G = C3⋊(C42⋊8C4) order 192 = 26·3
The semidirect product of C3 and C42⋊8C4 acting via C42⋊8C4/C2.C42=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊1(C42⋊8C4), (C4×Dic3)⋊12C4, C22.53(S3×D4), (C22×C4).23D6, C22.12(S3×Q8), Dic3.1(C4⋊C4), (C2×Dic3).10Q8, (C2×Dic3).83D4, C6.6(C42.C2), C6.1(C42⋊C2), C2.5(C42⋊2S3), C6.15(C4.4D4), C2.2(Dic3.Q8), C6.C42.1C2, C2.C42.2S3, (C22×C12).3C22, C22.27(C4○D12), (C22×C6).275C23, C23.257(C22×S3), C22.29(D4⋊2S3), C2.2(C23.11D6), C2.5(C23.16D6), (C22×Dic3).3C22, C6.3(C2×C4⋊C4), C2.6(S3×C4⋊C4), C22.82(S3×C2×C4), (C2×C6).59(C2×Q8), (C2×C4).122(C4×S3), (C2×C6).190(C2×D4), (C2×C4×Dic3).23C2, (C2×C12).139(C2×C4), (C2×C6).54(C4○D4), (C2×Dic3⋊C4).3C2, (C2×C6).41(C22×C4), (C2×Dic3).75(C2×C4), (C3×C2.C42).21C2, SmallGroup(192,209)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊(C42⋊8C4)
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 >
Subgroups: 352 in 154 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C4×Dic3, Dic3⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C42⋊8C4, C6.C42, C6.C42, C3×C2.C42, C2×C4×Dic3, C2×Dic3⋊C4, C3⋊(C42⋊8C4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, C22×S3, C2×C4⋊C4, C42⋊C2, C4.4D4, C42.C2, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, S3×Q8, C42⋊8C4, C42⋊2S3, C23.16D6, C23.11D6, Dic3.Q8, S3×C4⋊C4, C3⋊(C42⋊8C4)
►Regular action on 192 points
Generators in S192
(1 13 9)(2 10 14)(3 15 11)(4 12 16)(5 45 59)(6 60 46)(7 47 57)(8 58 48)(17 49 155)(18 156 50)(19 51 153)(20 154 52)(21 151 65)(22 66 152)(23 149 67)(24 68 150)(25 175 89)(26 90 176)(27 173 91)(28 92 174)(29 69 83)(30 84 70)(31 71 81)(32 82 72)(33 95 41)(34 42 96)(35 93 43)(36 44 94)(37 163 53)(38 54 164)(39 161 55)(40 56 162)(61 187 77)(62 78 188)(63 185 79)(64 80 186)(73 179 127)(74 128 180)(75 177 125)(76 126 178)(85 167 115)(86 116 168)(87 165 113)(88 114 166)(97 147 101)(98 102 148)(99 145 103)(100 104 146)(105 159 117)(106 118 160)(107 157 119)(108 120 158)(109 191 139)(110 140 192)(111 189 137)(112 138 190)(121 171 129)(122 130 172)(123 169 131)(124 132 170)(133 183 141)(134 142 184)(135 181 143)(136 144 182)
(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 25 19 171)(2 26 20 172)(3 27 17 169)(4 28 18 170)(5 187 87 141)(6 188 88 142)(7 185 85 143)(8 186 86 144)(9 89 153 121)(10 90 154 122)(11 91 155 123)(12 92 156 124)(13 175 51 129)(14 176 52 130)(15 173 49 131)(16 174 50 132)(21 95 147 125)(22 96 148 126)(23 93 145 127)(24 94 146 128)(29 117 111 163)(30 118 112 164)(31 119 109 161)(32 120 110 162)(33 97 177 65)(34 98 178 66)(35 99 179 67)(36 100 180 68)(37 83 159 137)(38 84 160 138)(39 81 157 139)(40 82 158 140)(41 101 75 151)(42 102 76 152)(43 103 73 149)(44 104 74 150)(45 77 165 133)(46 78 166 134)(47 79 167 135)(48 80 168 136)(53 69 105 189)(54 70 106 190)(55 71 107 191)(56 72 108 192)(57 63 115 181)(58 64 116 182)(59 61 113 183)(60 62 114 184)
