185)(18,111,94,186)(19,112,95,187)(20,109,96,188)(21,103,128,177)(22,104,125,178)(23,101,126,179)(24,102,127,180)(25,42,57,129)(26,43,58,130)(27,44,59,131)(28,41,60,132)(33,173,107,74)(34,174,108,75)(35,175,105,76)(36,176,106,73)(37,145,69,117)(38,146,70,118)(39,147,71,119)(40,148,72,120)(53,90,184,149)(54,91,181,150)(55,92,182,151)(56,89,183,152)(61,80,165,163)(62,77,166,164)(63,78,167,161)(64,79,168,162)(113,138,191,170)(114,139,192,171)(115,140,189,172)(116,137,190,169), (1,58,70,51)(2,27,71,10)(3,60,72,49)(4,25,69,12)(5,95,173,88)(6,20,174,15)(7,93,175,86)(8,18,176,13)(9,31,26,38)(11,29,28,40)(14,81,19,74)(16,83,17,76)(21,80,149,138)(22,164,150,171)(23,78,151,140)(24,162,152,169)(30,57,37,50)(32,59,39,52)(33,45,158,185)(34,100,159,111)(35,47,160,187)(36,98,157,109)(41,118,153,121)(42,147,154,144)(43,120,155,123)(44,145,156,142)(46,68,186,108)(48,66,188,106)(53,115,103,63)(54,190,104,168)(55,113,101,61)(56,192,102,166)(62,183,114,180)(64,181,116,178)(65,112,105,97)(67,110,107,99)(73,87,84,94)(75,85,82,96)(77,91,139,125)(79,89,137,127)(90,170,128,163)(92,172,126,161)(117,134,124,131)(119,136,122,129)(130,148,133,141)(132,146,135,143)(165,182,191,179)(167,184,189,177)>;
G:=Group( (1,182,110)(2,111,183)(3,184,112)(4,109,181)(5,155,80)(6,77,156)(7,153,78)(8,79,154)(9,61,158)(10,159,62)(11,63,160)(12,157,64)(13,127,147)(14,148,128)(15,125,145)(16,146,126)(17,143,92)(18,89,144)(19,141,90)(20,91,142)(21,88,120)(22,117,85)(23,86,118)(24,119,87)(25,36,116)(26,113,33)(27,34,114)(28,115,35)(29,53,187)(30,188,54)(31,55,185)(32,186,56)(37,48,104)(38,101,45)(39,46,102)(40,103,47)(41,140,175)(42,176,137)(43,138,173)(44,174,139)(49,167,65)(50,66,168)(51,165,67)(52,68,166)(57,106,190)(58,191,107)(59,108,192)(60,189,105)(69,98,178)(70,179,99)(71,100,180)(72,177,97)(73,169,129)(74,130,170)(75,171,131)(76,132,172)(81,133,163)(82,164,134)(83,135,161)(84,162,136)(93,121,151)(94,152,122)(95,123,149)(96,150,124), (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), (1,121,31,143)(2,122,32,144)(3,123,29,141)(4,124,30,142)(5,67,81,158)(6,68,82,159)(7,65,83,160)(8,66,84,157)(9,155,51,133)(10,156,52,134)(11,153,49,135)(12,154,50,136)(13,100,87,46)(14,97,88,47)(15,98,85,48)(16,99,86,45)(17,110,93,185)(18,111,94,186)(19,112,95,187)(20,109,96,188)(21,103,128,177)(22,104,125,178)(23,101,126,179)(24,102,127,180)(25,42,57,129)(26,43,58,130)(27,44,59,131)(28,41,60,132)(33,173,107,74)(34,174,108,75)(35,175,105,76)(36,176,106,73)(37,145,69,117)(38,146,70,118)(39,147,71,119)(40,148,72,120)(53,90,184,149)(54,91,181,150)(55,92,182,151)(56,89,183,152)(61,80,165,163)(62,77,166,164)(63,78,167,161)(64,79,168,162)(113,138,191,170)(114,139,192,171)(115,140,189,172)(116,137,190,169), (1,58,70,51)(2,27,71,10)(3,60,72,49)(4,25,69,12)(5,95,173,88)(6,20,174,15)(7,93,175,86)(8,18,176,13)(9,31,26,38)(11,29,28,40)(14,81,19,74)(16,83,17,76)(21,80,149,138)(22,164,150,171)(23,78,151,140)(24,162,152,169)(30,57,37,50)(32,59,39,52)(33,45,158,185)(34,100,159,111)(35,47,160,187)(36,98,157,109)(41,118,153,121)(42,147,154,144)(43,120,155,123)(44,145,156,142)(46,68,186,108)(48,66,188,106)(53,115,103,63)(54,190,104,168)(55,113,101,61)(56,192,102,166)(62,183,114,180)(64,181,116,178)(65,112,105,97)(67,110,107,99)(73,87,84,94)(75,85,82,96)(77,91,139,125)(79,89,137,127)(90,170,128,163)(92,172,126,161)(117,134,124,131)(119,136,122,129)(130,148,133,141)(132,146,135,143)(165,182,191,179)(167,184,189,177) );
