direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C42⋊4C4, C42⋊7C12, C12.36C42, (C4×C12)⋊8C4, C4.7(C4×C12), (C2×C42).5C6, C6.30(C2×C42), C23.53(C22×C6), C6.50(C42⋊C2), C2.C42.14C6, (C22×C6).440C23, C22.13(C22×C12), (C22×C12).487C22, (C2×C4×C12).4C2, C2.2(C2×C4×C12), (C2×C4).54(C2×C12), (C2×C12).283(C2×C4), (C22×C4).88(C2×C6), C2.1(C3×C42⋊C2), C22.12(C3×C4○D4), (C2×C6).202(C4○D4), (C2×C6).212(C22×C4), (C3×C2.C42).29C2, SmallGroup(192,809)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊4C4
G = < a,b,c,d | a3=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, cd=dc >
Subgroups: 226 in 178 conjugacy classes, 130 normal (10 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C23, C12, C12, C2×C6, C42, C22×C4, C22×C4, C2×C12, C2×C12, C22×C6, C2.C42, C2×C42, C4×C12, C22×C12, C22×C12, C42⋊4C4, C3×C2.C42, C2×C4×C12, C3×C42⋊4C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C42, C22×C4, C4○D4, C2×C12, C22×C6, C2×C42, C42⋊C2, C4×C12, C22×C12, C3×C4○D4, C42⋊4C4, C2×C4×C12, C3×C42⋊C2, C3×C42⋊4C4
►Regular action on 192 points
Generators in S192
(1 59 54)(2 60 55)(3 57 56)(4 58 53)(5 189 186)(6 190 187)(7 191 188)(8 192 185)(9 17 14)(10 18 15)(11 19 16)(12 20 13)(21 29 28)(22 30 25)(23 31 26)(24 32 27)(33 41 38)(34 42 39)(35 43 40)(36 44 37)(45 149 148)(46 150 145)(47 151 146)(48 152 147)(49 101 98)(50 102 99)(51 103 100)(52 104 97)(61 69 66)(62 70 67)(63 71 68)(64 72 65)(73 81 80)(74 82 77)(75 83 78)(76 84 79)(85 93 90)(86 94 91)(87 95 92)(88 96 89)(105 113 112)(106 114 109)(107 115 110)(108 116 111)(117 139 134)(118 140 135)(119 137 136)(120 138 133)(121 129 128)(122 130 125)(123 131 126)(124 132 127)(141 157 154)(142 158 155)(143 159 156)(144 160 153)(161 169 168)(162 170 165)(163 171 166)(164 172 167)(173 181 178)(174 182 179)(175 183 180)(176 184 177)
(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)
(1 188 50 46)(2 185 51 47)(3 186 52 48)(4 187 49 45)(5 104 152 57)(6 101 149 58)(7 102 150 59)(8 103 151 60)(9 87 143 135)(10 88 144 136)(11 85 141 133)(12 86 142 134)(13 91 155 139)(14 92 156 140)(15 89 153 137)(16 90 154 138)(17 95 159 118)(18 96 160 119)(19 93 157 120)(20 94 158 117)(21 123 163 75)(22 124 164 76)(23 121 161 73)(24 122 162 74)(25 127 167 79)(26 128 168 80)(27 125 165 77)(28 126 166 78)(29 131 171 83)(30 132 172 84)(31 129 169 81)(32 130 170 82)(33 107 175 62)(34 108 176 63)(35 105 173 64)(36 106 174 61)(37 109 179 66)(38 110 180 67)(39 111 177 68)(40 112 178 65)(41 115 183 70)(42 116 184 71)(43 113 181 72)(44 114 182 69)(53 190 98 148)(54 191 99 145)(55 192 100 146)(56 189 97 147)