(1 159 147 59)(2 38 148 114)(3 157 145 57)(4 40 146 116)(5 13 117 101)(6 52 118 152)(7 15 119 103)(8 50 120 150)(9 105 97 45)(10 54 98 166)(11 107 99 47)(12 56 100 168)(14 164 102 88)(16 162 104 86)(17 39 23 115)(18 158 24 58)(19 37 21 113)(20 160 22 60)(25 139 125 63)(26 82 126 182)(27 137 127 61)(28 84 128 184)(29 43 141 131)(30 74 142 174)(31 41 143 129)(32 76 144 176)(33 135 121 71)(34 80 122 192)(35 133 123 69)(36 78 124 190)(42 186 130 110)(44 188 132 112)(46 154 106 66)(48 156 108 68)(49 161 149 85)(51 163 151 87)(53 65 165 153)(55 67 167 155)(62 170 138 94)(64 172 140 96)(70 180 134 92)(72 178 136 90)(73 187 173 111)(75 185 175 109)(77 91 189 179)(79 89 191 177)(81 95 181 171)(83 93 183 169)
G:=sub<Sym(192)| (1,13,9)(2,10,14)(3,15,11)(4,12,16)(5,45,59)(6,60,46)(7,47,57)(8,58,48)(17,49,155)(18,156,50)(19,51,153)(20,154,52)(21,151,65)(22,66,152)(23,149,67)(24,68,150)(25,175,89)(26,90,176)(27,173,91)(28,92,174)(29,69,83)(30,84,70)(31,71,81)(32,82,72)(33,95,41)(34,42,96)(35,93,43)(36,44,94)(37,163,53)(38,54,164)(39,161,55)(40,56,162)(61,187,77)(62,78,188)(63,185,79)(64,80,186)(73,179,127)(74,128,180)(75,177,125)(76,126,178)(85,167,115)(86,116,168)(87,165,113)(88,114,166)(97,147,101)(98,102,148)(99,145,103)(100,104,146)(105,159,117)(106,118,160)(107,157,119)(108,120,158)(109,191,139)(110,140,192)(111,189,137)(112,138,190)(121,171,129)(122,130,172)(123,169,131)(124,132,170)(133,183,141)(134,142,184)(135,181,143)(136,144,182), (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), (1,25,19,171)(2,26,20,172)(3,27,17,169)(4,28,18,170)(5,187,87,141)(6,188,88,142)(7,185,85,143)(8,186,86,144)(9,89,153,121)(10,90,154,122)(11,91,155,123)(12,92,156,124)(13,175,51,129)(14,176,52,130)(15,173,49,131)(16,174,50,132)(21,95,147,125)(22,96,148,126)(23,93,145,127)(24,94,146,128)(29,117,111,163)(30,118,112,164)(31,119,109,161)(32,120,110,162)(33,97,177,65)(34,98,178,66)(35,99,179,67)(36,100,180,68)(37,83,159,137)(38,84,160,138)(39,81,157,139)(40,82,158,140)(41,101,75,151)(42,102,76,152)(43,103,73,149)(44,104,74,150)(45,77,165,133)(46,78,166,134)(47,79,167,135)(48,80,168,136)(53,69,105,189)(54,70,106,190)(55,71,107,191)(56,72,108,192)(57,63,115,181)(58,64,116,182)(59,61,113,183)(60,62,114,184), (1,159,147,59)(2,38,148,114)(3,157,145,57)(4,40,146,116)(5,13,117,101)(6,52,118,152)(7,15,119,103)(8,50,120,150)(9,105,97,45)(10,54,98,166)(11,107,99,47)(12,56,100,168)(14,164,102,88)(16,162,104,86)(17,39,23,115)(18,158,24,58)(19,37,21,113)(20,160,22,60)(25,139,125,63)(26,82,126,182)(27,137,127,61)(28,84,128,184)(29,43,141,131)(30,74,142,174)(31,41,143,129)(32,76,144,176)(33,135,121,71)(34,80,122,192)(35,133,123,69)(36,78,124,190)(42,186,130,110)(44,188,132,112)(46,154,106,66)(48,156,108,68)(49,161,149,85)(51,163,151,87)(53,65,165,153)(55,67,167,155)(62,170,138,94)(64,172,140,96)(70,180,134,92)(72,178,136,90)(73,187,173,111)(75,185,175,109)(77,91,189,179)(79,89,191,177)(81,95,181,171)(83,93,183,169)>;
