direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C42⋊5C4, C42⋊8C12, (C4×C12)⋊9C4, (C2×C42).7C6, C2.C42.5C6, C23.60(C22×C6), C6.56(C42⋊C2), C6.31(C42⋊2C2), C22.32(C22×C12), (C22×C6).447C23, (C22×C12).492C22, (C2×C4×C12).6C2, (C2×C4).57(C2×C12), (C2×C12).286(C2×C4), (C22×C4).92(C2×C6), C2.1(C3×C42⋊2C2), C2.8(C3×C42⋊C2), C22.18(C3×C4○D4), (C2×C6).208(C4○D4), (C2×C6).219(C22×C4), (C3×C2.C42).4C2, SmallGroup(192,816)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊5C4
G = < a,b,c,d | a3=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c-1 >
Subgroups: 210 in 138 conjugacy classes, 82 normal (10 characteristic)
C1, C2, C3, C4, C22, C22, C6, C2×C4, C2×C4, C23, C12, C2×C6, C2×C6, C42, C22×C4, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C4×C12, C22×C12, C42⋊5C4, C3×C2.C42, C2×C4×C12, C3×C42⋊5C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C4○D4, C2×C12, C22×C6, C42⋊C2, C42⋊2C2, C22×C12, C3×C4○D4, C42⋊5C4, C3×C42⋊C2, C3×C42⋊2C2, C3×C42⋊5C4
►Regular action on 192 points
Generators in S192
(1 44 38)(2 41 39)(3 42 40)(4 43 37)(5 190 188)(6 191 185)(7 192 186)(8 189 187)(9 17 15)(10 18 16)(11 19 13)(12 20 14)(21 36 27)(22 33 28)(23 34 25)(24 35 26)(29 61 59)(30 62 60)(31 63 57)(32 64 58)(45 97 103)(46 98 104)(47 99 101)(48 100 102)(49 138 55)(50 139 56)(51 140 53)(52 137 54)(65 73 71)(66 74 72)(67 75 69)(68 76 70)(77 85 83)(78 86 84)(79 87 81)(80 88 82)(89 145 95)(90 146 96)(91 147 93)(92 148 94)(105 111 119)(106 112 120)(107 109 117)(108 110 118)(113 121 127)(114 122 128)(115 123 125)(116 124 126)(129 135 143)(130 136 144)(131 133 141)(132 134 142)(149 157 155)(150 158 156)(151 159 153)(152 160 154)(161 169 167)(162 170 168)(163 171 165)(164 172 166)(173 181 179)(174 182 180)(175 183 177)(176 184 178)
(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 68 79 21)(2 65 80 22)(3 66 77 23)(4 67 78 24)(5 159 54 120)(6 160 55 117)(7 157 56 118)(8 158 53 119)(9 91 32 48)(10 92 29 45)(11 89 30 46)(12 90 31 47)(13 95 60 104)(14 96 57 101)(15 93 58 102)(16 94 59 103)(17 147 64 100)(18 148 61 97)(19 145 62 98)(20 146 63 99)(25 40 72 83)(26 37 69 84)(27 38 70 81)(28 39 71 82)(33 41 73 88)(34 42 74 85)(35 43 75 86)(36 44 76 87)(49 107 191 154)(50 108 192 155)(51 105 189 156)(52 106 190 153)(109 185 152 138)(110 186 149 139)(111 187 150 140)(112 188 151 137)(113 134 164 173)(114 135 161 174)(115 136 162 175)(116 133 163 176)(121 142 172 181)(122 143 169 182)(123 144 170 183)(124 141 171 184)(125 130 168 177)(126 131 165 178)(127 132 166 179)(128 129 167 180)