(1 124 33 9)(2 73 34 144)(3 122 35 11)(4 75 36 142)(5 170 113 93)(6 29 114 117)(7 172 115 95)(8 31 116 119)(10 51 121 176)(12 49 123 174)(13 98 126 179)(14 54 127 38)(15 100 128 177)(16 56 125 40)(17 59 132 41)(18 103 129 184)(19 57 130 43)(20 101 131 182)(21 106 134 187)(22 62 135 46)(23 108 136 185)(24 64 133 48)(25 67 140 145)(26 111 137 192)(27 65 138 147)(28 109 139 190)(30 70 118 150)(32 72 120 152)(37 155 53 78)(39 153 55 80)(42 160 60 81)(44 158 58 83)(45 163 61 86)(47 161 63 88)(50 76 175 143)(52 74 173 141)(66 91 148 166)(68 89 146 168)(69 94 149 171)(71 96 151 169)(77 178 154 97)(79 180 156 99)(82 181 157 104)(84 183 159 102)(85 186 162 105)(87 188 164 107)(90 189 165 112)(92 191 167 110)
G:=sub<Sym(192)| (1,59,54)(2,60,55)(3,57,56)(4,58,53)(5,189,186)(6,190,187)(7,191,188)(8,192,185)(9,17,14)(10,18,15)(11,19,16)(12,20,13)(21,29,28)(22,30,25)(23,31,26)(24,32,27)(33,41,38)(34,42,39)(35,43,40)(36,44,37)(45,149,148)(46,150,145)(47,151,146)(48,152,147)(49,101,98)(50,102,99)(51,103,100)(52,104,97)(61,69,66)(62,70,67)(63,71,68)(64,72,65)(73,81,80)(74,82,77)(75,83,78)(76,84,79)(85,93,90)(86,94,91)(87,95,92)(88,96,89)(105,113,112)(106,114,109)(107,115,110)(108,116,111)(117,139,134)(118,140,135)(119,137,136)(120,138,133)(121,129,128)(122,130,125)(123,131,126)(124,132,127)(141,157,154)(142,158,155)(143,159,156)(144,160,153)(161,169,168)(162,170,165)(163,171,166)(164,172,167)(173,181,178)(174,182,179)(175,183,180)(176,184,177), (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), (1,188,50,46)(2,185,51,47)(3,186,52,48)(4,187,49,45)(5,104,152,57)(6,101,149,58)(7,102,150,59)(8,103,151,60)(9,87,143,135)(10,88,144,136)(11,85,141,133)(12,86,142,134)(13,91,155,139)(14,92,156,140)(15,89,153,137)(16,90,154,138)(17,95,159,118)(18,96,160,119)(19,93,157,120)(20,94,158,117)(21,123,163,75)(22,124,164,76)(23,121,161,73)(24,122,162,74)(25,127,167,79)(26,128,168,80)(27,125,165,77)(28,126,166,78)(29,131,171,83)(30,132,172,84)(31,129,169,81)(32,130,170,82)(33,107,175,62)(34,108,176,63)(35,105,173,64)(36,106,174,61)(37,109,179,66)(38,110,180,67)(39,111,177,68)(40,112,178,65)(41,115,183,70)(42,116,184,71)(43,113,181,72)(44,114,182,69)(53,190,98,148)(54,191,99,145)(55,192,100,146)(56,189,97,147), (1,124,33,9)(2,73,34,144)(3,122,35,11)(4,75,36,142)(5,170,113,93)(6,29,114,117)(7,172,115,95)(8,31,116,119)(10,51,121,176)(12,49,123,174)(13,98,126,179)(14,54,127,38)(15,100,128,177)(16,56,125,40)(17,59,132,41)(18,103,129,184)(19,57,130,43)(20,101,131,182)(21,106,134,187)(22,62,135,46)(23,108,136,185)(24,64,133,48)(25,67,140,145)(26,111,137,192)(27,65,138,147)(28,109,139,190)(30,70,118,150)(32,72,120,152)(37,155,53,78)(39,153,55,80)(42,160,60,81)(44,158,58,83)(45,163,61,86)(47,161,63,88)(50,76,175,143)(52,74,173,141)(66,91,148,166)(68,89,146,168)(69,94,149,171)(71,96,151,169)(77,178,154,97)(79,180,156,99)(82,181,157,104)(84,183,159,102)(85,186,162,105)(87,188,164,107)(90,189,165,112)(92,191,167,110)>;
