direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3×C16, C48⋊9C4, C12.16C42, C3⋊C16⋊9C4, C3⋊2(C4×C16), C3⋊C8.5C8, C6.7(C4×C8), C4.20(S3×C8), C8.40(C4×S3), C2.2(S3×C16), C6.3(C2×C16), (C2×C48).15C2, C12.25(C2×C8), C24.61(C2×C4), (C2×C16).10S3, (C2×C8).332D6, C22.8(S3×C8), C2.2(C8×Dic3), (C2×Dic3).7C8, C8.23(C2×Dic3), C4.15(C4×Dic3), (C4×Dic3).22C4, (C8×Dic3).17C2, (C2×C24).417C22, (C2×C3⋊C8).17C4, (C2×C6).9(C2×C8), (C2×C3⋊C16).12C2, (C2×C4).166(C4×S3), (C2×C12).240(C2×C4), SmallGroup(192,59)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C16 |
Generators and relations for Dic3×C16
G = < a,b,c | a16=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C16
►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)
(1 126 134 90 161 188)(2 127 135 91 162 189)(3 128 136 92 163 190)(4 113 137 93 164 191)(5 114 138 94 165 192)(6 115 139 95 166 177)(7 116 140 96 167 178)(8 117 141 81 168 179)(9 118 142 82 169 180)(10 119 143 83 170 181)(11 120 144 84 171 182)(12 121 129 85 172 183)(13 122 130 86 173 184)(14 123 131 87 174 185)(15 124 132 88 175 186)(16 125 133 89 176 187)(17 101 64 152 77 44)(18 102 49 153 78 45)(19 103 50 154 79 46)(20 104 51 155 80 47)(21 105 52 156 65 48)(22 106 53 157 66 33)(23 107 54 158 67 34)(24 108 55 159 68 35)(25 109 56 160 69 36)(26 110 57 145 70 37)(27 111 58 146 71 38)(28 112 59 147 72 39)(29 97 60 148 73 40)(30 98 61 149 74 41)(31 99 62 150 75 42)(32 100 63 151 76 43)
(1 44 90 64)(2 45 91 49)(3 46 92 50)(4 47 93 51)(5 48 94 52)(6 33 95 53)(7 34 96 54)(8 35 81 55)(9 36 82 56)(10 37 83 57)(11 38 84 58)(12 39 85 59)(13 40 86 60)(14 41 87 61)(15 42 88 62)(16 43 89 63)(17 134 152 188)(18 135 153 189)(19 136 154 190)(20 137 155 191)(21 138 156 192)(22 139 157 177)(23 140 158 178)(24 141 159 179)(25 142 160 180)(26 143 145 181)(27 144 146 182)(28 129 147 183)(29 130 148 184)(30 131 149 185)(31 132 150 186)(32 133 151 187)(65 165 105 114)(66 166 106 115)(67 167 107 116)(68 168 108 117)(69 169 109 118)(70 170 110 119)(71 171 111 120)(72 172 112 121)(73 173 97 122)(74 174 98 123)(75 175 99 124)(76 176 100 125)(77 161 101 126)(78 162 102 127)(79 163 103 128)(80 164 104 113)
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), (1,126,134,90,161,188)(2,127,135,91,162,189)(3,128,136,92,163,190)(4,113,137,93,164,191)(5,114,138,94,165,192)(6,115,139,95,166,177)(7,116,140,96,167,178)(8,117,141,81,168,179)(9,118,142,82,169,180)(10,119,143,83,170,181)(11,120,144,84,171,182)(12,121,129,85,172,183)(13,122,130,86,173,184)(14,123,131,87,174,185)(15,124,132,88,175,186)(16,125,133,89,176,187)(17,101,64,152,77,44)(18,102,49,153,78,45)(19,103,50,154,79,46)(20,104,51,155,80,47)(21,105,52,156,65,48)(22,106,53,157,66,33)(23,107,54,158,67,34)(24,108,55,159,68,35)(25,109,56,160,69,36)(26,110,57,145,70,37)(27,111,58,146,71,38)(28,112,59,147,72,39)(29,97,60,148,73,40)(30,98,61,149,74,41)(31,99,62,150,75,42)(32,100,63,151,76,43), (1,44,90,64)(2,45,91,49)(3,46,92,50)(4,47,93,51)(5,48,94,52)(6,33,95,53)(7,34,96,54)(8,35,81,55)(9,36,82,56)(10,37,83,57)(11,38,84,58)(12,39,85,59)(13,40,86,60)(14,41,87,61)(15,42,88,62)(16,43,89,63)(17,134,152,188)(18,135,153,189)(19,136,154,190)(20,137,155,191)(21,138,156,192)(22,139,157,177)(23,140,158,178)(24,141,159,179)(25,142,160,180)(26,143,145,181)(27,144,146,182)(28,129,147,183)(29,130,148,184)(30,131,149,185)(31,132,150,186)(32,133,151,187)(65,165,105,114)(66,166,106,115)(67,167,107,116)(68,168,108,117)(69,169,109,118)(70,170,110,119)(71,171,111,120)(72,172,112,121)(73,173,97,122)(74,174,98,123)(75,175,99,124)(76,176,100,125)(77,161,101,126)(78,162,102,127)(79,163,103,128)(80,164,104,113)>;
