metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊1C16, C8.28D12, C24.71D4, C24.17Q8, C8.15Dic6, C6.5M5(2), C42.8Dic3, C12.28M4(2), C4⋊(C3⋊C16), C3⋊1(C4⋊C16), (C4×C8).3S3, C6.3(C4⋊C8), (C2×C12).4C8, C6.7(C2×C16), (C4×C24).26C2, (C2×C24).10C4, (C4×C12).13C4, (C2×C8).331D6, C12.38(C4⋊C4), (C2×C8).5Dic3, C2.1(C12⋊C8), C4.18(C4⋊Dic3), C2.2(C12.C8), (C2×C24).415C22, C4.10(C4.Dic3), C2.3(C2×C3⋊C16), (C2×C3⋊C16).8C2, (C2×C4).3(C3⋊C8), C22.9(C2×C3⋊C8), (C2×C6).27(C2×C8), (C2×C12).311(C2×C4), (C2×C4).92(C2×Dic3), SmallGroup(192,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊C16
G = < a,b | a12=b16=1, bab-1=a-1 >
Smallest permutation representation of C12⋊C16
►Regular action on 192 points
Generators in S192
(1 72 24 155 91 128 59 170 184 100 129 48)(2 33 130 101 185 171 60 113 92 156 25 73)(3 74 26 157 93 114 61 172 186 102 131 34)(4 35 132 103 187 173 62 115 94 158 27 75)(5 76 28 159 95 116 63 174 188 104 133 36)(6 37 134 105 189 175 64 117 96 160 29 77)(7 78 30 145 81 118 49 176 190 106 135 38)(8 39 136 107 191 161 50 119 82 146 31 79)(9 80 32 147 83 120 51 162 192 108 137 40)(10 41 138 109 177 163 52 121 84 148 17 65)(11 66 18 149 85 122 53 164 178 110 139 42)(12 43 140 111 179 165 54 123 86 150 19 67)(13 68 20 151 87 124 55 166 180 112 141 44)(14 45 142 97 181 167 56 125 88 152 21 69)(15 70 22 153 89 126 57 168 182 98 143 46)(16 47 144 99 183 169 58 127 90 154 23 71)
(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)
G:=sub<Sym(192)| (1,72,24,155,91,128,59,170,184,100,129,48)(2,33,130,101,185,171,60,113,92,156,25,73)(3,74,26,157,93,114,61,172,186,102,131,34)(4,35,132,103,187,173,62,115,94,158,27,75)(5,76,28,159,95,116,63,174,188,104,133,36)(6,37,134,105,189,175,64,117,96,160,29,77)(7,78,30,145,81,118,49,176,190,106,135,38)(8,39,136,107,191,161,50,119,82,146,31,79)(9,80,32,147,83,120,51,162,192,108,137,40)(10,41,138,109,177,163,52,121,84,148,17,65)(11,66,18,149,85,122,53,164,178,110,139,42)(12,43,140,111,179,165,54,123,86,150,19,67)(13,68,20,151,87,124,55,166,180,112,141,44)(14,45,142,97,181,167,56,125,88,152,21,69)(15,70,22,153,89,126,57,168,182,98,143,46)(16,47,144,99,183,169,58,127,90,154,23,71), (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)>;
G:=Group( (1,72,24,155,91,128,59,170,184,100,129,48)(2,33,130,101,185,171,60,113,92,156,25,73)(3,74,26,157,93,114,61,172,186,102,131,34)(4,35,132,103,187,173,62,115,94,158,27,75)(5,76,28,159,95,116,63,174,188,104,133,36)(6,37,134,105,189,175,64,117,96,160,29,77)(7,78,30,145,81,118,49,176,190,106,135,38)(8,39,136,107,191,161,50,119,82,146,31,79)(9,80,32,147,83,120,51,162,192,108,137,40)(10,41,138,109,177,163,52,121,84,148,17,65)(11,66,18,149,85,122,53,164,178,110,139,42)(12,43,140,111,179,165,54,123,86,150,19,67)(13,68,20,151,87,124,55,166,180,112,141,44)(14,45,142,97,181,167,56,125,88,152,21,69)(15,70,22,153,89,126,57,168,182,98,143,46)(16,47,144,99,183,169,58,127,90,154,23,71), (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) );
G=PermutationGroup([[(1,72,24,155,91,128,59,170,184,100,129,48),(2,33,130,101,185,171,60,113,92,156,25,73),(3,74,26,157,93,114,61,172,186,102,131,34),(4,35,132,103,187,173,62,115,94,158,27,75),(5,76,28,159,95,116,63,174,188,104,133,36),(6,37,134,105,189,175,64,117,96,160,29,77),(7,78,30,145,81,118,49,176,190,106,135,38),(8,39,136,107,191,161,50,119,82,146,31,79),(9,80,32,147,83,120,51,162,192,108,137,40),(10,41,138,109,177,163,52,121,84,148,17,65),(11,66,18,149,85,122,53,164,178,110,139,42),(12,43,140,111,179,165,54,123,86,150,19,67),(13,68,20,151,87,124,55,166,180,112,141,44),(14,45,142,97,181,167,56,125,88,152,21,69),(15,70,22,153,89,126,57,168,182,98,143,46),(16,47,144,99,183,169,58,127,90,154,23,71)], [(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)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12L | 16A | ··· | 16P | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | - | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C16 | S3 | D4 | Q8 | Dic3 | Dic3 | D6 | M4(2) | Dic6 | D12 | C3⋊C8 | M5(2) | C3⋊C16 | C4.Dic3 | C12.C8 |
| kernel | C12⋊C16 | C2×C3⋊C16 | C4×C24 | C4×C12 | C2×C24 | C2×C12 | C12 | C4×C8 | C24 | C24 | C42 | C2×C8 | C2×C8 | C12 | C8 | C8 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 16 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C12⋊C16 ►in GL3(𝔽97) generated by
| 96 | 0 | 0 |
| 0 | 39 | 68 |
| 0 | 29 | 68 |
| 18 | 0 | 0 |
| 0 | 39 | 68 |
| 0 | 29 | 58 |
G:=sub<GL(3,GF(97))| [96,0,0,0,39,29,0,68,68],[18,0,0,0,39,29,0,68,58] >;
C12⋊C16 in GAP, Magma, Sage, TeX
C_{12}\rtimes C_{16} % in TeX
G:=Group("C12:C16"); // GroupNames label
G:=SmallGroup(192,21);
// by ID
G=gap.SmallGroup(192,21);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,64,100,102,6278]);
// Polycyclic
G:=Group<a,b|a^12=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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