G:=Group( (1,13,9)(2,10,14)(3,15,11)(4,12,16)(5,45,59)(6,60,46)(7,47,57)(8,58,48)(17,49,155)(18,156,50)(19,51,153)(20,154,52)(21,151,65)(22,66,152)(23,149,67)(24,68,150)(25,175,89)(26,90,176)(27,173,91)(28,92,174)(29,69,83)(30,84,70)(31,71,81)(32,82,72)(33,95,41)(34,42,96)(35,93,43)(36,44,94)(37,163,53)(38,54,164)(39,161,55)(40,56,162)(61,187,77)(62,78,188)(63,185,79)(64,80,186)(73,179,127)(74,128,180)(75,177,125)(76,126,178)(85,167,115)(86,116,168)(87,165,113)(88,114,166)(97,147,101)(98,102,148)(99,145,103)(100,104,146)(105,159,117)(106,118,160)(107,157,119)(108,120,158)(109,191,139)(110,140,192)(111,189,137)(112,138,190)(121,171,129)(122,130,172)(123,169,131)(124,132,170)(133,183,141)(134,142,184)(135,181,143)(136,144,182), (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), (1,25,19,171)(2,26,20,172)(3,27,17,169)(4,28,18,170)(5,187,87,141)(6,188,88,142)(7,185,85,143)(8,186,86,144)(9,89,153,121)(10,90,154,122)(11,91,155,123)(12,92,156,124)(13,175,51,129)(14,176,52,130)(15,173,49,131)(16,174,50,132)(21,95,147,125)(22,96,148,126)(23,93,145,127)(24,94,146,128)(29,117,111,163)(30,118,112,164)(31,119,109,161)(32,120,110,162)(33,97,177,65)(34,98,178,66)(35,99,179,67)(36,100,180,68)(37,83,159,137)(38,84,160,138)(39,81,157,139)(40,82,158,140)(41,101,75,151)(42,102,76,152)(43,103,73,149)(44,104,74,150)(45,77,165,133)(46,78,166,134)(47,79,167,135)(48,80,168,136)(53,69,105,189)(54,70,106,190)(55,71,107,191)(56,72,108,192)(57,63,115,181)(58,64,116,182)(59,61,113,183)(60,62,114,184), (1,159,147,59)(2,38,148,114)(3,157,145,57)(4,40,146,116)(5,13,117,101)(6,52,118,152)(7,15,119,103)(8,50,120,150)(9,105,97,45)(10,54,98,166)(11,107,99,47)(12,56,100,168)(14,164,102,88)(16,162,104,86)(17,39,23,115)(18,158,24,58)(19,37,21,113)(20,160,22,60)(25,139,125,63)(26,82,126,182)(27,137,127,61)(28,84,128,184)(29,43,141,131)(30,74,142,174)(31,41,143,129)(32,76,144,176)(33,135,121,71)(34,80,122,192)(35,133,123,69)(36,78,124,190)(42,186,130,110)(44,188,132,112)(46,154,106,66)(48,156,108,68)(49,161,149,85)(51,163,151,87)(53,65,165,153)(55,67,167,155)(62,170,138,94)(64,172,140,96)(70,180,134,92)(72,178,136,90)(73,187,173,111)(75,185,175,109)(77,91,189,179)(79,89,191,177)(81,95,181,171)(83,93,183,169) );
G=PermutationGroup([[(1,13,9),(2,10,14),(3,15,11),(4,12,16),(5,45,59),(6,60,46),(7,47,57),(8,58,48),(17,49,155),(18,156,50),(19,51,153),(20,154,52),(21,151,65),(22,66,152),(23,149,67),(24,68,150),(25,175,89),(26,90,176),(27,173,91),(28,92,174),(29,69,83),(30,84,70),(31,71,81),(32,82,72),(33,95,41),(34,42,96),(35,93,43),(36,44,94),(37,163,53),(38,54,164),(39,161,55),(40,56,162),(61,187,77),(62,78,188),(63,185,79),(64,80,186),(73,179,127),(74,128,180),(75,177,125),(76,126,178),(85,167,115),(86,116,168),(87,165,113),(88,114,166),(97,147,101),(98,102,148),(99,145,103),(100,104,146),(105,159,117),(106,118,160),(107,157,119),(108,120,158),(109,191