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), (1,126,134,90,161,188)(2,127,135,91,162,189)(3,128,136,92,163,190)(4,113,137,93,164,191)(5,114,138,94,165,192)(6,115,139,95,166,177)(7,116,140,96,167,178)(8,117,141,81,168,179)(9,118,142,82,169,180)(10,119,143,83,170,181)(11,120,144,84,171,182)(12,121,129,85,172,183)(13,122,130,86,173,184)(14,123,131,87,174,185)(15,124,132,88,175,186)(16,125,133,89,176,187)(17,101,64,152,77,44)(18,102,49,153,78,45)(19,103,50,154,79,46)(20,104,51,155,80,47)(21,105,52,156,65,48)(22,106,53,157,66,33)(23,107,54,158,67,34)(24,108,55,159,68,35)(25,109,56,160,69,36)(26,110,57,145,70,37)(27,111,58,146,71,38)(28,112,59,147,72,39)(29,97,60,148,73,40)(30,98,61,149,74,41)(31,99,62,150,75,42)(32,100,63,151,76,43), (1,44,90,64)(2,45,91,49)(3,46,92,50)(4,47,93,51)(5,48,94,52)(6,33,95,53)(7,34,96,54)(8,35,81,55)(9,36,82,56)(10,37,83,57)(11,38,84,58)(12,39,85,59)(13,40,86,60)(14,41,87,61)(15,42,88,62)(16,43,89,63)(17,134,152,188)(18,135,153,189)(19,136,154,190)(20,137,155,191)(21,138,156,192)(22,139,157,177)(23,140,158,178)(24,141,159,179)(25,142,160,180)(26,143,145,181)(27,144,146,182)(28,129,147,183)(29,130,148,184)(30,131,149,185)(31,132,150,186)(32,133,151,187)(65,165,105,114)(66,166,106,115)(67,167,107,116)(68,168,108,117)(69,169,109,118)(70,170,110,119)(71,171,111,120)(72,172,112,121)(73,173,97,122)(74,174,98,123)(75,175,99,124)(76,176,100,125)(77,161,101,126)(78,162,102,127)(79,163,103,128)(80,164,104,113) );
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)], [(1,126,134,90,161,188),(2,127,135,91,162,189),(3,128,136,92,163,190),(4,113,137,93,164,191),(5,114,138,94,165,192),(6,115,139,95,166,177),(7,116,140,96,167,178),(8,117,141,81,168,179),(9,118,142,82,169,180),(10,119,143,83,170,181),(11,120,144,84,171,182),(12,121,129,85,172,183),(13,122,130,86,173,184),(14,123,131,87,174,185),(15,124,132,88,175,186),(16,125,133,89,176,187),(17,101,64,152,77,44),(18,102,49,153,78,45),(19,103,50,154,79,46),(20,104,51,155,80,47),(21,105,52,156,65,48),(22,106,53,157,66,33),(23,107,54,158,67,34),(24,108,55,159,68,35),(25,109,56,160,69,36),(26,110,57,145,70,37),(27,111,58,146,71,38),(28,112,59,147,72,39),(29,97,60,148,73,40),(30,98,61,149,74,41),(31,99,62,150,75,42),(32,100,63,151,76,43)], [(1,44,90,64),(2,45,91,49),(3,46,92,50),(4,47,93,51),(5,48,94,52),(6,33,95,53),(7,34,96,54),(8,35,81,55),(9,36,82,56),(10,37,83,57),(11,38,84,58),(12,39,85,59),(13,40,86,60),(14,41,87,61),(15,42,88,62),(16,43,89,63),(17,134,152,188),(18,135,153,189),(19,136,154,190),(20,137,155,191),(21,138,156,192),(22,139,157,177),(23,140,158,178),(24,141,159,179),(25,142,160,180),(26,143,145,181),(27,144,146,182),(28,129,147,183),(29,130,148,184),(30,131,149,185),(31,132,150,186),(32,133,151,187),(65,165,105,114),(66,166,106,115),(67,167,107,116),(68,168,108,117),(69,169,109,118),(70,170,110,119),(71,171,111,120),(72,172,112,121),(73,173,97,122),(74,174,98,123),(75,175,99,124),(76,176,100,125),(77,161,101,126),(78,162,102,127),(79,163,103,128),(80,164,104,113)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 12A | 12B | 12C | 12D | 16A | ··· | 16P | 16Q | ··· | 16AF | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 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 | C4 | C4 | C4 | C4 | C8 | C8 | C16 | S3 | Dic3 | D6 | C4×S3 | C4×S3 | S3×C8 | S3×C8 | S3×C16 |
| kernel | Dic3×C16 | C2×C3⋊C16 | C8×Dic3 | C2×C48 | C3⋊C16 | C48 | C2×C3⋊C8 | C4×Dic3 | C3⋊C8 | C2×Dic3 | Dic3 | C2×C16 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 32 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 |
Matrix representation of Dic3×C16 ►in GL3(𝔽97) generated by
| 22 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 96 | 0 | 0 |
| 0 | 96 | 1 |
| 0 | 96 | 0 |
| 75 | 0 | 0 |
| 0 | 44 | 73 |
| 0 | 20 | 53 |
G:=sub<GL(3,GF(97))| [22,0,0,0,18,0,0,0,18],[96,0,0,0,96,96,0,1,0],[75,0,0,0,44,20,0,73,53] >;
Dic3×C16 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{16} % in TeX
G:=Group("Dic3xC16"); // GroupNames label
G:=SmallGroup(192,59);
// by ID
G=gap.SmallGroup(192,59);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,64,80,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^16=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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