direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C12×Q16, (C4×C8).6C6, (C4×Q8).8C6, C2.3(C6×Q16), C2.D8.9C6, (C4×C24).17C2, C8.10(C2×C12), C24.56(C2×C4), C2.14(D4×C12), C6.116(C4×D4), (C2×Q16).7C6, Q8.6(C2×C12), C6.50(C2×Q16), (C2×C12).362D4, C42.72(C2×C6), Q8⋊C4.8C6, (Q8×C12).15C2, (C6×Q16).14C2, C22.53(C6×D4), C6.118(C4○D8), C4.11(C22×C12), C12.258(C4○D4), C12.156(C22×C4), (C2×C24).400C22, (C4×C12).357C22, (C2×C12).906C23, (C6×Q8).255C22, C4.3(C3×C4○D4), C2.5(C3×C4○D8), C4⋊C4.47(C2×C6), (C2×C8).67(C2×C6), (C2×C4).52(C3×D4), (C2×C6).629(C2×D4), (C3×Q8).19(C2×C4), (C2×Q8).52(C2×C6), (C3×C2.D8).18C2, (C2×C4).81(C22×C6), (C3×C4⋊C4).368C22, (C3×Q8⋊C4).17C2, SmallGroup(192,872)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12×Q16
G = < a,b,c | a12=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 154 in 110 conjugacy classes, 74 normal (34 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C8, C2×C4, C2×C4, Q8, Q8, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C24, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C2×Q16, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×Q16, C6×Q8, C4×Q16, C4×C24, C3×Q8⋊C4, C3×C2.D8, Q8×C12, C6×Q16, C12×Q16
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, Q16, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C4×D4, C2×Q16, C4○D8, C3×Q16, C22×C12, C6×D4, C3×C4○D4, C4×Q16, D4×C12, C6×Q16, C3×C4○D8, 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)
(1 112 130 167 156 53 101 178)(2 113 131 168 145 54 102 179)(3 114 132 157 146 55 103 180)(4 115 121 158 147 56 104 169)(5 116 122 159 148 57 105 170)(6 117 123 160 149 58 106 171)(7 118 124 161 150 59 107 172)(8 119 125 162 151 60 108 173)(9 120 126 163 152 49 97 174)(10 109 127 164 153 50 98 175)(11 110 128 165 154 51 99 176)(12 111 129 166 155 52 100 177)(13 27 63 37 80 185 136 90)(14 28 64 38 81 186 137 91)(15 29 65 39 82 187 138 92)(16 30 66 40 83 188 139 93)(17 31 67 41 84 189 140 94)(18 32 68 42 73 190 141 95)(19 33 69 43 74 191 142 96)(20 34 70 44 75 192 143 85)(21 35 71 45 76 181 144 86)(22 36 72 46 77 182 133 87)(23 25 61 47 78 183 134 88)(24 26 62 48 79 184 135 89)
(1 42 156 95)(2 43 145 96)(3 44 146 85)(4 45 147 86)(5 46 148 87)(6 47 149 88)(7 48 150 89)(8 37 151 90)(9 38 152 91)(10 39 153 92)(11 40 154 93)(12 41 155 94)(13 173 80 162)(14 174 81 163)(15 175 82 164)(16 176 83 165)(17 177 84 166)(18 178 73 167)(19 179 74 168)(20 180 75 157)(21 169 76 158)(22 170 77 159)(23 171 78 160)(24 172 79 161)(25 106 183 123)(26 107 184 124)(27 108 185 125)(28 97 186 126)(29 98 187 127)(30 99 188 128)(31 100 189 129)(32 101 190 130)(33 102 191 131)(34 103 192 132)(35 104 181 121)(36 105 182 122)(49 137 120 64)(50 138 109 65)(51 139 110 66)(52 140 111 67)(53 141 112 68)(54 142 113 69)(55 143 114 70)(56 144 115 71)(57 133 116 72)(58 134 117 61)(59 135 118 62)(60 136 119 63)
