metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.1C42, C4⋊C4⋊3Dic3, (C2×C6).35D8, C12.6(C4⋊C4), (C2×C12).4Q8, C4⋊Dic3⋊10C4, (C2×C6).14Q16, C4.1(C4×Dic3), C6.4(C4.Q8), C6.5(C2.D8), C4.31(D6⋊C4), (C2×C12).102D4, (C2×C4).125D12, C4.1(C4⋊Dic3), (C2×C4).23Dic6, (C2×C6).38SD16, C12.7(C22⋊C4), C4.6(Dic3⋊C4), C3⋊1(C22.4Q16), C6.4(Q8⋊C4), C2.2(C6.Q16), C2.2(C6.D8), (C22×C4).339D6, (C22×C6).178D4, C6.21(D4⋊C4), C22.16(D4⋊S3), C22.38(D6⋊C4), C2.2(C6.SD16), C2.1(Q8⋊2Dic3), C2.1(D4⋊Dic3), C22.9(D4.S3), C2.2(C12.Q8), C22.6(C3⋊Q16), C23.101(C3⋊D4), C6.5(C2.C42), C2.6(C6.C42), C22.9(Q8⋊2S3), C22.20(Dic3⋊C4), (C22×C12).118C22, C22.26(C6.D4), (C2×C3⋊C8)⋊4C4, (C3×C4⋊C4)⋊5C4, (C6×C4⋊C4).1C2, (C2×C4⋊C4).2S3, (C22×C3⋊C8).1C2, (C2×C6).35(C4⋊C4), (C2×C4).138(C4×S3), (C2×C12).54(C2×C4), (C2×C4⋊Dic3).27C2, (C2×C4).35(C2×Dic3), (C2×C4).174(C3⋊D4), (C2×C6).87(C22⋊C4), SmallGroup(192,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.C42
G = < a,b,c | a12=c4=1, b4=a6, bab-1=a5, cac-1=a7, cbc-1=a3b >
Subgroups: 264 in 114 conjugacy classes, 67 normal (59 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C2×C3⋊C8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C22×C12, C22.4Q16, C22×C3⋊C8, C2×C4⋊Dic3, C6×C4⋊C4, C12.C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C6.D4, C22.4Q16, C6.Q16, C12.Q8, C6.D8, C6.SD16, C6.C42, D4⋊Dic3, Q8⋊2Dic3, C12.C42
►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 98 192 69 7 104 186 63)(2 103 181 62 8 97 187 68)(3 108 182 67 9 102 188 61)(4 101 183 72 10 107 189 66)(5 106 184 65 11 100 190 71)(6 99 185 70 12 105 191 64)(13 26 73 175 19 32 79 169)(14 31 74 180 20 25 80 174)(15 36 75 173 21 30 81 179)(16 29 76 178 22 35 82 172)(17 34 77 171 23 28 83 177)(18 27 78 176 24 33 84 170)(37 165 146 134 43 159 152 140)(38 158 147 139 44 164 153 133)(39 163 148 144 45 157 154 138)(40 168 149 137 46 162 155 143)(41 161 150 142 47 167 156 136)(42 166 151 135 48 160 145 141)(49 131 117 86 55 125 111 92)(50 124 118 91 56 130 112 85)(51 129 119 96 57 123 113 90)(52 122 120 89 58 128 114 95)(53 127 109 94 59 121 115 88)(54 132 110 87 60 126 116 93)
(1 90 145 77)(2 85 146 84)(3 92 147 79)(4 87 148 74)(5 94 149 81)(6 89 150 76)(7 96 151 83)(8 91 152 78)(9 86 153 73)(10 93 154 80)(11 88 155 75)(12 95 156 82)(13 188 131 38)(14 183 132 45)(15 190 121 40)(16 185 122 47)(17 192 123 42)(18 187 124 37)(19 182 125 44)(20 189 126 39)(21 184 127 46)(22 191 128 41)(23 186 129 48)(24 181 130 43)(25 69 116 166)(26 64 117 161)(27 71 118 168)(28 66 119 163)(29 61 120 158)(30 68 109 165)(31 63 110 160)(32 70 111 167)(33 65 112 162)(34 72 113 157)(35 67 114 164)(36 62 115 159)(49 142 169 99)(50 137 170 106)(51 144 171 101)(52 139 172 108)(53 134 173 103)(54 141 174 98)(55 136 175 105)(56 143 176 100)(57 138 177 107)(58 133 178 102)(59 140 179 97)(60 135 180 104)