G:=Group( (1,59,54)(2,60,55)(3,57,56)(4,58,53)(5,189,186)(6,190,187)(7,191,188)(8,192,185)(9,17,14)(10,18,15)(11,19,16)(12,20,13)(21,29,28)(22,30,25)(23,31,26)(24,32,27)(33,41,38)(34,42,39)(35,43,40)(36,44,37)(45,149,148)(46,150,145)(47,151,146)(48,152,147)(49,101,98)(50,102,99)(51,103,100)(52,104,97)(61,69,66)(62,70,67)(63,71,68)(64,72,65)(73,81,80)(74,82,77)(75,83,78)(76,84,79)(85,93,90)(86,94,91)(87,95,92)(88,96,89)(105,113,112)(106,114,109)(107,115,110)(108,116,111)(117,139,134)(118,140,135)(119,137,136)(120,138,133)(121,129,128)(122,130,125)(123,131,126)(124,132,127)(141,157,154)(142,158,155)(143,159,156)(144,160,153)(161,169,168)(162,170,165)(163,171,166)(164,172,167)(173,181,178)(174,182,179)(175,183,180)(176,184,177), (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), (1,188,50,46)(2,185,51,47)(3,186,52,48)(4,187,49,45)(5,104,152,57)(6,101,149,58)(7,102,150,59)(8,103,151,60)(9,87,143,135)(10,88,144,136)(11,85,141,133)(12,86,142,134)(13,91,155,139)(14,92,156,140)(15,89,153,137)(16,90,154,138)(17,95,159,118)(18,96,160,119)(19,93,157,120)(20,94,158,117)(21,123,163,75)(22,124,164,76)(23,121,161,73)(24,122,162,74)(25,127,167,79)(26,128,168,80)(27,125,165,77)(28,126,166,78)(29,131,171,83)(30,132,172,84)(31,129,169,81)(32,130,170,82)(33,107,175,62)(34,108,176,63)(35,105,173,64)(36,106,174,61)(37,109,179,66)(38,110,180,67)(39,111,177,68)(40,112,178,65)(41,115,183,70)(42,116,184,71)(43,113,181,72)(44,114,182,69)(53,190,98,148)(54,191,99,145)(55,192,100,146)(56,189,97,147), (1,124,33,9)(2,73,34,144)(3,122,35,11)(4,75,36,142)(5,170,113,93)(6,29,114,117)(7,172,115,95)(8,31,116,119)(10,51,121,176)(12,49,123,174)(13,98,126,179)(14,54,127,38)(15,100,128,177)(16,56,125,40)(17,59,132,41)(18,103,129,184)(19,57,130,43)(20,101,131,182)(21,106,134,187)(22,62,135,46)(23,108,136,185)(24,64,133,48)(25,67,140,145)(26,111,137,192)(27,65,138,147)(28,109,139,190)(30,70,118,150)(32,72,120,152)(37,155,53,78)(39,153,55,80)(42,160,60,81)(44,158,58,83)(45,163,61,86)(47,161,63,88)(50,76,175,143)(52,74,173,141)(66,91,148,166)(68,89,146,168)(69,94,149,171)(71,96,151,169)(77,178,154,97)(79,180,156,99)(82,181,157,104)(84,183,159,102)(85,186,162,105)(87,188,164,107)(90,189,165,112)(92,191,167,110) );
G=PermutationGroup([[(1,59,54),(2,60,55),(3,57,56),(4,58,53),(5,189,186),(6,190,187),(7,191,188),(8,192,185),(9,17,14),(10,18,15),(11,19,16),(12,20,13),(21,29,28),(22,30,25),(23,31,26),(24,32,27),(33,41,38),(34,42,39),(35,43,40),(36,44,37),(45,149,148),(46,150,145),(47,151,146),(48,152,147),(49,101,98),(50,102,99),(51,103,100),(52,104,97),(61,69,66),(62,70,67),(63,71,68),(64,72,65),(73,81,80),(74,82,77),(75,83,78),(76,84,79),(85,93,90),(86,94,91),(87,95,92),(88,96,89),(105,113,112),(106,114,109),(107,115,110),(108,116,111),(117,139,134),(118,140,135),(119,137,136),(120,138,133),(121,129,128),(122,130,125),(123,131,126),(124,132,127),(141,157,154),(142,158,155),(143,159,156),(144,160,153),(161,169,168),(162,170,165),(163,171,166),(164