metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C51⋊2D4, C3⋊2D68, D34⋊2S3, Dic3⋊D17, C6.5D34, C34.5D6, D102⋊3C2, C102.5C22, (C6×D17)⋊2C2, C17⋊1(C3⋊D4), C2.5(S3×D17), (Dic3×C17)⋊3C2, SmallGroup(408,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D68
G = < a,b,c | a3=b68=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C3⋊D68
►On 204 points
Generators in S204
(1 85 191)(2 192 86)(3 87 193)(4 194 88)(5 89 195)(6 196 90)(7 91 197)(8 198 92)(9 93 199)(10 200 94)(11 95 201)(12 202 96)(13 97 203)(14 204 98)(15 99 137)(16 138 100)(17 101 139)(18 140 102)(19 103 141)(20 142 104)(21 105 143)(22 144 106)(23 107 145)(24 146 108)(25 109 147)(26 148 110)(27 111 149)(28 150 112)(29 113 151)(30 152 114)(31 115 153)(32 154 116)(33 117 155)(34 156 118)(35 119 157)(36 158 120)(37 121 159)(38 160 122)(39 123 161)(40 162 124)(41 125 163)(42 164 126)(43 127 165)(44 166 128)(45 129 167)(46 168 130)(47 131 169)(48 170 132)(49 133 171)(50 172 134)(51 135 173)(52 174 136)(53 69 175)(54 176 70)(55 71 177)(56 178 72)(57 73 179)(58 180 74)(59 75 181)(60 182 76)(61 77 183)(62 184 78)(63 79 185)(64 186 80)(65 81 187)(66 188 82)(67 83 189)(68 190 84)
(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)(137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204)
(1 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(69 189)(70 188)(71 187)(72 186)(73 185)(74 184)(75 183)(76 182)(77 181)(78 180)(79 179)(80 178)(81 177)(82 176)(83 175)(84 174)(85 173)(86 172)(87 171)(88 170)(89 169)(90 168)(91 167)(92 166)(93 165)(94 164)(95 163)(96 162)(97 161)(98 160)(99 159)(100 158)(101 157)(102 156)(103 155)(104 154)(105 153)(106 152)(107 151)(108 150)(109 149)(110 148)(111 147)(112 146)(113 145)(114 144)(115 143)(116 142)(117 141)(118 140)(119 139)(120 138)(121 137)(122 204)(123 203)(124 202)(125 201)(126 200)(127 199)(128 198)(129 197)(130 196)(131 195)(132 194)(133 193)(134 192)(135 191)(136 190)
G:=sub<Sym(204)| (1,85,191)(2,192,86)(3,87,193)(4,194,88)(5,89,195)(6,196,90)(7,91,197)(8,198,92)(9,93,199)(10,200,94)(11,95,201)(12,202,96)(13,97,203)(14,204,98)(15,99,137)(16,138,100)(17,101,139)(18,140,102)(19,103,141)(20,142,104)(21,105,143)(22,144,106)(23,107,145)(24,146,108)(25,109,147)(26,148,110)(27,111,149)(28,150,112)(29,113,151)(30,152,114)(31,115,153)(32,154,116)(33,117,155)(34,156,118)(35,119,157)(36,158,120)(37,121,159)(38,160,122)(39,123,161)(40,162,124)(41,125,163)(42,164,126)(43,127,165)(44,166,128)(45,129,167)(46,168,130)(47,131,169)(48,170,132)(49,133,171)(50,172,134)(51,135,173)(52,174,136)(53,69,175)(54,176,70)(55,71,177)(56,178,72)(57,73,179)(58,180,74)(59,75,181)(60,182,76)(61,77,183)(62,184,78)(63,79,185)(64,186,80)(65,81,187)(66,188,82)(67,83,189)(68,190,84), (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)(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(69,189)(70,188)(71,187)(72,186)(73,185)(74,184)(75,183)(76,182)(77,181)(78,180)(79,179)(80,178)(81,177)(82,176)(83,175)(84,174)(85,173)(86,172)(87,171)(88,170)(89,169)(90,168)(91,167)(92,166)(93,165)(94,164)(95,163)(96,162)(97,161)(98,160)(99,159)(100,158)(101,157)(102,156)(103,155)(104,154)(105,153)(106,152)(107,151)(108,150)(109,149)(110,148)(111,147)(112,146)(113,145)(114,144)(115,143)(116,142)(117,141)(118,140)(119,139)(120,138)(121,137)(122,204)(123,203)(124,202)(125,201)(126,200)(127,199)(128,198)(129,197)(130,196)(131,195)(132,194)(133,193)(134,192)(135,191)(136,190)>;
G:=Group( (1,85,191)(2,192,86)(3,87,193)(4,194,88)(5,89,195)(6,196,90)(7,91,197)(8,198,92)(9,93,199)(10,200,94)(11,95,201)(12,202,96)(13,97,203)(14,204,98)(15,99,137)(16,138,100)(17,101,139)(18,140,102)(19,103,141)(20,142,104)(21,105,143)(22,144,106)(23,107,145)(24,146,108)(25,109,147)(26,148,110)(27,111,149)(28,150,112)(29,113,151)(30,152,114)(31,115,153)(32,154,116)(33,117,155)(34,156,118)(35,119,157)(36,158,120)(37,121,159)(38,160,122)(39,123,161)(40,162,124)(41,125,163)(42,164,126)(43,127,165)(44,166,128)(45,129,167)(46,168,130)(47,131,169)(48,170,132)(49,133,171)(50,172,134)(51,135,173)(52,174,136)(53,69,175)(54,176,70)(55