G=PermutationGroup([[(1,182,110),(2,111,183),(3,184,112),(4,109,181),(5,155,80),(6,77,156),(7,153,78),(8,79,154),(9,61,158),(10,159,62),(11,63,160),(12,157,64),(13,127,147),(14,148,128),(15,125,145),(16,146,126),(17,143,92),(18,89,144),(19,141,90),(20,91,142),(21,88,120),(22,117,85),(23,86,118),(24,119,87),(25,36,116),(26,113,33),(27,34,114),(28,115,35),(29,53,187),(30,188,54),(31,55,185),(32,186,56),(37,48,104),(38,101,45),(39,46,102),(40,103,47),(41,140,175),(42,176,137),(43,138,173),(44,174,139),(49,167,65),(50,66,168),(51,165,67),(52,68,166),(57,106,190),(58,191,107),(59,108,192),(60,189,105),(69,98,178),(70,179,99),(71,100,180),(72,177,97),(73,169,129),(74,130,170),(75,171,131),(76,132,172),(81,133,163),(82,164,134),(83,135,161),(84,162,136),(93,121,151),(94,152,122),(95,123,149),(96,150,124)], [(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)], [(1,121,31,143),(2,122,32,144),(3,123,29,141),(4,124,30,142),(5,67,81,158),(6,68,82,159),(7,65,83,160),(8,66,84,157),(9,155,51,133),(10,156,52,134),(11,153,49,135),(12,154,50,136),(13,100,87,46),(14,97,88,47),(15,98,85,48),(16,99,86,45),(17,110,93,185),(18,111,94,186),(19,112,95,187),(20,109,96,188),(21,103,128,177),(22,104,125,178),(23,101,126,179),(24,102,127,180),(25,42,57,129),(26,43,58,130),(27,44,59,131),(28,41,60,132),(33,173,107,74),(34,174,108,75),(35,175,105,76),(36,176,106,73),(37,145,69,117),(38,146,70,118),(39,147,71,119),(40,148,72,120),(53,90,184,149),(54,91,181,150),(55,92,182,151),(56,89,183,152),(61,80,165,163),(62,77,166,164),(63,78,167,161),(64,79,168,162),(113,138,191,170),(114,139,192,171),(115,140,189,172),(116,137,190,169)], [(1,58,70,51),(2,27,71,10),(3,60,72,49),(4,25,69,12),(5,95,173,88),(6,20,174,15),(7,93,175,86),(8,18,176,13),(9,31,26,38),(11,29,28,40),(14,81,19,74),(16,83,17,76),(21,80,149,138),(22,164,150,171),(23,78,151,140),(24,162,152,169),(30,57,37,50),(32,59,39,52),(33,45,158,185),(34,100,159,111),(35,47,160,187),(36,98,157,109),(41,118,153,121),(42,147,154,144),(43,120,155,123),(44,145,156,142),(46,68,186,108),(48,66,188,106),(53,115,103,63),(54,190,104,168),(55,113,101,61),(56,192,102,166),(62,183,114,180),(64,181,116,178),(65,112,105,97),(67,110,107,99),(73,87,84,94),(75,85,82,96),(77,91,139,125),(79,89,137,127),(90,170,128,163),(92,172,126,161),(117,134,124,131),(119,136,122,129),(130,148,133,141),(132,146,135,143),(165,182,191,179),(167,184,189,177)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | C4○D4 | C4×S3 | C4○D12 | D4⋊2S3 | Q8⋊3S3 |
| kernel | C3⋊(C42⋊5C4) | C6.C42 | C3×C2.C42 | C2×C4×Dic3 | C4×Dic3 | C2.C42 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 5 | 1 | 1 | 8 | 1 | 3 | 12 | 4 | 8 | 3 | 1 |
Matrix representation of C3⋊(C42⋊5C4) ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 8 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 1 |
| 0 | 0 | 0 | 0 | 2 | 5 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 8 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 6 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 1 | 12 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,1,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,7,8,0,0,0,0,10,6,0,0,0,0,0,0,8,2,0,0,0,0,1,5],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,3,0,0,0,0,8,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,5,6,0,0,0,0,0,8,0,0,0,0,0,0,1,1,0,0,0,0,11,12] >;
C3⋊(C42⋊5C4) in GAP, Magma, Sage, TeX
C_3\rtimes (C_4^2\rtimes_5C_4)
% in TeX
G:=Group("C3:(C4^2:5C4)"); // GroupNames label
G:=SmallGroup(192,210);
// by ID
G=gap.SmallGroup(192,210);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,1094,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=d^4=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c^-1>;
// generators/relations