direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D68, C51⋊5D4, C68⋊1C6, C204⋊3C2, D34⋊1C6, C12⋊3D17, C6.15D34, C102.15C22, C4⋊(C3×D17), C17⋊1(C3×D4), (C6×D17)⋊4C2, C34.3(C2×C6), C2.4(C6×D17), SmallGroup(408,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D68
G = < a,b,c | a3=b68=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D68
►On 204 points
Generators in S204
(1 168 92)(2 169 93)(3 170 94)(4 171 95)(5 172 96)(6 173 97)(7 174 98)(8 175 99)(9 176 100)(10 177 101)(11 178 102)(12 179 103)(13 180 104)(14 181 105)(15 182 106)(16 183 107)(17 184 108)(18 185 109)(19 186 110)(20 187 111)(21 188 112)(22 189 113)(23 190 114)(24 191 115)(25 192 116)(26 193 117)(27 194 118)(28 195 119)(29 196 120)(30 197 121)(31 198 122)(32 199 123)(33 200 124)(34 201 125)(35 202 126)(36 203 127)(37 204 128)(38 137 129)(39 138 130)(40 139 131)(41 140 132)(42 141 133)(43 142 134)(44 143 135)(45 144 136)(46 145 69)(47 146 70)(48 147 71)(49 148 72)(50 149 73)(51 150 74)(52 151 75)(53 152 76)(54 153 77)(55 154 78)(56 155 79)(57 156 80)(58 157 81)(59 158 82)(60 159 83)(61 160 84)(62 161 85)(63 162 86)(64 163 87)(65 164 88)(66 165 89)(67 166 90)(68 167 91)
(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 97)(70 96)(71 95)(72 94)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(98 136)(99 135)(100 134)(101 133)(102 132)(103 131)(104 130)(105 129)(106 128)(107 127)(108 126)(109 125)(110 124)(111 123)(112 122)(113 121)(114 120)(115 119)(116 118)(137 181)(138 180)(139 179)(140 178)(141 177)(142 176)(143 175)(144 174)(145 173)(146 172)(147 171)(148 170)(149 169)(150 168)(151 167)(152 166)(153 165)(154 164)(155 163)(156 162)(157 161)(158 160)(182 204)(183 203)(184 202)(185 201)(186 200)(187 199)(188 198)(189 197)(190 196)(191 195)(192 194)
G:=sub<Sym(204)| (1,168,92)(2,169,93)(3,170,94)(4,171,95)(5,172,96)(6,173,97)(7,174,98)(8,175,99)(9,176,100)(10,177,101)(11,178,102)(12,179,103)(13,180,104)(14,181,105)(15,182,106)(16,183,107)(17,184,108)(18,185,109)(19,186,110)(20,187,111)(21,188,112)(22,189,113)(23,190,114)(24,191,115)(25,192,116)(26,193,117)(27,194,118)(28,195,119)(29,196,120)(30,197,121)(31,198,122)(32,199,123)(33,200,124)(34,201,125)(35,202,126)(36,203,127)(37,204,128)(38,137,129)(39,138,130)(40,139,131)(41,140,132)(42,141,133)(43,142,134)(44,143,135)(45,144,136)(46,145,69)(47,146,70)(48,147,71)(49,148,72)(50,149,73)(51,150,74)(52,151,75)(53,152,76)(54,153,77)(55,154,78)(56,155,79)(57,156,80)(58,157,81)(59,158,82)(60,159,83)(61,160,84)(62,161,85)(63,162,86)(64,163,87)(65,164,88)(66,165,89)(67,166,90)(68,167,91), (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)(69,97)(70,96)(71,95)(72,94)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(98,136)(99,135)(100,134)(101,133)(102,132)(103,131)(104,130)(105,129)(106,128)(107,127)(108,126)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(137,181)(138,180)(139,179)(140,178)(141,177)(142,176)(143,175)(144,174)(145,173)(146,172)(147,171)(148,170)(149,169)(150,168)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(182,204)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,196)(191,195)(192,194)>;