,71,177)(56,178,72)(57,73,179)(58,180,74)(59,75,181)(60,182,76)(61,77,183)(62,184,78)(63,79,185)(64,186,80)(65,81,187)(66,188,82)(67,83,189)(68,190,84), (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)(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,27)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(69,189)(70,188)(71,187)(72,186)(73,185)(74,184)(75,183)(76,182)(77,181)(78,180)(79,179)(80,178)(81,177)(82,176)(83,175)(84,174)(85,173)(86,172)(87,171)(88,170)(89,169)(90,168)(91,167)(92,166)(93,165)(94,164)(95,163)(96,162)(97,161)(98,160)(99,159)(100,158)(101,157)(102,156)(103,155)(104,154)(105,153)(106,152)(107,151)(108,150)(109,149)(110,148)(111,147)(112,146)(113,145)(114,144)(115,143)(116,142)(117,141)(118,140)(119,139)(120,138)(121,137)(122,204)(123,203)(124,202)(125,201)(126,200)(127,199)(128,198)(129,197)(130,196)(131,195)(132,194)(133,193)(134,192)(135,191)(136,190) );
G=PermutationGroup([[(1,85,191),(2,192,86),(3,87,193),(4,194,88),(5,89,195),(6,196,90),(7,91,197),(8,198,92),(9,93,199),(10,200,94),(11,95,201),(12,202,96),(13,97,203),(14,204,98),(15,99,137),(16,138,100),(17,101,139),(18,140,102),(19,103,141),(20,142,104),(21,105,143),(22,144,106),(23,107,145),(24,146,108),(25,109,147),(26,148,110),(27,111,149),(28,150,112),(29,113,151),(30,152,114),(31,115,153),(32,154,116),(33,117,155),(34,156,118),(35,119,157),(36,158,120),(37,121,159),(38,160,122),(39,123,161),(40,162,124),(41,125,163),(42,164,126),(43,127,165),(44,166,128),(45,129,167),(46,168,130),(47,131,169),(48,170,132),(49,133,171),(50,172,134),(51,135,173),(52,174,136),(53,69,175),(54,176,70),(55,71,177),(56,178,72),(57,73,179),(58,180,74),(59,75,181),(60,182,76),(61,77,183),(62,184,78),(63,79,185),(64,186,80),(65,81,187),(66,188,82),(67,83,189),(68,190,84)], [(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),(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)], [(1,51),(2,50),(3,49),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,40),(13,39),(14,38),(15,37),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28),(25,27),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(69,189),(70,188),(71,187),(72,186),(73,185),(74,184),(75,183),(76,182),(77,181),(78,180),(79,179),(80,178),(81,177),(82,176),(83,175),(84,174),(85,173),(86,172),(87,171),(88,170),(89,169),(90,168),(91,167),(92,166),(93,165),(94,164),(95,163),(96,162),(97,161),(98,160),(99,159),(100,158),(101,157),(102,156),(103,155),(104,154),(105,153),(106,152),(107,151),(108,150),(109,149),(110,148),(111,147),(112,146),(113,145),(114,144),(115,143),(116,142),(117,141),(118,140),(119,139),(120,138),(121,137),(122,204),(123,203),(124,202),(125,201),(126,200),(127,199),(128,198),(129,197),(130,196),(131,195),(132,194),(133,193),(134,192),(135,191),(136,190)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51H | 68A | ··· | 68P | 102A | ··· | 102H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 |
| size | 1 | 1 | 34 | 102 | 2 | 6 | 2 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | C3⋊D4 | D17 | D34 | D68 | S3×D17 | C3⋊D68 |
| kernel | C3⋊D68 | Dic3×C17 | C6×D17 | D102 | D34 | C51 | C34 | C17 | Dic3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 16 | 8 | 8 |
Matrix representation of C3⋊D68 ►in GL4(𝔽409) generated by
| 0 | 1 | 0 | 0 |
| 408 | 408 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 237 | 65 | 0 | 0 |
| 237 | 172 | 0 | 0 |
| 0 | 0 | 192 | 310 |
| 0 | 0 | 135 | 382 |
| 408 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 57 | 408 |
| 0 | 0 | 385 | 352 |
G:=sub<GL(4,GF(409))| [0,408,0,0,1,408,0,0,0,0,1,0,0,0,0,1],[237,237,0,0,65,172,0,0,0,0,192,135,0,0,310,382],[408,1,0,0,0,1,0,0,0,0,57,385,0,0,408,352] >;
C3⋊D68 in GAP, Magma, Sage, TeX
C_3\rtimes D_{68} % in TeX
G:=Group("C3:D68"); // GroupNames label
G:=SmallGroup(408,11);
// by ID
G=gap.SmallGroup(408,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-17,61,26,168,9604]);
// Polycyclic
G:=Group<a,b,c|a^3=b^68=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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