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), (1,98,192,69,7,104,186,63)(2,103,181,62,8,97,187,68)(3,108,182,67,9,102,188,61)(4,101,183,72,10,107,189,66)(5,106,184,65,11,100,190,71)(6,99,185,70,12,105,191,64)(13,26,73,175,19,32,79,169)(14,31,74,180,20,25,80,174)(15,36,75,173,21,30,81,179)(16,29,76,178,22,35,82,172)(17,34,77,171,23,28,83,177)(18,27,78,176,24,33,84,170)(37,165,146,134,43,159,152,140)(38,158,147,139,44,164,153,133)(39,163,148,144,45,157,154,138)(40,168,149,137,46,162,155,143)(41,161,150,142,47,167,156,136)(42,166,151,135,48,160,145,141)(49,131,117,86,55,125,111,92)(50,124,118,91,56,130,112,85)(51,129,119,96,57,123,113,90)(52,122,120,89,58,128,114,95)(53,127,109,94,59,121,115,88)(54,132,110,87,60,126,116,93), (1,90,145,77)(2,85,146,84)(3,92,147,79)(4,87,148,74)(5,94,149,81)(6,89,150,76)(7,96,151,83)(8,91,152,78)(9,86,153,73)(10,93,154,80)(11,88,155,75)(12,95,156,82)(13,188,131,38)(14,183,132,45)(15,190,121,40)(16,185,122,47)(17,192,123,42)(18,187,124,37)(19,182,125,44)(20,189,126,39)(21,184,127,46)(22,191,128,41)(23,186,129,48)(24,181,130,43)(25,69,116,166)(26,64,117,161)(27,71,118,168)(28,66,119,163)(29,61,120,158)(30,68,109,165)(31,63,110,160)(32,70,111,167)(33,65,112,162)(34,72,113,157)(35,67,114,164)(36,62,115,159)(49,142,169,99)(50,137,170,106)(51,144,171,101)(52,139,172,108)(53,134,173,103)(54,141,174,98)(55,136,175,105)(56,143,176,100)(57,138,177,107)(58,133,178,102)(59,140,179,97)(60,135,180,104)>;
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), (1,98,192,69,7,104,186,63)(2,103,181,62,8,97,187,68)(3,108,182,67,9,102,188,61)(4,101,183,72,10,107,189,66)(5,106,184,65,11,100,190,71)(6,99,185,70,12,105,191,64)(13,26,73,175,19,32,79,169)(14,31,74,180,20,25,80,174)(15,36,75,173,21,30,81,179)(16,29,76,178,22,35,82,172)(17,34,77,171,23,28,83,177)(18,27,78,176,24,33,84,170)(37,165,146,134,43,159,152,140)(38,158,147,139,44,164,153,133)(39,163,148,144,45,157,154,138)(40,168,149,137,46,162,155,143)(41,161,150,142,47,167,156,136)(42,166,151,135,48,160,145,141)(49,131,117,86,55,125,111,92)(50,124,118,91,56,130,112,85)(51,129,119,96,57,123,113,90)(52,122,120,89,58,128,114,95)(53,127,109,94,59,121,115,88)(54,132,110,87,60,126,116,93), (1,90,145,77)(2,85,146,84)(3,92,147,79)(4,87,148,74)(5,94,149,81)(6,89,150,76)(7,96,151,83)(8,91,152,78)(9,86,153,73)(10,93,154,80)(11,88,155,75)(12,95,156,82)(13,188,131,38)(14,183,132,45)(15,190,121,40)(16,185,122,47)(17,192,123,42)(18,187,124,37)(19,182,125,44)(20,189,126,39)(21,184,127,46)(22,191,128,41)(23,186,129,48)(24,181,130,43)(25,69,116,166)(26,64,117,161)(27,71,118,168)(28,66,119,163)(29,61,120,158)(30,68,109,165)(31,63,110,160)(32,70,111,167)(33,65,112,162)(34,72,113,157)(35,67,114,164)(36,62,115,159)(49,142,169,99)(50,137,170,106)(51,144,171,101)(52,139,172,108)(53,134,173,103)(54,141,174,98)(55,136,175,105)(56,143,176,100)(57,138,177,107)(58,133,178,102)(59,140,179,97)(60,135,180,104) );
