direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C42.C2, C4.9(C6×Q8), (C2×C12).80Q8, C12.98(C2×Q8), (C2×C42).21C6, C42.88(C2×C6), C22.18(C6×Q8), C6.58(C22×Q8), (C2×C6).347C24, (C2×C12).660C23, (C4×C12).372C22, C23.75(C22×C6), C22.21(C23×C6), (C22×C6).469C23, (C22×C12).509C22, C2.4(Q8×C2×C6), (C2×C4×C12).41C2, (C6×C4⋊C4).47C2, (C2×C4⋊C4).18C6, 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
Generators and relations for C6×C42.C2
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 >
Subgroups: 274 in 226 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C2×C12, C2×C12, C22×C6, C2×C42, C2×C4⋊C4, C42.C2, C4×C12, C3×C4⋊C4, C22×C12, C22×C12, C2×C42.C2, C2×C4×C12, C6×C4⋊C4, C3×C42.C2, C6×C42.C2
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, C2×Q8, C4○D4, C24, C3×Q8, C22×C6, C42.C2, C22×Q8, C2×C4○D4, C6×Q8, C3×C4○D4, C23×C6, C2×C42.C2, C3×C42.C2, Q8×C2×C6, C6×C4○D4, C6×C42.C2
►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 83 37 89)(2 84 38 90)(3 79 39 85)(4 80 40 86)(5 81 41 87)(6 82 42 88)(7 160 13 154)(8 161 14 155)(9 162 15 156)(10 157 16 151)(11 158 17 152)(12 159 18 153)(19 141 192 147)(20 142 187 148)(21 143 188 149)(22 144 189 150)(23 139 190 145)(24 140 191 146)(25 70 31 76)(26 71 32 77)(27 72 33 78)(28 67 34 73)(29 68 35 74)(30 69 36 75)(43 112 49 106)(44 113 50 107)(45 114 51 108)(46 109 52 103)(47 110 53 104)(48 111 54 105)(55 100 61 94)(56 101 62 95)(57 102 63 96)(58 97 64 91)(59 98 65 92)(60 99 66 93)(115 166 121 172)(116 167 122 173)(117 168 123 174)(118 163 124 169)(119 164 125 170)(120 165 126 171)(127 178 133 184)(128 179 134 185)(129 180 135 186)(130 175 136 181)(131 176 137 182)(132 177 138 183)
(1 47 32 65)(2 48 33 66)(3 43 34 61)(4 44 35 62)(5 45 36 63)(6 46 31 64)(7 172 20 178)(8 173 21 179)(9 174 22 180)(10 169 23 175)(11 170 24 176)(12 171 19 177)(13 166 187 184)(14 167 188 185)(15 168 189 186)(16 163 190 181)(17 164 191 182)(18 165 192 183)(25 58 42 52)(26 59 37 53)(27 60 38 54)(28 55 39 49)(29 56 40 50)(30 57 41 51)(67 100 85 106)(68 101 86 107)(69 102 87 108)(70 97 88 103)(71 98 89 104)(72 99 90 105)(73 94 79 112)(74 95 80 113)(75 96 81 114)(76 91 82 109)(77 92 83 110)(78 93 84 111)(115 142 133 160)(116 143 134 161)(117 144 135 162)(118 139 136 157)(119 140 137 158)(120 141 138 159)(121 148 127 154)(122 149 128 155)(123 150 129 156)(124 145 130 151)(125 146 131 152)(126 147 132 153)
(1 119 32 137)(2 120 33 138)(3 115 34 133)(4 116 35 134)(5 117 36 135)(6 118 31 136)(7 94 20 112)(8 95 21 113)(9 96 22 114)(10 91 23 109)(11 92 24 110)(12 93 19 111)(13 100 187 106)(14 101 188 107)(15 102 189 108)(16 97 190 103)(17 98 191 104)(18 99 192 105)(25 130 42 124)(26 131 37 125)(27 132 38 126)(28 127 39 121)(29 128 40 122)(30 129 41 123)(43 148 61 154)(44 149 62 155)(45 150 63 156)(46 145 64 151)(47 146 65 152)(48 147 66 153)(49 142 55 160)(50 143 56 161)(51 144 57 162)(52 139 58 157)(53 140 59 158)(54 141 60 159)(67 172 85 178)(68 173 86 179)(69 174 87 180)(70 169 88 175)(71 170 89 176)(72 171 90 177)(73 166 79 184)(74 167 80 185)(75 168 81 186)(76 163 82 181)(77 164 83 182)(78 165 84 183)
