metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.11Q16, C12.20SD16, C42.226D6, C4⋊C4.87D6, C4⋊Q8.12S3, C6.45(C2×Q16), (C2×C12).162D4, C4.5(D4.S3), C6.61(C2×SD16), C4.5(C3⋊Q16), C3⋊4(C4.SD16), C12.87(C4○D4), C12⋊2Q8.22C2, (C4×C12).140C22, (C2×C12).411C23, C4.18(Q8⋊3S3), C6.SD16.15C2, C6.60(C4.4D4), C2.13(C12.23D4), (C2×Dic6).116C22, (C4×C3⋊C8).15C2, (C3×C4⋊Q8).12C2, (C2×C6).542(C2×D4), C2.15(C2×D4.S3), C2.16(C2×C3⋊Q16), (C2×C3⋊C8).265C22, (C2×C4).139(C3⋊D4), (C3×C4⋊C4).134C22, (C2×C4).508(C22×S3), C22.214(C2×C3⋊D4), SmallGroup(192,652)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.Q16
G = < a,b,c | a12=b8=1, c2=a6b4, bab-1=a5, cac-1=a7, cbc-1=a6b-1 >
Subgroups: 240 in 98 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×C8, Q8⋊C4, C4⋊Q8, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, C4.SD16, C4×C3⋊C8, C6.SD16, C12⋊2Q8, C3×C4⋊Q8, C12.Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, Q16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, C2×SD16, C2×Q16, D4.S3, C3⋊Q16, Q8⋊3S3, C2×C3⋊D4, C4.SD16, C2×D4.S3, C2×C3⋊Q16, C12.23D4, 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)
(1 176 166 128 53 28 151 70)(2 169 167 121 54 33 152 63)(3 174 168 126 55 26 153 68)(4 179 157 131 56 31 154 61)(5 172 158 124 57 36 155 66)(6 177 159 129 58 29 156 71)(7 170 160 122 59 34 145 64)(8 175 161 127 60 27 146 69)(9 180 162 132 49 32 147 62)(10 173 163 125 50 25 148 67)(11 178 164 130 51 30 149 72)(12 171 165 123 52 35 150 65)(13 112 133 91 77 181 100 43)(14 117 134 96 78 186 101 48)(15 110 135 89 79 191 102 41)(16 115 136 94 80 184 103 46)(17 120 137 87 81 189 104 39)(18 113 138 92 82 182 105 44)(19 118 139 85 83 187 106 37)(20 111 140 90 84 192 107 42)(21 116 141 95 73 185 108 47)(22 109 142 88 74 190 97 40)(23 114 143 93 75 183 98 45)(24 119 144 86 76 188 99 38)
(1 90 59 48)(2 85 60 43)(3 92 49 38)(4 87 50 45)(5 94 51 40)(6 89 52 47)(7 96 53 42)(8 91 54 37)(9 86 55 44)(10 93 56 39)(11 88 57 46)(12 95 58 41)(13 69 83 121)(14 64 84 128)(15 71 73 123)(16 66 74 130)(17 61 75 125)(18 68 76 132)(19 63 77 127)(20 70 78 122)(21 65 79 129)(22 72 80 124)(23 67 81 131)(24 62 82 126)(25 104 179 143)(26 99 180 138)(27 106 169 133)(28 101 170 140)(29 108 171 135)(30 103 172 142)(31 98 173 137)(32 105 174 144)(33 100 175 139)(34 107 176 134)(35 102 177 141)(36 97 178 136)(109 155 184 164)(110 150 185 159)(111 145 186 166)(112 152 187 161)(113 147 188 168)(114 154 189 163)(115 149 190 158)(116 156 191 165)(117 151 192 160)(118 146 181 167)(119 153 182 162)(120 148 183 157)
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), (1,176,166,128,53,28,151,70)(2,169,167,121,54,33,152,63)(3,174,168,126,55,26,153,68)(4,179,157,131,56,31,154,61)(5,172,158,124,57,36,155,66)(6,177,159,129,58,29,156,71)(7,170,160,122,59,34,145,64)(8,175,161,127,60,27,146,69)(9,180,162,132,49,32,147,62)(10,173,163,125,50,25,148,67)(11,178,164,130,51,30,149,72)(12,171,165,123,52,35,150,65)(13,112,133,91,77,181,100,43)(14,117,134,96,78,186,101,48)(15,110,135,89,79,191,102,41)(16,115,136,94,80,184,103,46)(17,120,137,87,81,189,104,39)(18,113,138,92,82,182,105,44)(19,118,139,85,83,187,106,37)(20,111,140,90,84,192,107,42)(21,116,141,95,73,185,108,47)(22,109,142,88,74,190,97,40)(23,114,143,93,75,183,98,45)(24,119,144,86,76,188,99,38), (1,90,59,48)(2,85,60,43)(3,92,49,38)(4,87,50,45)(5,94,51,40)(6,89,52,47)(7,96,53,42)(8,91,54,37)(9,86,55,44)(10,93,56,39)(11,88,57,46)(12,95,58,41)(13,69,83,121)(14,64,84,128)(15,71,73,123)(16,66,74,130)(17,61,75,125)(18,68,76,132)(19,63,77,127)(20,70,78,122)(21,65,79,129)(22,72,80,124)(23,67,81,131)(24,62,82,126)(25,104,179,143)(26,99,180,138)(27,106,169,133)(28,101,170,140)(29,108,171,135)(30,103,172,142)(31,98,173,137)(32,105,174,144)(33,100,175,139)(34,107,176,134)(35,102,177,141)(36,97,178,136)(109,155,184,164)(110,150,185,159)(111,145,186,166)(112,152,187,161)(113,147,188,168)(114,154,189,163)(115,149,190,158)(116,156,191,165)(117,151,192,160)(118,146,181,167)(119,153,182,162)(120,148,183,157)>;
