direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C42.C2, C4.9(C6×Q8), C12.98(C2×Q8), (C2×C12).80Q8, (C2×C42).21C6, C42.88(C2×C6), C6.58(C22×Q8), C22.18(C6×Q8), (C2×C6).347C24, (C2×C12).660C23, (C4×C12).372C22, C22.21(C23×C6), C23.75(C22×C6), (C22×C6).469C23, (C22×C12).509C22, C2.4(Q8×C2×C6), (C2×C4×C12).41C2, (C2×C4⋊C4).18C6, (C6×C4⋊C4).47C2, C4⋊C4.63(C2×C6), C2.10(C6×C4○D4), (C2×C4).22(C3×Q8), C6.229(C2×C4○D4), (C2×C6).116(C2×Q8), (C2×C4).15(C22×C6), C22.33(C3×C4○D4), (C2×C6).233(C4○D4), (C3×C4⋊C4).386C22, (C22×C4).108(C2×C6), SmallGroup(192,1416)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 274 in 226 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C4 [×12], C22, C22 [×6], C6, C6 [×6], C2×C4 [×18], C2×C4 [×12], C23, C12 [×4], C12 [×12], C2×C6, C2×C6 [×6], C42 [×4], C4⋊C4 [×24], C22×C4, C22×C4 [×6], C2×C12 [×18], C2×C12 [×12], C22×C6, C2×C42, C2×C4⋊C4 [×6], C42.C2 [×8], C4×C12 [×4], C3×C4⋊C4 [×24], C22×C12, C22×C12 [×6], C2×C42.C2, C2×C4×C12, C6×C4⋊C4 [×6], C3×C42.C2 [×8], C6×C42.C2
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], Q8 [×4], C23 [×15], C2×C6 [×35], C2×Q8 [×6], C4○D4 [×4], C24, C3×Q8 [×4], C22×C6 [×15], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], C6×Q8 [×6], C3×C4○D4 [×4], C23×C6, C2×C42.C2, C3×C42.C2 [×4], Q8×C2×C6, C6×C4○D4 [×2], C6×C42.C2
Generators and relations
G = < a,b,c,d | a6=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >
►Regular action on 192 points
Generators in S192
(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 90 26 82)(2 85 27 83)(3 86 28 84)(4 87 29 79)(5 88 30 80)(6 89 25 81)(7 153 13 161)(8 154 14 162)(9 155 15 157)(10 156 16 158)(11 151 17 159)(12 152 18 160)(19 141 192 147)(20 142 187 148)(21 143 188 149)(22 144 189 150)(23 139 190 145)(24 140 191 146)(31 71 39 77)(32 72 40 78)(33 67 41 73)(34 68 42 74)(35 69 37 75)(36 70 38 76)(43 105 49 113)(44 106 50 114)(45 107 51 109)(46 108 52 110)(47 103 53 111)(48 104 54 112)(55 95 63 101)(56 96 64 102)(57 91 65 97)(58 92 66 98)(59 93 61 99)(60 94 62 100)(115 166 121 172)(116 167 122 173)(117 168 123 174)(118 163 124 169)(119 164 125 170)(120 165 126 171)(127 178 135 186)(128 179 136 181)(129 180 137 182)(130 175 138 183)(131 176 133 184)(132 177 134 185)
(1 47 39 58)(2 48 40 59)(3 43 41 60)(4 44 42 55)(5 45 37 56)(6 46 38 57)(7 172 20 185)(8 173 21 186)(9 174 22 181)(10 169 23 182)(11 170 24 183)(12 171 19 184)(13 166 187 177)(14 167 188 178)(15 168 189 179)(16 163 190 180)(17 164 191 175)(18 165 192 176)(25 52 36 65)(26 53 31 66)(27 54 32 61)(28 49 33 62)(29 50 34 63)(30 51 35 64)(67 100 84 113)(68 101 79 114)(69 102 80 109)(70 97 81 110)(71 98 82 111)(72 99 83 112)(73 94 86 105)(74 95 87 106)(75 96 88 107)(76 91 89 108)(77 92 90 103)(78 93 85 104)(115 142 132 153)(116 143 127 154)(117 144 128 155)(118 139 129 156)(119 140 130 151)(120 141 131 152)(121 148 134 161)(122 149 135 162)(123 150 136 157)(124 145 137 158)(125 146 138 159)(126 147 133 160)
