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G = S3×C74order 444 = 22·3·37

Direct product of C74 and S3

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

Aliases: S3×C74, C6⋊C74, C2223C2, C1114C22, C3⋊(C2×C74), SmallGroup(444,16)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C74
C1C3C111S3×C37 — S3×C74
C3 — S3×C74
C1C74

Generators and relations for S3×C74
 G = < a,b,c | a74=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C74
3C74
3C2×C74

Smallest permutation representation of S3×C74
On 222 points
Generators in S222
(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)(149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222)
(1 79 179)(2 80 180)(3 81 181)(4 82 182)(5 83 183)(6 84 184)(7 85 185)(8 86 186)(9 87 187)(10 88 188)(11 89 189)(12 90 190)(13 91 191)(14 92 192)(15 93 193)(16 94 194)(17 95 195)(18 96 196)(19 97 197)(20 98 198)(21 99 199)(22 100 200)(23 101 201)(24 102 202)(25 103 203)(26 104 204)(27 105 205)(28 106 206)(29 107 207)(30 108 208)(31 109 209)(32 110 210)(33 111 211)(34 112 212)(35 113 213)(36 114 214)(37 115 215)(38 116 216)(39 117 217)(40 118 218)(41 119 219)(42 120 220)(43 121 221)(44 122 222)(45 123 149)(46 124 150)(47 125 151)(48 126 152)(49 127 153)(50 128 154)(51 129 155)(52 130 156)(53 131 157)(54 132 158)(55 133 159)(56 134 160)(57 135 161)(58 136 162)(59 137 163)(60 138 164)(61 139 165)(62 140 166)(63 141 167)(64 142 168)(65 143 169)(66 144 170)(67 145 171)(68 146 172)(69 147 173)(70 148 174)(71 75 175)(72 76 176)(73 77 177)(74 78 178)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(75 212)(76 213)(77 214)(78 215)(79 216)(80 217)(81 218)(82 219)(83 220)(84 221)(85 222)(86 149)(87 150)(88 151)(89 152)(90 153)(91 154)(92 155)(93 156)(94 157)(95 158)(96 159)(97 160)(98 161)(99 162)(100 163)(101 164)(102 165)(103 166)(104 167)(105 168)(106 169)(107 170)(108 171)(109 172)(110 173)(111 174)(112 175)(113 176)(114 177)(115 178)(116 179)(117 180)(118 181)(119 182)(120 183)(121 184)(122 185)(123 186)(124 187)(125 188)(126 189)(127 190)(128 191)(129 192)(130 193)(131 194)(132 195)(133 196)(134 197)(135 198)(136 199)(137 200)(138 201)(139 202)(140 203)(141 204)(142 205)(143 206)(144 207)(145 208)(146 209)(147 210)(148 211)

G:=sub<Sym(222)| (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)(149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222), (1,79,179)(2,80,180)(3,81,181)(4,82,182)(5,83,183)(6,84,184)(7,85,185)(8,86,186)(9,87,187)(10,88,188)(11,89,189)(12,90,190)(13,91,191)(14,92,192)(15,93,193)(16,94,194)(17,95,195)(18,96,196)(19,97,197)(20,98,198)(21,99,199)(22,100,200)(23,101,201)(24,102,202)(25,103,203)(26,104,204)(27,105,205)(28,106,206)(29,107,207)(30,108,208)(31,109,209)(32,110,210)(33,111,211)(34,112,212)(35,113,213)(36,114,214)(37,115,215)(38,116,216)(39,117,217)(40,118,218)(41,119,219)(42,120,220)(43,121,221)(44,122,222)(45,123,149)(46,124,150)(47,125,151)(48,126,152)(49,127,153)(50,128,154)(51,129,155)(52,130,156)(53,131,157)(54,132,158)(55,133,159)(56,134,160)(57,135,161)(58,136,162)(59,137,163)(60,138,164)(61,139,165)(62,140,166)(63,141,167)(64,142,168)(65,143,169)(66,144,170)(67,145,171)(68,146,172)(69,147,173)(70,148,174)(71,75,175)(72,76,176)(73,77,177)(74,78,178), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(75,212)(76,213)(77,214)(78,215)(79,216)(80,217)(81,218)(82,219)(83,220)(84,221)(85,222)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)(133,196)(134,197)(135,198)(136,199)(137,200)(138,201)(139,202)(140,203)(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,211)>;

