metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊2Q16, C42.83D6, Dic6.24D4, C4⋊Q8.9S3, C4.57(S3×D4), C4⋊2(C3⋊Q16), C12.39(C2×D4), C3⋊4(C4⋊2Q16), (C2×Q8).71D6, C6.42(C2×Q16), (C2×C12).159D4, C12⋊C8.21C2, C12.85(C4○D4), C4.6(D4⋊2S3), (C4×Dic6).17C2, (C6×Q8).65C22, C2.15(D6⋊3D4), C6.106(C4⋊D4), (C2×C12).408C23, (C4×C12).137C22, Q8⋊2Dic3.13C2, C6.99(C8.C22), C4⋊Dic3.349C22, C2.20(Q8.11D6), (C2×Dic6).275C22, (C3×C4⋊Q8).9C2, (C2×C6).539(C2×D4), (C2×C3⋊Q16).6C2, C2.13(C2×C3⋊Q16), (C2×C3⋊C8).140C22, (C2×C4).190(C3⋊D4), (C2×C4).505(C22×S3), C22.211(C2×C3⋊D4), SmallGroup(192,649)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊Q16
G = < a,b,c | a12=b8=1, c2=b4, bab-1=a-1, cac-1=a7, cbc-1=b-1 >
Subgroups: 256 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3⋊Q16, C4×C12, C3×C4⋊C4, C2×Dic6, C6×Q8, C4⋊2Q16, C12⋊C8, Q8⋊2Dic3, C4×Dic6, C2×C3⋊Q16, C3×C4⋊Q8, C12⋊Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×Q16, C8.C22, C3⋊Q16, S3×D4, D4⋊2S3, C2×C3⋊D4, C4⋊2Q16, D6⋊3D4, Q8.11D6, C2×C3⋊Q16, C12⋊Q16
►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 58 133 29 21 170 75 93)(2 57 134 28 22 169 76 92)(3 56 135 27 23 180 77 91)(4 55 136 26 24 179 78 90)(5 54 137 25 13 178 79 89)(6 53 138 36 14 177 80 88)(7 52 139 35 15 176 81 87)(8 51 140 34 16 175 82 86)(9 50 141 33 17 174 83 85)(10 49 142 32 18 173 84 96)(11 60 143 31 19 172 73 95)(12 59 144 30 20 171 74 94)(37 104 163 125 153 113 189 70)(38 103 164 124 154 112 190 69)(39 102 165 123 155 111 191 68)(40 101 166 122 156 110 192 67)(41 100 167 121 145 109 181 66)(42 99 168 132 146 120 182 65)(43 98 157 131 147 119 183 64)(44 97 158 130 148 118 184 63)(45 108 159 129 149 117 185 62)(46 107 160 128 150 116 186 61)(47 106 161 127 151 115 187 72)(48 105 162 126 152 114 188 71)
(1 166 21 192)(2 161 22 187)(3 168 23 182)(4 163 24 189)(5 158 13 184)(6 165 14 191)(7 160 15 186)(8 167 16 181)(9 162 17 188)(10 157 18 183)(11 164 19 190)(12 159 20 185)(25 63 89 130)(26 70 90 125)(27 65 91 132)(28 72 92 127)(29 67 93 122)(30 62 94 129)(31 69 95 124)(32 64 96 131)(33 71 85 126)(34 66 86 121)(35 61 87 128)(36 68 88 123)(37 78 153 136)(38 73 154 143)(39 80 155 138)(40 75 156 133)(41 82 145 140)(42 77 146 135)(43 84 147 142)(44 79 148 137)(45 74 149 144)(46 81 150 139)(47 76 151 134)(48 83 152 141)(49 98 173 119)(50 105 174 114)(51 100 175 109)(52 107 176 116)(53 102 177 111)(54 97 178 118)(55 104 179 113)(56 99 180 120)(57 106 169 115)(58 101 170 110)(59 108 171 117)(60 103 172 112)
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), (1,58,133,29,21,170,75,93)(2,57,134,28,22,169,76,92)(3,56,135,27,23,180,77,91)(4,55,136,26,24,179,78,90)(5,54,137,25,13,178,79,89)(6,53,138,36,14,177,80,88)(7,52,139,35,15,176,81,87)(8,51,140,34,16,175,82,86)(9,50,141,33,17,174,83,85)(10,49,142,32,18,173,84,96)(11,60,143,31,19,172,73,95)(12,59,144,30,20,171,74,94)(37,104,163,125,153,113,189,70)(38,103,164,124,154,112,190,69)(39,102,165,123,155,111,191,68)(40,101,166,122,156,110,192,67)(41,100,167,121,145,109,181,66)(42,99,168,132,146,120,182,65)(43,98,157,131,147,119,183,64)(44,97,158,130,148,118,184,63)(45,108,159,129,149,117,185,62)(46,107,160,128,150,116,186,61)(47,106,161,127,151,115,187,72)(48,105,162,126,152,114,188,71), (1,166,21,192)(2,161,22,187)(3,168,23,182)(4,163,24,189)(5,158,13,184)(6,165,14,191)(7,160,15,186)(8,167,16,181)(9,162,17,188)(10,157,18,183)(11,164,19,190)(12,159,20,185)(25,63,89,130)(26,70,90,125)(27,65,91,132)(28,72,92,127)(29,67,93,122)(30,62,94,129)(31,69,95,124)(32,64,96,131)(33,71,85,126)(34,66,86,121)(35,61,87,128)(36,68,88,123)(37,78,153,136)(38,73,154,143)(39,80,155,138)(40,75,156,133)(41,82,145,140)(42,77,146,135)(43,84,147,142)(44,79,148,137)(45,74,149,144)(46,81,150,139)(47,76,151,134)(48,83,152,141)(49,98,173,119)(50,105,174,114)(51,100,175,109)(52,107,176,116)(53,102,177,111)(54,97,178,118)(55,104,179,113)(56,99,180,120)(57,106,169,115)(58,101,170,110)(59,108,171,117)(60,103,172,112)>;