(1 119 39 130)(2 120 40 131)(3 115 41 132)(4 116 42 127)(5 117 37 128)(6 118 38 129)(7 94 20 105)(8 95 21 106)(9 96 22 107)(10 91 23 108)(11 92 24 103)(12 93 19 104)(13 100 187 113)(14 101 188 114)(15 102 189 109)(16 97 190 110)(17 98 191 111)(18 99 192 112)(25 124 36 137)(26 125 31 138)(27 126 32 133)(28 121 33 134)(29 122 34 135)(30 123 35 136)(43 148 60 161)(44 149 55 162)(45 150 56 157)(46 145 57 158)(47 146 58 159)(48 147 59 160)(49 142 62 153)(50 143 63 154)(51 144 64 155)(52 139 65 156)(53 140 66 151)(54 141 61 152)(67 172 84 185)(68 173 79 186)(69 174 80 181)(70 169 81 182)(71 170 82 183)(72 171 83 184)(73 166 86 177)(74 167 87 178)(75 168 88 179)(76 163 89 180)(77 164 90 175)(78 165 85 176)
G:=sub<Sym(192)| (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), (1,90,26,82)(2,85,27,83)(3,86,28,84)(4,87,29,79)(5,88,30,80)(6,89,25,81)(7,153,13,161)(8,154,14,162)(9,155,15,157)(10,156,16,158)(11,151,17,159)(12,152,18,160)(19,141,192,147)(20,142,187,148)(21,143,188,149)(22,144,189,150)(23,139,190,145)(24,140,191,146)(31,71,39,77)(32,72,40,78)(33,67,41,73)(34,68,42,74)(35,69,37,75)(36,70,38,76)(43,105,49,113)(44,106,50,114)(45,107,51,109)(46,108,52,110)(47,103,53,111)(48,104,54,112)(55,95,63,101)(56,96,64,102)(57,91,65,97)(58,92,66,98)(59,93,61,99)(60,94,62,100)(115,166,121,172)(116,167,122,173)(117,168,123,174)(118,163,124,169)(119,164,125,170)(120,165,126,171)(127,178,135,186)(128,179,136,181)(129,180,137,182)(130,175,138,183)(131,176,133,184)(132,177,134,185), (1,47,39,58)(2,48,40,59)(3,43,41,60)(4,44,42,55)(5,45,37,56)(6,46,38,57)(7,172,20,185)(8,173,21,186)(9,174,22,181)(10,169,23,182)(11,170,24,183)(12,171,19,184)(13,166,187,177)(14,167,188,178)(15,168,189,179)(16,163,190,180)(17,164,191,175)(18,165,192,176)(25,52,36,65)(26,53,31,66)(27,54,32,61)(28,49,33,62)(29,50,34,63)(30,51,35,64)(67,100,84,113)(68,101,79,114)(69,102,80,109)(70,97,81,110)(71,98,82,111)(72,99,83,112)(73,94,86,105)(74,95,87,106)(75,96,88,107)(76,91,89,108)(77,92,90,103)(78,93,85,104)(115,142,132,153)(116,143,127,154)(117,144,128,155)(118,139,129,156)(119,140,130,151)(120,141,131,152)(121,148,134,161)(122,149,135,162)(123,150,136,157)(124,145,137,158)(125,146,138,159)(126,147,133,160), (1,119,39,130)(2,120,40,131)(3,115,41,132)(4,116,42,127)(5,117,37,128)(6,118,38,129)(7,94,20,105)(8,95,21,106)(9,96,22,107)(10,91,23,108)(11,92,24,103)(12,93,19,104)(13,100,187,113)(14,101,188,114)(15,102,189,109)(16,97,190,110)(17,98,191,111)(18,99,192,112)(25,124,36,137)(26,125,31,138)(27,126,32,133)(28,121,33,134)(29,122,34,135)(30,123,35,136)(43,148,60,161)(44,149,55,162)(45,150,56,157)(46,145,57,158)(47,146,58,159)(48,147,59,160)(49,142,62,153)(50,143,63,154)(51,144,64,155)(52,139,65,156)(53,140,66,151)(54,141,61,152)(67,172,84,185)(68,173,79,186)(69,174,80,181)(70,169,81,182)(71,170,82,183)(72,171,83,184)(73,166,86,177)(74,167,87,178)(75,168,88,179)(76,163,89,180)(77,164,90,175)(78,165,85,176)>;
