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G = S3×C68order 408 = 23·3·17

Direct product of C68 and S3

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×C68, D6.C34, C2046C2, C122C34, C34.14D6, Dic32C34, C102.19C22, C31(C2×C68), C519(C2×C4), C2.1(S3×C34), C6.2(C2×C34), (S3×C34).2C2, (Dic3×C17)⋊5C2, SmallGroup(408,21)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C68
C1C3C6C102S3×C34 — S3×C68
C3 — S3×C68
C1C68

Generators and relations for S3×C68
 G = < a,b,c | a68=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C4
3C22
3C34
3C34
3C2×C4
3C68
3C2×C34
3C2×C68

Smallest permutation representation of S3×C68
On 204 points
Generators in S204
(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 98 170)(2 99 171)(3 100 172)(4 101 173)(5 102 174)(6 103 175)(7 104 176)(8 105 177)(9 106 178)(10 107 179)(11 108 180)(12 109 181)(13 110 182)(14 111 183)(15 112 184)(16 113 185)(17 114 186)(18 115 187)(19 116 188)(20 117 189)(21 118 190)(22 119 191)(23 120 192)(24 121 193)(25 122 194)(26 123 195)(27 124 196)(28 125 197)(29 126 198)(30 127 199)(31 128 200)(32 129 201)(33 130 202)(34 131 203)(35 132 204)(36 133 137)(37 134 138)(38 135 139)(39 136 140)(40 69 141)(41 70 142)(42 71 143)(43 72 144)(44 73 145)(45 74 146)(46 75 147)(47 76 148)(48 77 149)(49 78 150)(50 79 151)(51 80 152)(52 81 153)(53 82 154)(54 83 155)(55 84 156)(56 85 157)(57 86 158)(58 87 159)(59 88 160)(60 89 161)(61 90 162)(62 91 163)(63 92 164)(64 93 165)(65 94 166)(66 95 167)(67 96 168)(68 97 169)
(69 141)(70 142)(71 143)(72 144)(73 145)(74 146)(75 147)(76 148)(77 149)(78 150)(79 151)(80 152)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 161)(90 162)(91 163)(92 164)(93 165)(94 166)(95 167)(96 168)(97 169)(98 170)(99 171)(100 172)(101 173)(102 174)(103 175)(104 176)(105 177)(106 178)(107 179)(108 180)(109 181)(110 182)(111 183)(112 184)(113 185)(114 186)(115 187)(116 188)(117 189)(118 190)(119 191)(120 192)(121 193)(122 194)(123 195)(124 196)(125 197)(126 198)(127 199)(128 200)(129 201)(130 202)(131 203)(132 204)(133 137)(134 138)(135 139)(136 140)

G:=sub<Sym(204)| (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), (1,98,170)(2,99,171)(3,100,172)(4,101,173)(5,102,174)(6,103,175)(7,104,176)(8,105,177)(9,106,178)(10,107,179)(11,108,180)(12,109,181)(13,110,182)(14,111,183)(15,112,184)(16,113,185)(17,114,186)(18,115,187)(19,116,188)(20,117,189)(21,118,190)(22,119,191)(23,120,192)(24,121,193)(25,122,194)(26,123,195)(27,124,196)(28,125,197)(29,126,198)(30,127,199)(31,128,200)(32,129,201)(33,130,202)(34,131,203)(35,132,204)(36,133,137)(37,134,138)(38,135,139)(39,136,140)(40,69,141)(41,70,142)(42,71,143)(43,72,144)(44,73,145)(45,74,146)(46,75,147)(47,76,148)(48,77,149)(49,78,150)(50,79,151)(51,80,152)(52,81,153)(53,82,154)(54,83,155)(55,84,156)(56,85,157)(57,86,158)(58,87,159)(59,88,160)(60,89,161)(61,90,162)(62,91,163)(63,92,164)(64,93,165)(65,94,166)(66,95,167)(67,96,168)(68,97,169), (69,141)(70,142)(71,143)(72,144)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,169)(98,170)(99,171)(100,172)(101,173)(102,174)(103,175)(104,176)(105,177)(106,178)(107,179)(108,180)(109,181)(110,182)(111,183)(112,184)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,197)(126,198)(127,199)(128,200)(129,201)(130,202)(131,203)(132,204)(133,137)(134,138)(135,139)(136,140)>;