(1 111 11 175)(2 151 12 133)(3 109 9 173)(4 149 10 135)(5 148 124 75)(6 98 121 36)(7 146 122 73)(8 100 123 34)(13 177 38 105)(14 131 39 153)(15 179 40 107)(16 129 37 155)(17 181 42 117)(18 143 43 157)(19 183 44 119)(20 141 41 159)(21 185 46 113)(22 139 47 161)(23 187 48 115)(24 137 45 163)(25 189 102 125)(26 52 103 165)(27 191 104 127)(28 50 101 167)(29 174 78 110)(30 136 79 150)(31 176 80 112)(32 134 77 152)(33 56 99 169)(35 54 97 171)(49 95 166 70)(51 93 168 72)(53 147 170 74)(55 145 172 76)(57 178 82 106)(58 132 83 154)(59 180 84 108)(60 130 81 156)(61 182 86 118)(62 144 87 158)(63 184 88 120)(64 142 85 160)(65 186 90 114)(66 140 91 162)(67 188 92 116)(68 138 89 164)(69 190 94 126)(71 192 96 128)
G:=sub<Sym(192)| (1,44,38)(2,41,39)(3,42,40)(4,43,37)(5,190,188)(6,191,185)(7,192,186)(8,189,187)(9,17,15)(10,18,16)(11,19,13)(12,20,14)(21,36,27)(22,33,28)(23,34,25)(24,35,26)(29,61,59)(30,62,60)(31,63,57)(32,64,58)(45,97,103)(46,98,104)(47,99,101)(48,100,102)(49,138,55)(50,139,56)(51,140,53)(52,137,54)(65,73,71)(66,74,72)(67,75,69)(68,76,70)(77,85,83)(78,86,84)(79,87,81)(80,88,82)(89,145,95)(90,146,96)(91,147,93)(92,148,94)(105,111,119)(106,112,120)(107,109,117)(108,110,118)(113,121,127)(114,122,128)(115,123,125)(116,124,126)(129,135,143)(130,136,144)(131,133,141)(132,134,142)(149,157,155)(150,158,156)(151,159,153)(152,160,154)(161,169,167)(162,170,168)(163,171,165)(164,172,166)(173,181,179)(174,182,180)(175,183,177)(176,184,178), (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), (1,68,79,21)(2,65,80,22)(3,66,77,23)(4,67,78,24)(5,159,54,120)(6,160,55,117)(7,157,56,118)(8,158,53,119)(9,91,32,48)(10,92,29,45)(11,89,30,46)(12,90,31,47)(13,95,60,104)(14,96,57,101)(15,93,58,102)(16,94,59,103)(17,147,64,100)(18,148,61,97)(19,145,62,98)(20,146,63,99)(25,40,72,83)(26,37,69,84)(27,38,70,81)(28,39,71,82)(33,41,73,88)(34,42,74,85)(35,43,75,86)(36,44,76,87)(49,107,191,154)(50,108,192,155)(51,105,189,156)(52,106,190,153)(109,185,152,138)(110,186,149,139)(111,187,150,140)(112,188,151,137)(113,134,164,173)(114,135,161,174)(115,136,162,175)(116,133,163,176)(121,142,172,181)(122,143,169,182)(123,144,170,183)(124,141,171,184)(125,130,168,177)(126,131,165,178)(127,132,166,179)(128,129,167,180), (1,111,11,175)(2,151,12,133)(3,109,9,173)(4,149,10,135)(5,148,124,75)(6,98,121,36)(7,146,122,73)(8,100,123,34)(13,177,38,105)(14,131,39,153)(15,179,40,107)(16,129,37,155)(17,181,42,117)(18,143,43,157)(19,183,44,119)(20,141,41,159)(21,185,46,113)(22,139,47,161)(23,187,48,115)(24,137,45,163)(25,189,102,125)(26,52,103,165)(27,191,104,127)(28,50,101,167)(29,174,78,110)(30,136,79,150)(31,176,80,112)(32,134,77,152)(33,56,99,169)(35,54,97,171)(49,95,166,70)(51,93,168,72)(53,147,170,74)(55,145,172,76)(57,178,82,106)(58,132,83,154)(59,180,84,108)(60,130,81,156)(61,182,86,118)(62,144,87,158)(63,184,88,120)(64,142,85,160)(65,186,90,114)(66,140,91,162)(67,188,92,116)(68,138,89,164)(69,190,94,126)(71,192,96,128)>;