,139),(110,140,192),(111,189,137),(112,138,190),(121,171,129),(122,130,172),(123,169,131),(124,132,170),(133,183,141),(134,142,184),(135,181,143),(136,144,182)], [(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)], [(1,25,19,171),(2,26,20,172),(3,27,17,169),(4,28,18,170),(5,187,87,141),(6,188,88,142),(7,185,85,143),(8,186,86,144),(9,89,153,121),(10,90,154,122),(11,91,155,123),(12,92,156,124),(13,175,51,129),(14,176,52,130),(15,173,49,131),(16,174,50,132),(21,95,147,125),(22,96,148,126),(23,93,145,127),(24,94,146,128),(29,117,111,163),(30,118,112,164),(31,119,109,161),(32,120,110,162),(33,97,177,65),(34,98,178,66),(35,99,179,67),(36,100,180,68),(37,83,159,137),(38,84,160,138),(39,81,157,139),(40,82,158,140),(41,101,75,151),(42,102,76,152),(43,103,73,149),(44,104,74,150),(45,77,165,133),(46,78,166,134),(47,79,167,135),(48,80,168,136),(53,69,105,189),(54,70,106,190),(55,71,107,191),(56,72,108,192),(57,63,115,181),(58,64,116,182),(59,61,113,183),(60,62,114,184)], [(1,159,147,59),(2,38,148,114),(3,157,145,57),(4,40,146,116),(5,13,117,101),(6,52,118,152),(7,15,119,103),(8,50,120,150),(9,105,97,45),(10,54,98,166),(11,107,99,47),(12,56,100,168),(14,164,102,88),(16,162,104,86),(17,39,23,115),(18,158,24,58),(19,37,21,113),(20,160,22,60),(25,139,125,63),(26,82,126,182),(27,137,127,61),(28,84,128,184),(29,43,141,131),(30,74,142,174),(31,41,143,129),(32,76,144,176),(33,135,121,71),(34,80,122,192),(35,133,123,69),(36,78,124,190),(42,186,130,110),(44,188,132,112),(46,154,106,66),(48,156,108,68),(49,161,149,85),(51,163,151,87),(53,65,165,153),(55,67,167,155),(62,170,138,94),(64,172,140,96),(70,180,134,92),(72,178,136,90),(73,187,173,111),(75,185,175,109),(77,91,189,179),(79,89,191,177),(81,95,181,171),(83,93,183,169)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | C4○D4 | C4×S3 | C4○D12 | S3×D4 | D4⋊2S3 | S3×Q8 |
| kernel | C3⋊(C42⋊8C4) | C6.C42 | C3×C2.C42 | C2×C4×Dic3 | C2×Dic3⋊C4 | C4×Dic3 | C2.C42 | C2×Dic3 | C2×Dic3 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 1 | 1 | 2 | 8 | 1 | 2 | 2 | 3 | 8 | 4 | 8 | 1 | 2 | 1 |
Matrix representation of C3⋊(C42⋊8C4) ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 10 | 0 | 0 |
| 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 3 | 6 | 0 | 0 |
| 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 9 | 3 |
| 0 | 0 | 0 | 8 | 4 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,7,3,0,0,0,10,6,0,0,0,0,0,8,0,0,0,0,0,8],[12,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,1,0,0,0,0,1,12],[8,0,0,0,0,0,3,7,0,0,0,6,10,0,0,0,0,0,9,8,0,0,0,3,4] >;
C3⋊(C42⋊8C4) in GAP, Magma, Sage, TeX
C_3\rtimes (C_4^2\rtimes_8C_4)
% in TeX
G:=Group("C3:(C4^2:8C4)"); // GroupNames label
G:=SmallGroup(192,209);
// by ID
G=gap.SmallGroup(192,209);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,422,387,58,6278]);
// 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>;
// generators/relations