Copied to
clipboard

G = S3×C67order 402 = 2·3·67

Direct product of C67 and S3

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

Aliases: S3×C67, C3⋊C134, C2013C2, SmallGroup(402,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C67
C1C3C201 — S3×C67
C3 — S3×C67
C1C67

Generators and relations for S3×C67
 G = < a,b,c | a67=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C134

Smallest permutation representation of S3×C67
On 201 points
Generators in S201
(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)
(1 107 153)(2 108 154)(3 109 155)(4 110 156)(5 111 157)(6 112 158)(7 113 159)(8 114 160)(9 115 161)(10 116 162)(11 117 163)(12 118 164)(13 119 165)(14 120 166)(15 121 167)(16 122 168)(17 123 169)(18 124 170)(19 125 171)(20 126 172)(21 127 173)(22 128 174)(23 129 175)(24 130 176)(25 131 177)(26 132 178)(27 133 179)(28 134 180)(29 68 181)(30 69 182)(31 70 183)(32 71 184)(33 72 185)(34 73 186)(35 74 187)(36 75 188)(37 76 189)(38 77 190)(39 78 191)(40 79 192)(41 80 193)(42 81 194)(43 82 195)(44 83 196)(45 84 197)(46 85 198)(47 86 199)(48 87 200)(49 88 201)(50 89 135)(51 90 136)(52 91 137)(53 92 138)(54 93 139)(55 94 140)(56 95 141)(57 96 142)(58 97 143)(59 98 144)(60 99 145)(61 100 146)(62 101 147)(63 102 148)(64 103 149)(65 104 150)(66 105 151)(67 106 152)
(68 181)(69 182)(70 183)(71 184)(72 185)(73 186)(74 187)(75 188)(76 189)(77 190)(78 191)(79 192)(80 193)(81 194)(82 195)(83 196)(84 197)(85 198)(86 199)(87 200)(88 201)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)(113 159)(114 160)(115 161)(116 162)(117 163)(118 164)(119 165)(120 166)(121 167)(122 168)(123 169)(124 170)(125 171)(126 172)(127 173)(128 174)(129 175)(130 176)(131 177)(132 178)(133 179)(134 180)

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

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

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

201 conjugacy classes

class 1  2  3 67A···67BN134A···134BN201A···201BN
order12367···67134···134201···201
size1321···13···32···2

201 irreducible representations

dim111122
type+++
imageC1C2C67C134S3S3×C67
kernelS3×C67C201S3C3C67C1
# reps116666166

Matrix representation of S3×C67 in GL2(𝔽1609) generated by

8930
0893
,
16081608
10
,
10
16081608
G:=sub<GL(2,GF(1609))| [893,0,0,893],[1608,1,1608,0],[1,1608,0,1608] >;

S3×C67 in GAP, Magma, Sage, TeX

S_3\times C_{67}
% in TeX

G:=Group("S3xC67");
// GroupNames label

G:=SmallGroup(402,3);
// by ID

G=gap.SmallGroup(402,3);
# by ID

G:=PCGroup([3,-2,-67,-3,2414]);
// Polycyclic

G:=Group<a,b,c|a^67=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×C67 in TeX

׿
×
𝔽