G:=Group( (1,44,38)(2,41,39)(3,42,40)(4,43,37)(5,190,188)(6,191,185)(7,192,186)(8,189,187)(9,17,15)(10,18,16)(11,19,13)(12,20,14)(21,36,27)(22,33,28)(23,34,25)(24,35,26)(29,61,59)(30,62,60)(31,63,57)(32,64,58)(45,97,103)(46,98,104)(47,99,101)(48,100,102)(49,138,55)(50,139,56)(51,140,53)(52,137,54)(65,73,71)(66,74,72)(67,75,69)(68,76,70)(77,85,83)(78,86,84)(79,87,81)(80,88,82)(89,145,95)(90,146,96)(91,147,93)(92,148,94)(105,111,119)(106,112,120)(107,109,117)(108,110,118)(113,121,127)(114,122,128)(115,123,125)(116,124,126)(129,135,143)(130,136,144)(131,133,141)(132,134,142)(149,157,155)(150,158,156)(151,159,153)(152,160,154)(161,169,167)(162,170,168)(163,171,165)(164,172,166)(173,181,179)(174,182,180)(175,183,177)(176,184,178), (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), (1,68,79,21)(2,65,80,22)(3,66,77,23)(4,67,78,24)(5,159,54,120)(6,160,55,117)(7,157,56,118)(8,158,53,119)(9,91,32,48)(10,92,29,45)(11,89,30,46)(12,90,31,47)(13,95,60,104)(14,96,57,101)(15,93,58,102)(16,94,59,103)(17,147,64,100)(18,148,61,97)(19,145,62,98)(20,146,63,99)(25,40,72,83)(26,37,69,84)(27,38,70,81)(28,39,71,82)(33,41,73,88)(34,42,74,85)(35,43,75,86)(36,44,76,87)(49,107,191,154)(50,108,192,155)(51,105,189,156)(52,106,190,153)(109,185,152,138)(110,186,149,139)(111,187,150,140)(112,188,151,137)(113,134,164,173)(114,135,161,174)(115,136,162,175)(116,133,163,176)(121,142,172,181)(122,143,169,182)(123,144,170,183)(124,141,171,184)(125,130,168,177)(126,131,165,178)(127,132,166,179)(128,129,167,180), (1,111,11,175)(2,151,12,133)(3,109,9,173)(4,149,10,135)(5,148,124,75)(6,98,121,36)(7,146,122,73)(8,100,123,34)(13,177,38,105)(14,131,39,153)(15,179,40,107)(16,129,37,155)(17,181,42,117)(18,143,43,157)(19,183,44,119)(20,141,41,159)(21,185,46,113)(22,139,47,161)(23,187,48,115)(24,137,45,163)(25,189,102,125)(26,52,103,165)(27,191,104,127)(28,50,101,167)(29,174,78,110)(30,136,79,150)(31,176,80,112)(32,134,77,152)(33,56,99,169)(35,54,97,171)(49,95,166,70)(51,93,168,72)(53,147,170,74)(55,145,172,76)(57,178,82,106)(58,132,83,154)(59,180,84,108)(60,130,81,156)(61,182,86,118)(62,144,87,158)(63,184,88,120)(64,142,85,160)(65,186,90,114)(66,140,91,162)(67,188,92,116)(68,138,89,164)(69,190,94,126)(71,192,96,128) );
G=PermutationGroup([[(1,44,38),(2,41,39),(3,42,40),(4,43,37),(5,190,188),(6,191,185),(7,192,186),(8,189,187),(9,17,15),(10,18,16),(11,19,13),(12,20,14),(21,36,27),(22,33,28),(23,34,25),(24,35,26),(29,61,59),(30,62,60),(31,63,57),(32,64,58),(45,97,103),(46,98,104),(47,99,101),(48,100,102),(49,138,55),(50,139,56),(51,140,53),(52,137,54),(65,73,71),(66,74,72),(67,75,69),(68,76,70),(77,85,83),(78,86,84),(79,87,81),(80,88,82),(89,145,95),(90,146,96),(91,147,93),(92,148,94),(105,111,119),(106,112,120),(107,109,117),(108,110,118),(113,121,127),(114,122,128),(115,123,125),(116,124,126),(129,135,143),(130,136,144),(131,133,141),(132,134,142),(149,157,155),(150,158,156),(151,159,153),(152,160,154),(161,169,167),(162,170,168),(163,171