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), (1,83,37,89)(2,84,38,90)(3,79,39,85)(4,80,40,86)(5,81,41,87)(6,82,42,88)(7,160,13,154)(8,161,14,155)(9,162,15,156)(10,157,16,151)(11,158,17,152)(12,159,18,153)(19,141,192,147)(20,142,187,148)(21,143,188,149)(22,144,189,150)(23,139,190,145)(24,140,191,146)(25,70,31,76)(26,71,32,77)(27,72,33,78)(28,67,34,73)(29,68,35,74)(30,69,36,75)(43,112,49,106)(44,113,50,107)(45,114,51,108)(46,109,52,103)(47,110,53,104)(48,111,54,105)(55,100,61,94)(56,101,62,95)(57,102,63,96)(58,97,64,91)(59,98,65,92)(60,99,66,93)(115,166,121,172)(116,167,122,173)(117,168,123,174)(118,163,124,169)(119,164,125,170)(120,165,126,171)(127,178,133,184)(128,179,134,185)(129,180,135,186)(130,175,136,181)(131,176,137,182)(132,177,138,183), (1,47,32,65)(2,48,33,66)(3,43,34,61)(4,44,35,62)(5,45,36,63)(6,46,31,64)(7,172,20,178)(8,173,21,179)(9,174,22,180)(10,169,23,175)(11,170,24,176)(12,171,19,177)(13,166,187,184)(14,167,188,185)(15,168,189,186)(16,163,190,181)(17,164,191,182)(18,165,192,183)(25,58,42,52)(26,59,37,53)(27,60,38,54)(28,55,39,49)(29,56,40,50)(30,57,41,51)(67,100,85,106)(68,101,86,107)(69,102,87,108)(70,97,88,103)(71,98,89,104)(72,99,90,105)(73,94,79,112)(74,95,80,113)(75,96,81,114)(76,91,82,109)(77,92,83,110)(78,93,84,111)(115,142,133,160)(116,143,134,161)(117,144,135,162)(118,139,136,157)(119,140,137,158)(120,141,138,159)(121,148,127,154)(122,149,128,155)(123,150,129,156)(124,145,130,151)(125,146,131,152)(126,147,132,153), (1,119,32,137)(2,120,33,138)(3,115,34,133)(4,116,35,134)(5,117,36,135)(6,118,31,136)(7,94,20,112)(8,95,21,113)(9,96,22,114)(10,91,23,109)(11,92,24,110)(12,93,19,111)(13,100,187,106)(14,101,188,107)(15,102,189,108)(16,97,190,103)(17,98,191,104)(18,99,192,105)(25,130,42,124)(26,131,37,125)(27,132,38,126)(28,127,39,121)(29,128,40,122)(30,129,41,123)(43,148,61,154)(44,149,62,155)(45,150,63,156)(46,145,64,151)(47,146,65,152)(48,147,66,153)(49,142,55,160)(50,143,56,161)(51,144,57,162)(52,139,58,157)(53,140,59,158)(54,141,60,159)(67,172,85,178)(68,173,86,179)(69,174,87,180)(70,169,88,175)(71,170,89,176)(72,171,90,177)(73,166,79,184)(74,167,80,185)(75,168,81,186)(76,163,82,181)(77,164,83,182)(78,165,84,183)>;
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), (1,83,37,89)(2,84,38,90)(3,79,39,85)(4,80,40,86)(5,81,41,87)(6,82,42,88)(7,160,13,154)(8,161,14,155)(9,162,15,156)(10,157,16,151)(11,158,17,152)(12,159,18,153)(19,141,192,147)(20,142,187,148)(21,143,188,149)(22,144,189,150)(23,139,190,145)(24,140,191,146)(25,70,31,76)(26,71,32,77)(27,72,33,78)(28,67,34,73)(29,68,35,74)(30,69,36,75)(43,112,49,106)(44,113,50,107)(45,114,51,108)(46,109,52,103)(47,110,53,104)(48,111,54,105)(55,100,61,94)(56,101,62,95)(57,102,63,96)(58,97,64,91)(59,98,65,92)(60,99,66,93)(115,166,121,172)(116,167,122,173)(117,168,123,174)(118,163,124,169)(119,164,125,170)(120,165,126,171)(127,178,133,184)(128,179,134,185)(129,180,135,186)(130,175,136,181)(131,176,137,182)(132,177,138,183), (1,47,32,65)(2,48,33,66)(3,43,34,61)(4,44,35,62)(5,45,36,63)(6,46,31,64)(7,172,20,178)(8,173,21,179)(9,174,22,180)(10,169,23,175)(11,170,24,176)(12,171,19,177)(13,166,187,184)(14,167,188,185)(15,168,189,186)(16,163,190,181)(17,164,191,182)(18,165,192,183)(25,58,42,52)(26,59,37,53)(27,60,38,54)(28,55,39,49)(29,56,40,50)(30,57,41,51)(67,100,85,106)(68,101,86,107)(69,102,87,108)(70,97,88,103)(71,98,89,104)(72,99,90,105)(73,94,79,112)(74,95,80,113)(75,96,81,114)(76,91,82,109)(77,92,83,110)(78,93,84,111)(115,142,133,160)(116,143,134,161)(117,144,135,162)(118,139,136,157)(119,140,137,158)(120,141,138,159)(121,148,127,154)(122,149,128