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)(149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222), (1,79,179)(2,80,180)(3,81,181)(4,82,182)(5,83,183)(6,84,184)(7,85,185)(8,86,186)(9,87,187)(10,88,188)(11,89,189)(12,90,190)(13,91,191)(14,92,192)(15,93,193)(16,94,194)(17,95,195)(18,96,196)(19,97,197)(20,98,198)(21,99,199)(22,100,200)(23,101,201)(24,102,202)(25,103,203)(26,104,204)(27,105,205)(28,106,206)(29,107,207)(30,108,208)(31,109,209)(32,110,210)(33,111,211)(34,112,212)(35,113,213)(36,114,214)(37,115,215)(38,116,216)(39,117,217)(40,118,218)(41,119,219)(42,120,220)(43,121,221)(44,122,222)(45,123,149)(46,124,150)(47,125,151)(48,126,152)(49,127,153)(50,128,154)(51,129,155)(52,130,156)(53,131,157)(54,132,158)(55,133,159)(56,134,160)(57,135,161)(58,136,162)(59,137,163)(60,138,164)(61,139,165)(62,140,166)(63,141,167)(64,142,168)(65,143,169)(66,144,170)(67,145,171)(68,146,172)(69,147,173)(70,148,174)(71,75,175)(72,76,176)(73,77,177)(74,78,178), (1,38)(2,39)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(75,212)(76,213)(77,214)(78,215)(79,216)(80,217)(81,218)(82,219)(83,220)(84,221)(85,222)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)(133,196)(134,197)(135,198)(136,199)(137,200)(138,201)(139,202)(140,203)(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,211) );

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),(149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222)], [(1,79,179),(2,80,180),(3,81,181),(4,82,182),(5,83,183),(6,84,184),(7,85,185),(8,86,186),(9,87,187),(10,88,188),(11,89,189),(12,90,190),(13,91,191),(14,92,192),(15,93,193),(16,94,194),(17,95,195),(18,96,196),(19,97,197),(20,98,198),(21,99,199),(22,100,200),(23,101,201),(24,102,202),(25,103,203),(26,104,204),(27,105,205),(28,106,206),(29,107,207),(30,108,208),(31,109,209),(32,110,210),(33,111,211),(34,112,212),(35,113,213),(36,114,214),(37,115,215),(38,116,216),(39,117,217),(40,118,218),(41,119,219),(42,120,220),(43,121,221),(44,122,222),(45,123,149),(46,124,150),(47,125,151),(48,126,152),(49,127,153),(50,128,154),(51,129,155),(52,130,156),(53,131,157),(54,132,158),(55,133,159),(56,134,160),(57,135,161),(58,136,162),(59,137,163),(60,138,164),(61,139,165),(62,140,166),(63,141,167),(64,142,168),(65,143,169),(66,144,170),(67,145,171),(68,146,172),(69,147,173),(70,148,174),(71,75,175),(72,76,176),(73,77,177),(74,78,178)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,73),(37,74),(75,212),(76,213),(77,214),(78,215),(79,216),(80,217),(81,218),(82,219),(83,220),(84,221),(85,222),(86,149),(87,150),(88,151),(89,152),(90,153),(91,154),(92,155),(93,156),(94,157),(95,158),(96,159),(97,160),(98,161),(99,162),(100,163),(101,164),(102,165),(103,166),(104,167),(105,168),(106,169),(107,170),(108,171),(109,172),(110,173),(111,174),(112,175),(113,176),(114,177),(115,178),(116,179),(117,180),(118,181),(119,182),(120,183),(121,184),(122,185),(123,186),(124,187),(125,188),(126,189),(127,190),(128,191),(129,192),(130,193),(131,194),(132,195),(133,196),(134,197),(135,198),(136,199),(137,200),(138,201),(139,202),(140,203),(141,204),(142,205),(143,206),(144,207),(145,208),(146,209),(147,210),(148,211)])

222 conjugacy classes

class 1 2A2B2C 3  6 37A···37AJ74A···74AJ74AK···74DD111A···111AJ222A···222AJ
order12223637···3774···7474···74111···111222···222
size1133221···11···13···32···22···2

222 irreducible representations

dim1111112222
type+++++
imageC1C2C2C37C74C74S3D6S3×C37S3×C74
kernelS3×C74S3×C37C222D6S3C6C74C37C2C1
# reps121367236113636

Matrix representation of S3×C74 in GL3(𝔽223) generated by

22200
0680
0068
,
100
0222222
010
,
100
010
0222222
G:=sub<GL(3,GF(223))| [222,0,0,0,68,0,0,0,68],[1,0,0,0,222,1,0,222,0],[1,0,0,0,1,222,0,0,222] >;

S3×C74 in GAP, Magma, Sage, TeX

S_3\times C_{74}
% in TeX

G:=Group("S3xC74");
// GroupNames label

G:=SmallGroup(444,16);
// by ID

G=gap.SmallGroup(444,16);
# by ID

G:=PCGroup([4,-2,-2,-37,-3,4739]);
// Polycyclic

G:=Group<a,b,c|a^74=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×C74 in TeX

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