G:=Group( (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), (1,90,26,82)(2,85,27,83)(3,86,28,84)(4,87,29,79)(5,88,30,80)(6,89,25,81)(7,153,13,161)(8,154,14,162)(9,155,15,157)(10,156,16,158)(11,151,17,159)(12,152,18,160)(19,141,192,147)(20,142,187,148)(21,143,188,149)(22,144,189,150)(23,139,190,145)(24,140,191,146)(31,71,39,77)(32,72,40,78)(33,67,41,73)(34,68,42,74)(35,69,37,75)(36,70,38,76)(43,105,49,113)(44,106,50,114)(45,107,51,109)(46,108,52,110)(47,103,53,111)(48,104,54,112)(55,95,63,101)(56,96,64,102)(57,91,65,97)(58,92,66,98)(59,93,61,99)(60,94,62,100)(115,166,121,172)(116,167,122,173)(117,168,123,174)(118,163,124,169)(119,164,125,170)(120,165,126,171)(127,178,135,186)(128,179,136,181)(129,180,137,182)(130,175,138,183)(131,176,133,184)(132,177,134,185), (1,47,39,58)(2,48,40,59)(3,43,41,60)(4,44,42,55)(5,45,37,56)(6,46,38,57)(7,172,20,185)(8,173,21,186)(9,174,22,181)(10,169,23,182)(11,170,24,183)(12,171,19,184)(13,166,187,177)(14,167,188,178)(15,168,189,179)(16,163,190,180)(17,164,191,175)(18,165,192,176)(25,52,36,65)(26,53,31,66)(27,54,32,61)(28,49,33,62)(29,50,34,63)(30,51,35,64)(67,100,84,113)(68,101,79,114)(69,102,80,109)(70,97,81,110)(71,98,82,111)(72,99,83,112)(73,94,86,105)(74,95,87,106)(75,96,88,107)(76,91,89,108)(77,92,90,103)(78,93,85,104)(115,142,132,153)(116,143,127,154)(117,144,128,155)(118,139,129,156)(119,140,130,151)(120,141,131,152)(121,148,134,161)(122,149,135,162)(123,150,136,157)(124,145,137,158)(125,146,138,159)(126,147,133,160), (1,119,39,130)(2,120,40,131)(3,115,41,132)(4,116,42,127)(5,117,37,128)(6,118,38,129)(7,94,20,105)(8,95,21,106)(9,96,22,107)(10,91,23,108)(11,92,24,103)(12,93,19,104)(13,100,187,113)(14,101,188,114)(15,102,189,109)(16,97,190,110)(17,98,191,111)(18,99,192,112)(25,124,36,137)(26,125,31,138)(27,126,32,133)(28,121,33,134)(29,122,34,135)(30,123,35,136)(43,148,60,161)(44,149,55,162)(45,150,56,157)(46,145,57,158)(47,146,58,159)(48,147,59,160)(49,142,62,153)(50,143,63,154)(51,144,64,155)(52,139,65,156)(53,140,66,151)(54,141,61,152)(67,172,84,185)(68,173,79,186)(69,174,80,181)(70,169,81,182)(71,170,82,183)(72,171,83,184)(73,166,86,177)(74,167,87,178)(75,168,88,179)(76,163,89,180)(77,164,90,175)(78,165,85,176) );
G=PermutationGroup([(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)], [(1,90,26,82),(2,85,27,83),(3,86,28,84),(4,87,29,79),(5,88,30,80),(6,89,25,81),(7,153,13,161),(8,154,14,162),(9,155,15,157),(10,156,16,158),(11,151,17,159),(12,152,18,160),(19,141,192,147),(20,142,187,148),(21,143,188,149),(22,144,189,150),(23,139,190,145),(24,140,191,146),(31,71,39,77),(32,72,40,78),(33,67,41,73),(34,68,42,74),(35,69,37,75),(36,70,38,76),(43,105,49,113),(44,106,50,114),(45,107,51,109),(46,108,52,110),(47,103,53,111),(48,104,54,112),(55,95,63,101),(56,96,64,102),(57,91,65,97),(58,92,66,98),(59,93,61,99),(60