G:=Group( (1,168,92)(2,169,93)(3,170,94)(4,171,95)(5,172,96)(6,173,97)(7,174,98)(8,175,99)(9,176,100)(10,177,101)(11,178,102)(12,179,103)(13,180,104)(14,181,105)(15,182,106)(16,183,107)(17,184,108)(18,185,109)(19,186,110)(20,187,111)(21,188,112)(22,189,113)(23,190,114)(24,191,115)(25,192,116)(26,193,117)(27,194,118)(28,195,119)(29,196,120)(30,197,121)(31,198,122)(32,199,123)(33,200,124)(34,201,125)(35,202,126)(36,203,127)(37,204,128)(38,137,129)(39,138,130)(40,139,131)(41,140,132)(42,141,133)(43,142,134)(44,143,135)(45,144,136)(46,145,69)(47,146,70)(48,147,71)(49,148,72)(50,149,73)(51,150,74)(52,151,75)(53,152,76)(54,153,77)(55,154,78)(56,155,79)(57,156,80)(58,157,81)(59,158,82)(60,159,83)(61,160,84)(62,161,85)(63,162,86)(64,163,87)(65,164,88)(66,165,89)(67,166,90)(68,167,91), (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)(69,97)(70,96)(71,95)(72,94)(73,93)(74,92)(75,91)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(98,136)(99,135)(100,134)(101,133)(102,132)(103,131)(104,130)(105,129)(106,128)(107,127)(108,126)(109,125)(110,124)(111,123)(112,122)(113,121)(114,120)(115,119)(116,118)(137,181)(138,180)(139,179)(140,178)(141,177)(142,176)(143,175)(144,174)(145,173)(146,172)(147,171)(148,170)(149,169)(150,168)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(182,204)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,196)(191,195)(192,194) );
G=PermutationGroup([[(1,168,92),(2,169,93),(3,170,94),(4,171,95),(5,172,96),(6,173,97),(7,174,98),(8,175,99),(9,176,100),(10,177,101),(11,178,102),(12,179,103),(13,180,104),(14,181,105),(15,182,106),(16,183,107),(17,184,108),(18,185,109),(19,186,110),(20,187,111),(21,188,112),(22,189,113),(23,190,114),(24,191,115),(25,192,116),(26,193,117),(27,194,118),(28,195,119),(29,196,120),(30,197,121),(31,198,122),(32,199,123),(33,200,124),(34,201,125),(35,202,126),(36,203,127),(37,204,128),(38,137,129),(39,138,130),(40,139,131),(41,140,132),(42,141,133),(43,142,134),(44,143,135),(45,144,136),(46,145,69),(47,146,70),(48,147,71),(49,148,72),(50,149,73),(51,150,74),(52,151,75),(53,152,76),(54,153,77),(55,154,78),(56,155,79),(57,156,80),(58,157,81),(59,158,82),(60,159,83),(61,160,84),(62,161,85),(63,162,86),(64,163,87),(65,164,88),(66,165,89),(67,166,90),(68,167,91)], [(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),(69,97),(70,96),(71,95),(72,94),(73,93),(74,92),(75,91),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(98,136),(99,135),(100,134),(101,133),(102,132),(103,131),(104,130),(105,129),(106,128),(107,127),(108,126),(109,125),(110,124),(111,123),(112,122),(113,121),(114,120),(115,119),(116,118),(137,181),(138,180),(139,179),(140,178),(141,177),(142,176),(143,175),(144,174),(145,173),(146,172),(147,171),(148,170),(149,169),(150,168),(151,167),(152,166),(153,165),(154,164),(155,163),(156,162),(157,161),(158,160),(182,204),(183,203),(184,202),(185,201),(186,200),(187,199),(188,198),(189,197),(190,196),(191,195),(192,194)]])
111 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51P | 68A | ··· | 68P | 102A | ··· | 102P | 204A | ··· | 204AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 | 204 | ··· | 204 |
| size | 1 | 1 | 34 | 34 | 1 | 1 | 2 | 1 | 1 | 34 | 34 | 34 | 34 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
111 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | C3×D4 | D17 | D34 | C3×D17 | D68 | C6×D17 | C3×D68 |
| kernel | C3×D68 | C204 | C6×D17 | D68 | C68 | D34 | C51 | C17 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 8 | 8 | 16 | 16 | 16 | 32 |
Matrix representation of C3×D68 ►in GL2(𝔽409) generated by
| 53 | 0 |
| 0 | 53 |
| 294 | 230 |
| 179 | 83 |
| 181 | 305 |
| 315 | 228 |
G:=sub<GL(2,GF(409))| [53,0,0,53],[294,179,230,83],[181,315,305,228] >;
C3×D68 in GAP, Magma, Sage, TeX
C_3\times D_{68} % in TeX
G:=Group("C3xD68"); // GroupNames label
G:=SmallGroup(408,17);
// by ID
G=gap.SmallGroup(408,17);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-17,141,66,9604]);
// Polycyclic
G:=Group<a,b,c|a^3=b^68=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export