,155)(123,150,129,156)(124,145,130,151)(125,146,131,152)(126,147,132,153), (1,119,32,137)(2,120,33,138)(3,115,34,133)(4,116,35,134)(5,117,36,135)(6,118,31,136)(7,94,20,112)(8,95,21,113)(9,96,22,114)(10,91,23,109)(11,92,24,110)(12,93,19,111)(13,100,187,106)(14,101,188,107)(15,102,189,108)(16,97,190,103)(17,98,191,104)(18,99,192,105)(25,130,42,124)(26,131,37,125)(27,132,38,126)(28,127,39,121)(29,128,40,122)(30,129,41,123)(43,148,61,154)(44,149,62,155)(45,150,63,156)(46,145,64,151)(47,146,65,152)(48,147,66,153)(49,142,55,160)(50,143,56,161)(51,144,57,162)(52,139,58,157)(53,140,59,158)(54,141,60,159)(67,172,85,178)(68,173,86,179)(69,174,87,180)(70,169,88,175)(71,170,89,176)(72,171,90,177)(73,166,79,184)(74,167,80,185)(75,168,81,186)(76,163,82,181)(77,164,83,182)(78,165,84,183) );
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)], [(1,83,37,89),(2,84,38,90),(3,79,39,85),(4,80,40,86),(5,81,41,87),(6,82,42,88),(7,160,13,154),(8,161,14,155),(9,162,15,156),(10,157,16,151),(11,158,17,152),(12,159,18,153),(19,141,192,147),(20,142,187,148),(21,143,188,149),(22,144,189,150),(23,139,190,145),(24,140,191,146),(25,70,31,76),(26,71,32,77),(27,72,33,78),(28,67,34,73),(29,68,35,74),(30,69,36,75),(43,112,49,106),(44,113,50,107),(45,114,51,108),(46,109,52,103),(47,110,53,104),(48,111,54,105),(55,100,61,94),(56,101,62,95),(57,102,63,96),(58,97,64,91),(59,98,65,92),(60,99,66,93),(115,166,121,172),(116,167,122,173),(117,168,123,174),(118,163,124,169),(119,164,125,170),(120,165,126,171),(127,178,133,184),(128,179,134,185),(129,180,135,186),(130,175,136,181),(131,176,137,182),(132,177,138,183)], [(1,47,32,65),(2,48,33,66),(3,43,34,61),(4,44,35,62),(5,45,36,63),(6,46,31,64),(7,172,20,178),(8,173,21,179),(9,174,22,180),(10,169,23,175),(11,170,24,176),(12,171,19,177),(13,166,187,184),(14,167,188,185),(15,168,189,186),(16,163,190,181),(17,164,191,182),(18,165,192,183),(25,58,42,52),(26,59,37,53),(27,60,38,54),(28,55,39,49),(29,56,40,50),(30,57,41,51),(67,100,85,106),(68,101,86,107),(69,102,87,108),(70,97,88,103),(71,98,89,104),(72,99,90,105),(73,94,79,112),(74,95,80,113),(75,96,81,114),(76,91,82,109),(77,92,83,110),(78,93,84,111),(115,142,133,160),(116,143,134,161),(117,144,135,162),(118,139,136,157),(119,140,137,158),(120,141,138,159),(121,148,127,154),(122,149,128,155),(123,150,129,156),(124,145,130,151),(125,146,131,152),(126,147,132,153)], [(1,119,32,137),(2,120,33,138),(3,115,34,133),(4,116,35,134),(5,117,36,135),(6,118,31,136),(7,94,20,112),(8,95,21,113),(9,96,22,114),(10,91,23,109),(11,92,24,110),(12,93,19,111),(13,100,187,106),(14,101,188,107),(15,102,189,108),(16,97,190,103),(17,98,191,104),(18,99,192,105),(25,130,42,124),(26,131,37,125),(27,132,38,126),(28,127,39,121),(29,128,40,122),(30,129,41,123),(43,148,61,154),(44,149,62,155),(45,150,63,156),(46,145,64,151),(47,146,65,152),(48,147,66,153),(49,142,55,160),(50,143,56,161),(51,144,57,162),(52,139,58,157),(53,140,59,158),(54,141,60,159),(67,172,85,178),(68,173,86,179),(69,174,87,180),(70,169,88,175),(71,170,89,176),(72,171,90,177),(73,166,79,184),(74,167,80,185),(75,168,81,186),(76,163,82,181),(77,164,83,182),(78,165,84,183)]])
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 |
Matrix representation of C6×C42.C2 ►in 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] >;
C6×C42.C2 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