,94,62,100),(115,166,121,172),(116,167,122,173),(117,168,123,174),(118,163,124,169),(119,164,125,170),(120,165,126,171),(127,178,135,186),(128,179,136,181),(129,180,137,182),(130,175,138,183),(131,176,133,184),(132,177,134,185)], [(1,47,39,58),(2,48,40,59),(3,43,41,60),(4,44,42,55),(5,45,37,56),(6,46,38,57),(7,172,20,185),(8,173,21,186),(9,174,22,181),(10,169,23,182),(11,170,24,183),(12,171,19,184),(13,166,187,177),(14,167,188,178),(15,168,189,179),(16,163,190,180),(17,164,191,175),(18,165,192,176),(25,52,36,65),(26,53,31,66),(27,54,32,61),(28,49,33,62),(29,50,34,63),(30,51,35,64),(67,100,84,113),(68,101,79,114),(69,102,80,109),(70,97,81,110),(71,98,82,111),(72,99,83,112),(73,94,86,105),(74,95,87,106),(75,96,88,107),(76,91,89,108),(77,92,90,103),(78,93,85,104),(115,142,132,153),(116,143,127,154),(117,144,128,155),(118,139,129,156),(119,140,130,151),(120,141,131,152),(121,148,134,161),(122,149,135,162),(123,150,136,157),(124,145,137,158),(125,146,138,159),(126,147,133,160)], [(1,119,39,130),(2,120,40,131),(3,115,41,132),(4,116,42,127),(5,117,37,128),(6,118,38,129),(7,94,20,105),(8,95,21,106),(9,96,22,107),(10,91,23,108),(11,92,24,103),(12,93,19,104),(13,100,187,113),(14,101,188,114),(15,102,189,109),(16,97,190,110),(17,98,191,111),(18,99,192,112),(25,124,36,137),(26,125,31,138),(27,126,32,133),(28,121,33,134),(29,122,34,135),(30,123,35,136),(43,148,60,161),(44,149,55,162),(45,150,56,157),(46,145,57,158),(47,146,58,159),(48,147,59,160),(49,142,62,153),(50,143,63,154),(51,144,64,155),(52,139,65,156),(53,140,66,151),(54,141,61,152),(67,172,84,185),(68,173,79,186),(69,174,80,181),(70,169,81,182),(71,170,82,183),(72,171,83,184),(73,166,86,177),(74,167,87,178),(75,168,88,179),(76,163,89,180),(77,164,90,175),(78,165,85,176)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 12 | 0 | 0 |
| 0 | 4 | 9 | 0 | 0 |
| 0 | 0 | 0 | 2 | 9 |
| 0 | 0 | 0 | 4 | 11 |
G:=sub<GL(5,GF(13))| [4,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,0,0,0,0,11,12,0,0,0,0,0,5,0,0,0,0,0,5],[1,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,12,0],[1,0,0,0,0,0,4,4,0,0,0,12,9,0,0,0,0,0,2,4,0,0,0,9,11] >;
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 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | Q8 | C4○D4 | C3×Q8 | C3×C4○D4 |
| kernel | C6×C42.C2 | C2×C4×C12 | C6×C4⋊C4 | C3×C42.C2 | C2×C42.C2 | C2×C42 | C2×C4⋊C4 | C42.C2 | C2×C12 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 1 | 6 | 8 | 2 | 2 | 12 | 16 | 4 | 8 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_6\times C_4^2.C_2
% in TeX
G:=Group("C6xC4^2.C2"); // GroupNames label
G:=SmallGroup(192,1416);
// by ID
G=gap.SmallGroup(192,1416);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,680,2102,268]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^4=c^4=1,d^2=c^2,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>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