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), (1,176,166,128,53,28,151,70)(2,169,167,121,54,33,152,63)(3,174,168,126,55,26,153,68)(4,179,157,131,56,31,154,61)(5,172,158,124,57,36,155,66)(6,177,159,129,58,29,156,71)(7,170,160,122,59,34,145,64)(8,175,161,127,60,27,146,69)(9,180,162,132,49,32,147,62)(10,173,163,125,50,25,148,67)(11,178,164,130,51,30,149,72)(12,171,165,123,52,35,150,65)(13,112,133,91,77,181,100,43)(14,117,134,96,78,186,101,48)(15,110,135,89,79,191,102,41)(16,115,136,94,80,184,103,46)(17,120,137,87,81,189,104,39)(18,113,138,92,82,182,105,44)(19,118,139,85,83,187,106,37)(20,111,140,90,84,192,107,42)(21,116,141,95,73,185,108,47)(22,109,142,88,74,190,97,40)(23,114,143,93,75,183,98,45)(24,119,144,86,76,188,99,38), (1,90,59,48)(2,85,60,43)(3,92,49,38)(4,87,50,45)(5,94,51,40)(6,89,52,47)(7,96,53,42)(8,91,54,37)(9,86,55,44)(10,93,56,39)(11,88,57,46)(12,95,58,41)(13,69,83,121)(14,64,84,128)(15,71,73,123)(16,66,74,130)(17,61,75,125)(18,68,76,132)(19,63,77,127)(20,70,78,122)(21,65,79,129)(22,72,80,124)(23,67,81,131)(24,62,82,126)(25,104,179,143)(26,99,180,138)(27,106,169,133)(28,101,170,140)(29,108,171,135)(30,103,172,142)(31,98,173,137)(32,105,174,144)(33,100,175,139)(34,107,176,134)(35,102,177,141)(36,97,178,136)(109,155,184,164)(110,150,185,159)(111,145,186,166)(112,152,187,161)(113,147,188,168)(114,154,189,163)(115,149,190,158)(116,156,191,165)(117,151,192,160)(118,146,181,167)(119,153,182,162)(120,148,183,157) );
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)], [(1,176,166,128,53,28,151,70),(2,169,167,121,54,33,152,63),(3,174,168,126,55,26,153,68),(4,179,157,131,56,31,154,61),(5,172,158,124,57,36,155,66),(6,177,159,129,58,29,156,71),(7,170,160,122,59,34,145,64),(8,175,161,127,60,27,146,69),(9,180,162,132,49,32,147,62),(10,173,163,125,50,25,148,67),(11,178,164,130,51,30,149,72),(12,171,165,123,52,35,150,65),(13,112,133,91,77,181,100,43),(14,117,134,96,78,186,101,48),(15,110,135,89,79,191,102,41),(16,115,136,94,80,184,103,46),(17,120,137,87,81,189,104,39),(18,113,138,92,82,182,105,44),(19,118,139,85,83,187,106,37),(20,111,140,90,84,192,107,42),(21,116,141,95,73,185,108,47),(22,109,142,88,74,190,97,40),(23,114,143,93,75,183,98,45),(24,119,144,86,76,188,99,38)], [(1,90,59,48),(2,85,60,43),(3,92,49,38),(4,87,50,45),(5,94,51,40),(6,89,52,47),(7,96,53,42),(8,91,54,37),(9,86,55,44),(10,93,56,39),(11,88,57,46),(12,95,58,41),(13,69,83,121),(14,64,84,128),(15,71,73,123),(16,66,74,130),(17,61,75,125),(18,68,76,132),(19,63,77,127),(20,70,78,122),(21,65,79,129),(22,72,80,124),(23,67,81,131),(24,62,82,126),(25,104,179,143),(26,99,180,138),(27,106,169,133),(28,101,170,140),(29,108,171,135),(30,103,172,142),(31,98,173,137),(32,105,174,144),(33,100,175,139),(34,107,176,134),(35,102,177,141),(36,97,178,136),(109,155,184,164),(110,150,185,159),(111,145,186,166),(112,152,187,161),(113,147,188,168),(114,154,189,163),(115,149,190,158),(116,156,191,165),(117,151,192,160),(118,146,181,167),(119,153,182,162),(120,148,183,157)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | SD16 | Q16 | C4○D4 | C3⋊D4 | D4.S3 | C3⋊Q16 | Q8⋊3S3 |
| kernel | C12.Q16 | C4×C3⋊C8 | C6.SD16 | C12⋊2Q8 | C3×C4⋊Q8 | C4⋊Q8 | C2×C12 | C42 | C4⋊C4 | C12 | C12 | C12 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 |
Matrix representation of C12.Q16 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 48 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 2 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 5 | 59 | 0 | 0 | 0 | 0 |
| 54 | 68 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 67 | 12 |
| 30 | 60 | 0 | 0 | 0 | 0 |
| 13 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 18 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 48 |
| 0 | 0 | 0 | 0 | 3 | 21 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,48,0,0,0,0,3,72,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[5,54,0,0,0,0,59,68,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,67,0,0,0,0,12,12],[30,13,0,0,0,0,60,43,0,0,0,0,0,0,46,18,0,0,0,0,0,27,0,0,0,0,0,0,52,3,0,0,0,0,48,21] >;
C12.Q16 in GAP, Magma, Sage, TeX
C_{12}.Q_{16} % in TeX
G:=Group("C12.Q16"); // GroupNames label
G:=SmallGroup(192,652);
// by ID
G=gap.SmallGroup(192,652);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,219,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^6*b^4,b*a*b^-1=a^5,c*a*c^-1=a^7,c*b*c^-1=a^6*b^-1>;
// generators/relations