,165),(164,172,166),(173,181,179),(174,182,180),(175,183,177),(176,184,178)], [(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)], [(1,68,79,21),(2,65,80,22),(3,66,77,23),(4,67,78,24),(5,159,54,120),(6,160,55,117),(7,157,56,118),(8,158,53,119),(9,91,32,48),(10,92,29,45),(11,89,30,46),(12,90,31,47),(13,95,60,104),(14,96,57,101),(15,93,58,102),(16,94,59,103),(17,147,64,100),(18,148,61,97),(19,145,62,98),(20,146,63,99),(25,40,72,83),(26,37,69,84),(27,38,70,81),(28,39,71,82),(33,41,73,88),(34,42,74,85),(35,43,75,86),(36,44,76,87),(49,107,191,154),(50,108,192,155),(51,105,189,156),(52,106,190,153),(109,185,152,138),(110,186,149,139),(111,187,150,140),(112,188,151,137),(113,134,164,173),(114,135,161,174),(115,136,162,175),(116,133,163,176),(121,142,172,181),(122,143,169,182),(123,144,170,183),(124,141,171,184),(125,130,168,177),(126,131,165,178),(127,132,166,179),(128,129,167,180)], [(1,111,11,175),(2,151,12,133),(3,109,9,173),(4,149,10,135),(5,148,124,75),(6,98,121,36),(7,146,122,73),(8,100,123,34),(13,177,38,105),(14,131,39,153),(15,179,40,107),(16,129,37,155),(17,181,42,117),(18,143,43,157),(19,183,44,119),(20,141,41,159),(21,185,46,113),(22,139,47,161),(23,187,48,115),(24,137,45,163),(25,189,102,125),(26,52,103,165),(27,191,104,127),(28,50,101,167),(29,174,78,110),(30,136,79,150),(31,176,80,112),(32,134,77,152),(33,56,99,169),(35,54,97,171),(49,95,166,70),(51,93,168,72),(53,147,170,74),(55,145,172,76),(57,178,82,106),(58,132,83,154),(59,180,84,108),(60,130,81,156),(61,182,86,118),(62,144,87,158),(63,184,88,120),(64,142,85,160),(65,186,90,114),(66,140,91,162),(67,188,92,116),(68,138,89,164),(69,190,94,126),(71,192,96,128)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4L | 4M | ··· | 4T | 6A | ··· | 6N | 12A | ··· | 12X | 12Y | ··· | 12AN |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | C4○D4 | C3×C4○D4 |
| kernel | C3×C42⋊5C4 | C3×C2.C42 | C2×C4×C12 | C42⋊5C4 | C4×C12 | C2.C42 | C2×C42 | C42 | C2×C6 | C22 |
| # reps | 1 | 6 | 1 | 2 | 8 | 12 | 2 | 16 | 12 | 24 |
Matrix representation of C3×C42⋊5C4 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 8 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 11 | 7 |
| 0 | 0 | 0 | 7 | 2 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[12,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,5,0],[1,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,1,0],[5,0,0,0,0,0,5,0,0,0,0,0,8,0,0,0,0,0,11,7,0,0,0,7,2] >;
C3×C42⋊5C4 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes_5C_4
% in TeX
G:=Group("C3xC4^2:5C4"); // GroupNames label
G:=SmallGroup(192,816);
// by ID
G=gap.SmallGroup(192,816);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,512,1094,142]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=d^4=1,a*b=b*a,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