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), (1,58,133,29,21,170,75,93)(2,57,134,28,22,169,76,92)(3,56,135,27,23,180,77,91)(4,55,136,26,24,179,78,90)(5,54,137,25,13,178,79,89)(6,53,138,36,14,177,80,88)(7,52,139,35,15,176,81,87)(8,51,140,34,16,175,82,86)(9,50,141,33,17,174,83,85)(10,49,142,32,18,173,84,96)(11,60,143,31,19,172,73,95)(12,59,144,30,20,171,74,94)(37,104,163,125,153,113,189,70)(38,103,164,124,154,112,190,69)(39,102,165,123,155,111,191,68)(40,101,166,122,156,110,192,67)(41,100,167,121,145,109,181,66)(42,99,168,132,146,120,182,65)(43,98,157,131,147,119,183,64)(44,97,158,130,148,118,184,63)(45,108,159,129,149,117,185,62)(46,107,160,128,150,116,186,61)(47,106,161,127,151,115,187,72)(48,105,162,126,152,114,188,71), (1,166,21,192)(2,161,22,187)(3,168,23,182)(4,163,24,189)(5,158,13,184)(6,165,14,191)(7,160,15,186)(8,167,16,181)(9,162,17,188)(10,157,18,183)(11,164,19,190)(12,159,20,185)(25,63,89,130)(26,70,90,125)(27,65,91,132)(28,72,92,127)(29,67,93,122)(30,62,94,129)(31,69,95,124)(32,64,96,131)(33,71,85,126)(34,66,86,121)(35,61,87,128)(36,68,88,123)(37,78,153,136)(38,73,154,143)(39,80,155,138)(40,75,156,133)(41,82,145,140)(42,77,146,135)(43,84,147,142)(44,79,148,137)(45,74,149,144)(46,81,150,139)(47,76,151,134)(48,83,152,141)(49,98,173,119)(50,105,174,114)(51,100,175,109)(52,107,176,116)(53,102,177,111)(54,97,178,118)(55,104,179,113)(56,99,180,120)(57,106,169,115)(58,101,170,110)(59,108,171,117)(60,103,172,112) );
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)], [(1,58,133,29,21,170,75,93),(2,57,134,28,22,169,76,92),(3,56,135,27,23,180,77,91),(4,55,136,26,24,179,78,90),(5,54,137,25,13,178,79,89),(6,53,138,36,14,177,80,88),(7,52,139,35,15,176,81,87),(8,51,140,34,16,175,82,86),(9,50,141,33,17,174,83,85),(10,49,142,32,18,173,84,96),(11,60,143,31,19,172,73,95),(12,59,144,30,20,171,74,94),(37,104,163,125,153,113,189,70),(38,103,164,124,154,112,190,69),(39,102,165,123,155,111,191,68),(40,101,166,122,156,110,192,67),(41,100,167,121,145,109,181,66),(42,99,168,132,146,120,182,65),(43,98,157,131,147,119,183,64),(44,97,158,130,148,118,184,63),(45,108,159,129,149,117,185,62),(46,107,160,128,150,116,186,61),(47,106,161,127,151,115,187,72),(48,105,162,126,152,114,188,71)], [(1,166,21,192),(2,161,22,187),(3,168,23,182),(4,163,24,189),(5,158,13,184),(6,165,14,191),(7,160,15,186),(8,167,16,181),(9,162,17,188),(10,157,18,183),(11,164,19,190),(12,159,20,185),(25,63,89,130),(26,70,90,125),(27,65,91,132),(28,72,92,127),(29,67,93,122),(30,62,94,129),(31,69,95,124),(32,64,96,131),(33,71,85,126),(34,66,86,121),(35,61,87,128),(36,68,88,123),(37,78,153,136),(38,73,154,143),(39,80,155,138),(40,75,156,133),(41,82,145,140),(42,77,146,135),(43,84,147,142),(44,79,148,137),(45,74,149,144),(46,81,150,139),(47,76,151,134),(48,83,152,141),(49,98,173,119),(50,105,174,114),(51,100,175,109),(52,107,176,116),(53,102,177,111),(54,97,178,118),(55,104,179,113),(56,99,180,120),(57,106,169,115),(58,101,170,110),(59,108,171,117),(60,103,172,112)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | Q16 | C4○D4 | C3⋊D4 | C8.C22 | C3⋊Q16 | S3×D4 | D4⋊2S3 | Q8.11D6 |
| kernel | C12⋊Q16 | C12⋊C8 | Q8⋊2Dic3 | C4×Dic6 | C2×C3⋊Q16 | C3×C4⋊Q8 | C4⋊Q8 | Dic6 | C2×C12 | C42 | C2×Q8 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 1 | 2 | 1 | 1 | 2 |
Matrix representation of C12⋊Q16 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 52 | 0 | 0 |
| 0 | 0 | 21 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 16 | 57 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 6 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,41,21,0,0,0,0,52,32,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,1,1],[6,6,0,0,0,0,6,67,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C12⋊Q16 in GAP, Magma, Sage, TeX
C_{12}\rtimes Q_{16} % in TeX
G:=Group("C12:Q16"); // GroupNames label
G:=SmallGroup(192,649);
// by ID
G=gap.SmallGroup(192,649);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=b^4,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=b^-1>;
// generators/relations