metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic48, C32.S3, C3⋊1Q64, C96.1C2, C4.3D24, C6.3D16, C2.5D48, C8.7D12, C12.28D8, C24.57D4, C16.15D6, C48.16C22, Dic24.1C2, SmallGroup(192,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic48
G = < a,b | a96=1, b2=a48, bab-1=a-1 >
Smallest permutation representation of Dic48
►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)
(1 145 49 97)(2 144 50 192)(3 143 51 191)(4 142 52 190)(5 141 53 189)(6 140 54 188)(7 139 55 187)(8 138 56 186)(9 137 57 185)(10 136 58 184)(11 135 59 183)(12 134 60 182)(13 133 61 181)(14 132 62 180)(15 131 63 179)(16 130 64 178)(17 129 65 177)(18 128 66 176)(19 127 67 175)(20 126 68 174)(21 125 69 173)(22 124 70 172)(23 123 71 171)(24 122 72 170)(25 121 73 169)(26 120 74 168)(27 119 75 167)(28 118 76 166)(29 117 77 165)(30 116 78 164)(31 115 79 163)(32 114 80 162)(33 113 81 161)(34 112 82 160)(35 111 83 159)(36 110 84 158)(37 109 85 157)(38 108 86 156)(39 107 87 155)(40 106 88 154)(41 105 89 153)(42 104 90 152)(43 103 91 151)(44 102 92 150)(45 101 93 149)(46 100 94 148)(47 99 95 147)(48 98 96 146)
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), (1,145,49,97)(2,144,50,192)(3,143,51,191)(4,142,52,190)(5,141,53,189)(6,140,54,188)(7,139,55,187)(8,138,56,186)(9,137,57,185)(10,136,58,184)(11,135,59,183)(12,134,60,182)(13,133,61,181)(14,132,62,180)(15,131,63,179)(16,130,64,178)(17,129,65,177)(18,128,66,176)(19,127,67,175)(20,126,68,174)(21,125,69,173)(22,124,70,172)(23,123,71,171)(24,122,72,170)(25,121,73,169)(26,120,74,168)(27,119,75,167)(28,118,76,166)(29,117,77,165)(30,116,78,164)(31,115,79,163)(32,114,80,162)(33,113,81,161)(34,112,82,160)(35,111,83,159)(36,110,84,158)(37,109,85,157)(38,108,86,156)(39,107,87,155)(40,106,88,154)(41,105,89,153)(42,104,90,152)(43,103,91,151)(44,102,92,150)(45,101,93,149)(46,100,94,148)(47,99,95,147)(48,98,96,146)>;
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), (1,145,49,97)(2,144,50,192)(3,143,51,191)(4,142,52,190)(5,141,53,189)(6,140,54,188)(7,139,55,187)(8,138,56,186)(9,137,57,185)(10,136,58,184)(11,135,59,183)(12,134,60,182)(13,133,61,181)(14,132,62,180)(15,131,63,179)(16,130,64,178)(17,129,65,177)(18,128,66,176)(19,127,67,175)(20,126,68,174)(21,125,69,173)(22,124,70,172)(23,123,71,171)(24,122,72,170)(25,121,73,169)(26,120,74,168)(27,119,75,167)(28,118,76,166)(29,117,77,165)(30,116,78,164)(31,115,79,163)(32,114,80,162)(33,113,81,161)(34,112,82,160)(35,111,83,159)(36,110,84,158)(37,109,85,157)(38,108,86,156)(39,107,87,155)(40,106,88,154)(41,105,89,153)(42,104,90,152)(43,103,91,151)(44,102,92,150)(45,101,93,149)(46,100,94,148)(47,99,95,147)(48,98,96,146) );
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)], [(1,145,49,97),(2,144,50,192),(3,143,51,191),(4,142,52,190),(5,141,53,189),(6,140,54,188),(7,139,55,187),(8,138,56,186),(9,137,57,185),(10,136,58,184),(11,135,59,183),(12,134,60,182),(13,133,61,181),(14,132,62,180),(15,131,63,179),(16,130,64,178),(17,129,65,177),(18,128,66,176),(19,127,67,175),(20,126,68,174),(21,125,69,173),(22,124,70,172),(23,123,71,171),(24,122,72,170),(25,121,73,169),(26,120,74,168),(27,119,75,167),(28,118,76,166),(29,117,77,165),(30,116,78,164),(31,115,79,163),(32,114,80,162),(33,113,81,161),(34,112,82,160),(35,111,83,159),(36,110,84,158),(37,109,85,157),(38,108,86,156),(39,107,87,155),(40,106,88,154),(41,105,89,153),(42,104,90,152),(43,103,91,151),(44,102,92,150),(45,101,93,149),(46,100,94,148),(47,99,95,147),(48,98,96,146)]])
51 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 12A | 12B | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 32A | ··· | 32H | 48A | ··· | 48H | 96A | ··· | 96P |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 8 | 8 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 32 | ··· | 32 | 48 | ··· | 48 | 96 | ··· | 96 |
| size | 1 | 1 | 2 | 2 | 48 | 48 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
51 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D8 | D12 | D16 | D24 | Q64 | D48 | Dic48 |
| kernel | Dic48 | C96 | Dic24 | C32 | C24 | C16 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of Dic48 ►in GL2(𝔽97) generated by
| 14 | 50 |
| 47 | 64 |
| 0 | 22 |
| 22 | 0 |
G:=sub<GL(2,GF(97))| [14,47,50,64],[0,22,22,0] >;
Dic48 in GAP, Magma, Sage, TeX
{\rm Dic}_{48} % in TeX
G:=Group("Dic48"); // GroupNames label
G:=SmallGroup(192,9);
// by ID
G=gap.SmallGroup(192,9);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,85,92,254,142,675,192,1684,102,6278]);
// Polycyclic
G:=Group<a,b|a^96=1,b^2=a^48,b*a*b^-1=a^-1>;
// generators/relations
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