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)(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,98,170)(2,99,171)(3,100,172)(4,101,173)(5,102,174)(6,103,175)(7,104,176)(8,105,177)(9,106,178)(10,107,179)(11,108,180)(12,109,181)(13,110,182)(14,111,183)(15,112,184)(16,113,185)(17,114,186)(18,115,187)(19,116,188)(20,117,189)(21,118,190)(22,119,191)(23,120,192)(24,121,193)(25,122,194)(26,123,195)(27,124,196)(28,125,197)(29,126,198)(30,127,199)(31,128,200)(32,129,201)(33,130,202)(34,131,203)(35,132,204)(36,133,137)(37,134,138)(38,135,139)(39,136,140)(40,69,141)(41,70,142)(42,71,143)(43,72,144)(44,73,145)(45,74,146)(46,75,147)(47,76,148)(48,77,149)(49,78,150)(50,79,151)(51,80,152)(52,81,153)(53,82,154)(54,83,155)(55,84,156)(56,85,157)(57,86,158)(58,87,159)(59,88,160)(60,89,161)(61,90,162)(62,91,163)(63,92,164)(64,93,165)(65,94,166)(66,95,167)(67,96,168)(68,97,169), (69,141)(70,142)(71,143)(72,144)(73,145)(74,146)(75,147)(76,148)(77,149)(78,150)(79,151)(80,152)(81,153)(82,154)(83,155)(84,156)(85,157)(86,158)(87,159)(88,160)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,169)(98,170)(99,171)(100,172)(101,173)(102,174)(103,175)(104,176)(105,177)(106,178)(107,179)(108,180)(109,181)(110,182)(111,183)(112,184)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,197)(126,198)(127,199)(128,200)(129,201)(130,202)(131,203)(132,204)(133,137)(134,138)(135,139)(136,140) );

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),(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)], [(1,98,170),(2,99,171),(3,100,172),(4,101,173),(5,102,174),(6,103,175),(7,104,176),(8,105,177),(9,106,178),(10,107,179),(11,108,180),(12,109,181),(13,110,182),(14,111,183),(15,112,184),(16,113,185),(17,114,186),(18,115,187),(19,116,188),(20,117,189),(21,118,190),(22,119,191),(23,120,192),(24,121,193),(25,122,194),(26,123,195),(27,124,196),(28,125,197),(29,126,198),(30,127,199),(31,128,200),(32,129,201),(33,130,202),(34,131,203),(35,132,204),(36,133,137),(37,134,138),(38,135,139),(39,136,140),(40,69,141),(41,70,142),(42,71,143),(43,72,144),(44,73,145),(45,74,146),(46,75,147),(47,76,148),(48,77,149),(49,78,150),(50,79,151),(51,80,152),(52,81,153),(53,82,154),(54,83,155),(55,84,156),(56,85,157),(57,86,158),(58,87,159),(59,88,160),(60,89,161),(61,90,162),(62,91,163),(63,92,164),(64,93,165),(65,94,166),(66,95,167),(67,96,168),(68,97,169)], [(69,141),(70,142),(71,143),(72,144),(73,145),(74,146),(75,147),(76,148),(77,149),(78,150),(79,151),(80,152),(81,153),(82,154),(83,155),(84,156),(85,157),(86,158),(87,159),(88,160),(89,161),(90,162),(91,163),(92,164),(93,165),(94,166),(95,167),(96,168),(97,169),(98,170),(99,171),(100,172),(101,173),(102,174),(103,175),(104,176),(105,177),(106,178),(107,179),(108,180),(109,181),(110,182),(111,183),(112,184),(113,185),(114,186),(115,187),(116,188),(117,189),(118,190),(119,191),(120,192),(121,193),(122,194),(123,195),(124,196),(125,197),(126,198),(127,199),(128,200),(129,201),(130,202),(131,203),(132,204),(133,137),(134,138),(135,139),(136,140)])

204 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 12A12B17A···17P34A···34P34Q···34AV51A···51P68A···68AF68AG···68BL102A···102P204A···204AF
order1222344446121217···1734···3434···3451···5168···6868···68102···102204···204
size1133211332221···11···13···32···21···13···32···22···2

204 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C4C17C34C34C34C68S3D6C4×S3S3×C17S3×C34S3×C68
kernelS3×C68Dic3×C17C204S3×C34S3×C17C4×S3Dic3C12D6S3C68C34C17C4C2C1
# reps111141616161664112161632

Matrix representation of S3×C68 in GL3(𝔽409) generated by

14300
03400
00340
,
100
00408
01408
,
40800
01408
00408
G:=sub<GL(3,GF(409))| [143,0,0,0,340,0,0,0,340],[1,0,0,0,0,1,0,408,408],[408,0,0,0,1,0,0,408,408] >;

S3×C68 in GAP, Magma, Sage, TeX

S_3\times C_{68}
% in TeX

G:=Group("S3xC68");
// GroupNames label

G:=SmallGroup(408,21);
// by ID

G=gap.SmallGroup(408,21);
# by ID

G:=PCGroup([5,-2,-2,-17,-2,-3,346,6804]);
// Polycyclic

G:=Group<a,b,c|a^68=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C68 in TeX

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