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), (1,112,130,167,156,53,101,178)(2,113,131,168,145,54,102,179)(3,114,132,157,146,55,103,180)(4,115,121,158,147,56,104,169)(5,116,122,159,148,57,105,170)(6,117,123,160,149,58,106,171)(7,118,124,161,150,59,107,172)(8,119,125,162,151,60,108,173)(9,120,126,163,152,49,97,174)(10,109,127,164,153,50,98,175)(11,110,128,165,154,51,99,176)(12,111,129,166,155,52,100,177)(13,27,63,37,80,185,136,90)(14,28,64,38,81,186,137,91)(15,29,65,39,82,187,138,92)(16,30,66,40,83,188,139,93)(17,31,67,41,84,189,140,94)(18,32,68,42,73,190,141,95)(19,33,69,43,74,191,142,96)(20,34,70,44,75,192,143,85)(21,35,71,45,76,181,144,86)(22,36,72,46,77,182,133,87)(23,25,61,47,78,183,134,88)(24,26,62,48,79,184,135,89), (1,42,156,95)(2,43,145,96)(3,44,146,85)(4,45,147,86)(5,46,148,87)(6,47,149,88)(7,48,150,89)(8,37,151,90)(9,38,152,91)(10,39,153,92)(11,40,154,93)(12,41,155,94)(13,173,80,162)(14,174,81,163)(15,175,82,164)(16,176,83,165)(17,177,84,166)(18,178,73,167)(19,179,74,168)(20,180,75,157)(21,169,76,158)(22,170,77,159)(23,171,78,160)(24,172,79,161)(25,106,183,123)(26,107,184,124)(27,108,185,125)(28,97,186,126)(29,98,187,127)(30,99,188,128)(31,100,189,129)(32,101,190,130)(33,102,191,131)(34,103,192,132)(35,104,181,121)(36,105,182,122)(49,137,120,64)(50,138,109,65)(51,139,110,66)(52,140,111,67)(53,141,112,68)(54,142,113,69)(55,143,114,70)(56,144,115,71)(57,133,116,72)(58,134,117,61)(59,135,118,62)(60,136,119,63)>;
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), (1,112,130,167,156,53,101,178)(2,113,131,168,145,54,102,179)(3,114,132,157,146,55,103,180)(4,115,121,158,147,56,104,169)(5,116,122,159,148,57,105,170)(6,117,123,160,149,58,106,171)(7,118,124,161,150,59,107,172)(8,119,125,162,151,60,108,173)(9,120,126,163,152,49,97,174)(10,109,127,164,153,50,98,175)(11,110,128,165,154,51,99,176)(12,111,129,166,155,52,100,177)(13,27,63,37,80,185,136,90)(14,28,64,38,81,186,137,91)(15,29,65,39,82,187,138,92)(16,30,66,40,83,188,139,93)(17,31,67,41,84,189,140,94)(18,32,68,42,73,190,141,95)(19,33,69,43,74,191,142,96)(20,34,70,44,75,192,143,85)(21,35,71,45,76,181,144,86)(22,36,72,46,77,182,133,87)(23,25,61,47,78,183,134,88)(24,26,62,48,79,184,135,89), (1,42,156,95)(2,43,145,96)(3,44,146,85)(4,45,147,86)(5,46,148,87)(6,47,149,88)(7,48,150,89)(8,37,151,90)(9,38,152,91)(10,39,153,92)(11,40,154,93)(12,41,155,94)(13,173,80,162)(14,174,81,163)(15,175,82,164)(16,176,83,165)(17,177,84,166)(18,178,73,167)(19,179,74,168)(20,180,75,157)(21,169,76,158)(22,170,77,159)(23,171,78,160)(24,172,79,161)(25,106,183,123)(26,107,184,124)(27,108,185,125)(28,97,186,126)(29,98,187,127)(30,99,188,128)(31,100,189,129)(32,101,190,130)(33,102,191,131)(34,103,192,132)(35,104,181,121)(36,105,182,122)(49,137,120,64)(50,138,109,65)(51,139,110,66)(52,140,111,67)(53,141,112,68)(54,142,113,69)(55,143,114,70)(56,144,115,71)(57,133,116,72)(58,134,117,61)(59,135,118,62)(60,136,119,63) );