,172,167),(173,181,178),(174,182,179),(175,183,180),(176,184,177)], [(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)], [(1,188,50,46),(2,185,51,47),(3,186,52,48),(4,187,49,45),(5,104,152,57),(6,101,149,58),(7,102,150,59),(8,103,151,60),(9,87,143,135),(10,88,144,136),(11,85,141,133),(12,86,142,134),(13,91,155,139),(14,92,156,140),(15,89,153,137),(16,90,154,138),(17,95,159,118),(18,96,160,119),(19,93,157,120),(20,94,158,117),(21,123,163,75),(22,124,164,76),(23,121,161,73),(24,122,162,74),(25,127,167,79),(26,128,168,80),(27,125,165,77),(28,126,166,78),(29,131,171,83),(30,132,172,84),(31,129,169,81),(32,130,170,82),(33,107,175,62),(34,108,176,63),(35,105,173,64),(36,106,174,61),(37,109,179,66),(38,110,180,67),(39,111,177,68),(40,112,178,65),(41,115,183,70),(42,116,184,71),(43,113,181,72),(44,114,182,69),(53,190,98,148),(54,191,99,145),(55,192,100,146),(56,189,97,147)], [(1,124,33,9),(2,73,34,144),(3,122,35,11),(4,75,36,142),(5,170,113,93),(6,29,114,117),(7,172,115,95),(8,31,116,119),(10,51,121,176),(12,49,123,174),(13,98,126,179),(14,54,127,38),(15,100,128,177),(16,56,125,40),(17,59,132,41),(18,103,129,184),(19,57,130,43),(20,101,131,182),(21,106,134,187),(22,62,135,46),(23,108,136,185),(24,64,133,48),(25,67,140,145),(26,111,137,192),(27,65,138,147),(28,109,139,190),(30,70,118,150),(32,72,120,152),(37,155,53,78),(39,153,55,80),(42,160,60,81),(44,158,58,83),(45,163,61,86),(47,161,63,88),(50,76,175,143),(52,74,173,141),(66,91,148,166),(68,89,146,168),(69,94,149,171),(71,96,151,169),(77,178,154,97),(79,180,156,99),(82,181,157,104),(84,183,159,102),(85,186,162,105),(87,188,164,107),(90,189,165,112),(92,191,167,110)]])
120 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4AF | 6A | ··· | 6N | 12A | ··· | 12P | 12Q | ··· | 12BL |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
120 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⋊4C4 | C3×C2.C42 | C2×C4×C12 | C42⋊4C4 | C4×C12 | C2.C42 | C2×C42 | C42 | C2×C6 | C22 |
| # reps | 1 | 4 | 3 | 2 | 24 | 8 | 6 | 48 | 8 | 16 |
Matrix representation of C3×C42⋊4C4 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 8 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 8 |
| 0 | 0 | 3 | 4 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 5 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 5 | 2 |
| 0 | 0 | 1 | 8 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[8,0,0,0,0,1,0,0,0,0,9,3,0,0,8,4],[12,0,0,0,0,12,0,0,0,0,8,0,0,0,0,8],[5,0,0,0,0,8,0,0,0,0,5,1,0,0,2,8] >;
C3×C42⋊4C4 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes_4C_4
% in TeX
G:=Group("C3xC4^2:4C4"); // GroupNames label
G:=SmallGroup(192,809);
// by ID
G=gap.SmallGroup(192,809);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,680,142]);
// 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,c*d=d*c>;
// generators/relations