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)], [(1,98,192,69,7,104,186,63),(2,103,181,62,8,97,187,68),(3,108,182,67,9,102,188,61),(4,101,183,72,10,107,189,66),(5,106,184,65,11,100,190,71),(6,99,185,70,12,105,191,64),(13,26,73,175,19,32,79,169),(14,31,74,180,20,25,80,174),(15,36,75,173,21,30,81,179),(16,29,76,178,22,35,82,172),(17,34,77,171,23,28,83,177),(18,27,78,176,24,33,84,170),(37,165,146,134,43,159,152,140),(38,158,147,139,44,164,153,133),(39,163,148,144,45,157,154,138),(40,168,149,137,46,162,155,143),(41,161,150,142,47,167,156,136),(42,166,151,135,48,160,145,141),(49,131,117,86,55,125,111,92),(50,124,118,91,56,130,112,85),(51,129,119,96,57,123,113,90),(52,122,120,89,58,128,114,95),(53,127,109,94,59,121,115,88),(54,132,110,87,60,126,116,93)], [(1,90,145,77),(2,85,146,84),(3,92,147,79),(4,87,148,74),(5,94,149,81),(6,89,150,76),(7,96,151,83),(8,91,152,78),(9,86,153,73),(10,93,154,80),(11,88,155,75),(12,95,156,82),(13,188,131,38),(14,183,132,45),(15,190,121,40),(16,185,122,47),(17,192,123,42),(18,187,124,37),(19,182,125,44),(20,189,126,39),(21,184,127,46),(22,191,128,41),(23,186,129,48),(24,181,130,43),(25,69,116,166),(26,64,117,161),(27,71,118,168),(28,66,119,163),(29,61,120,158),(30,68,109,165),(31,63,110,160),(32,70,111,167),(33,65,112,162),(34,72,113,157),(35,67,114,164),(36,62,115,159),(49,142,169,99),(50,137,170,106),(51,144,171,101),(52,139,172,108),(53,134,173,103),(54,141,174,98),(55,136,175,105),(56,143,176,100),(57,138,177,107),(58,133,178,102),(59,140,179,97),(60,135,180,104)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6G | 8A | ··· | 8H | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | - | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | D4 | Dic3 | D6 | D8 | SD16 | Q16 | Dic6 | C4×S3 | D12 | C3⋊D4 | C3⋊D4 | D4⋊S3 | D4.S3 | Q8⋊2S3 | C3⋊Q16 |
| kernel | C12.C42 | C22×C3⋊C8 | C2×C4⋊Dic3 | C6×C4⋊C4 | C2×C3⋊C8 | C4⋊Dic3 | C3×C4⋊C4 | C2×C4⋊C4 | C2×C12 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×C6 | C2×C6 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 2 | 2 | 1 | 1 | 1 | 1 |
Matrix representation of C12.C42 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 71 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 28 | 0 | 0 |
| 0 | 0 | 70 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 32 |
| 0 | 0 | 0 | 0 | 57 | 0 |
| 7 | 14 | 0 | 0 | 0 | 0 |
| 59 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 60 | 0 | 0 |
| 0 | 0 | 13 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 18 |
| 0 | 0 | 0 | 0 | 8 | 1 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,1,0,0,0,0,71,72],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,31,70,0,0,0,0,28,42,0,0,0,0,0,0,41,57,0,0,0,0,32,0],[7,59,0,0,0,0,14,66,0,0,0,0,0,0,30,13,0,0,0,0,60,43,0,0,0,0,0,0,72,8,0,0,0,0,18,1] >;
C12.C42 in GAP, Magma, Sage, TeX
C_{12}.C_4^2 % in TeX
G:=Group("C12.C4^2"); // GroupNames label
G:=SmallGroup(192,88);
// by ID
G=gap.SmallGroup(192,88);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,365,36,570,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=c^4=1,b^4=a^6,b*a*b^-1=a^5,c*a*c^-1=a^7,c*b*c^-1=a^3*b>;
// generators/relations