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)], [(1,112,130,167,156,53,101,178),(2,113,131,168,145,54,102,179),(3,114,132,157,146,55,103,180),(4,115,121,158,147,56,104,169),(5,116,122,159,148,57,105,170),(6,117,123,160,149,58,106,171),(7,118,124,161,150,59,107,172),(8,119,125,162,151,60,108,173),(9,120,126,163,152,49,97,174),(10,109,127,164,153,50,98,175),(11,110,128,165,154,51,99,176),(12,111,129,166,155,52,100,177),(13,27,63,37,80,185,136,90),(14,28,64,38,81,186,137,91),(15,29,65,39,82,187,138,92),(16,30,66,40,83,188,139,93),(17,31,67,41,84,189,140,94),(18,32,68,42,73,190,141,95),(19,33,69,43,74,191,142,96),(20,34,70,44,75,192,143,85),(21,35,71,45,76,181,144,86),(22,36,72,46,77,182,133,87),(23,25,61,47,78,183,134,88),(24,26,62,48,79,184,135,89)], [(1,42,156,95),(2,43,145,96),(3,44,146,85),(4,45,147,86),(5,46,148,87),(6,47,149,88),(7,48,150,89),(8,37,151,90),(9,38,152,91),(10,39,153,92),(11,40,154,93),(12,41,155,94),(13,173,80,162),(14,174,81,163),(15,175,82,164),(16,176,83,165),(17,177,84,166),(18,178,73,167),(19,179,74,168),(20,180,75,157),(21,169,76,158),(22,170,77,159),(23,171,78,160),(24,172,79,161),(25,106,183,123),(26,107,184,124),(27,108,185,125),(28,97,186,126),(29,98,187,127),(30,99,188,128),(31,100,189,129),(32,101,190,130),(33,102,191,131),(34,103,192,132),(35,104,181,121),(36,105,182,122),(49,137,120,64),(50,138,109,65),(51,139,110,66),(52,140,111,67),(53,141,112,68),(54,142,113,69),(55,143,114,70),(56,144,115,71),(57,133,116,72),(58,134,117,61),(59,135,118,62),(60,136,119,63)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 6A | ··· | 6F | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12AF | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | Q16 | C4○D4 | C3×D4 | C4○D8 | C3×Q16 | C3×C4○D4 | C3×C4○D8 |
| kernel | C12×Q16 | C4×C24 | C3×Q8⋊C4 | C3×C2.D8 | Q8×C12 | C6×Q16 | C4×Q16 | C3×Q16 | C4×C8 | Q8⋊C4 | C2.D8 | C4×Q8 | C2×Q16 | Q16 | C2×C12 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 16 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C12×Q16 ►in GL3(𝔽73) generated by
| 27 | 0 | 0 |
| 0 | 49 | 0 |
| 0 | 0 | 49 |
| 72 | 0 | 0 |
| 0 | 16 | 57 |
| 0 | 16 | 16 |
| 72 | 0 | 0 |
| 0 | 65 | 64 |
| 0 | 64 | 8 |
G:=sub<GL(3,GF(73))| [27,0,0,0,49,0,0,0,49],[72,0,0,0,16,16,0,57,16],[72,0,0,0,65,64,0,64,8] >;
C12×Q16 in GAP, Magma, Sage, TeX
C_{12}\times Q_{16} % in TeX
G:=Group("C12xQ16"); // GroupNames label
G:=SmallGroup(192,872);
// by ID
G=gap.SmallGroup(192,872);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,680,772,4204,2111,172]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations