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G = C6×D37order 444 = 22·3·37

Direct product of C6 and D37

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

Aliases: C6×D37, C743C6, C2222C2, C1113C22, C373(C2×C6), SmallGroup(444,15)

Series: Derived Chief Lower central Upper central

C1C37 — C6×D37
C1C37C111C3×D37 — C6×D37
C37 — C6×D37
C1C6

Generators and relations for C6×D37
 G = < a,b,c | a6=b37=c2=1, ab=ba, ac=ca, cbc=b-1 >

37C2
37C2
37C22
37C6
37C6
37C2×C6

Smallest permutation representation of C6×D37
On 222 points
Generators in S222
(1 183 91 142 54 192)(2 184 92 143 55 193)(3 185 93 144 56 194)(4 149 94 145 57 195)(5 150 95 146 58 196)(6 151 96 147 59 197)(7 152 97 148 60 198)(8 153 98 112 61 199)(9 154 99 113 62 200)(10 155 100 114 63 201)(11 156 101 115 64 202)(12 157 102 116 65 203)(13 158 103 117 66 204)(14 159 104 118 67 205)(15 160 105 119 68 206)(16 161 106 120 69 207)(17 162 107 121 70 208)(18 163 108 122 71 209)(19 164 109 123 72 210)(20 165 110 124 73 211)(21 166 111 125 74 212)(22 167 75 126 38 213)(23 168 76 127 39 214)(24 169 77 128 40 215)(25 170 78 129 41 216)(26 171 79 130 42 217)(27 172 80 131 43 218)(28 173 81 132 44 219)(29 174 82 133 45 220)(30 175 83 134 46 221)(31 176 84 135 47 222)(32 177 85 136 48 186)(33 178 86 137 49 187)(34 179 87 138 50 188)(35 180 88 139 51 189)(36 181 89 140 52 190)(37 182 90 141 53 191)
(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)
(1 141)(2 140)(3 139)(4 138)(5 137)(6 136)(7 135)(8 134)(9 133)(10 132)(11 131)(12 130)(13 129)(14 128)(15 127)(16 126)(17 125)(18 124)(19 123)(20 122)(21 121)(22 120)(23 119)(24 118)(25 117)(26 116)(27 115)(28 114)(29 113)(30 112)(31 148)(32 147)(33 146)(34 145)(35 144)(36 143)(37 142)(38 161)(39 160)(40 159)(41 158)(42 157)(43 156)(44 155)(45 154)(46 153)(47 152)(48 151)(49 150)(50 149)(51 185)(52 184)(53 183)(54 182)(55 181)(56 180)(57 179)(58 178)(59 177)(60 176)(61 175)(62 174)(63 173)(64 172)(65 171)(66 170)(67 169)(68 168)(69 167)(70 166)(71 165)(72 164)(73 163)(74 162)(75 207)(76 206)(77 205)(78 204)(79 203)(80 202)(81 201)(82 200)(83 199)(84 198)(85 197)(86 196)(87 195)(88 194)(89 193)(90 192)(91 191)(92 190)(93 189)(94 188)(95 187)(96 186)(97 222)(98 221)(99 220)(100 219)(101 218)(102 217)(103 216)(104 215)(105 214)(106 213)(107 212)(108 211)(109 210)(110 209)(111 208)

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

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

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

120 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F37A···37R74A···74R111A···111AJ222A···222AJ
order12223366666637···3774···74111···111222···222
size1137371111373737372···22···22···22···2

120 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D37D74C3×D37C6×D37
kernelC6×D37C3×D37C222D74D37C74C6C3C2C1
# reps12124218183636

Matrix representation of C6×D37 in GL3(𝔽223) generated by

4000
010
001
,
100
01151
05949
,
100
0216164
01527
G:=sub<GL(3,GF(223))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,115,59,0,1,49],[1,0,0,0,216,152,0,164,7] >;

C6×D37 in GAP, Magma, Sage, TeX

C_6\times D_{37}
% in TeX

G:=Group("C6xD37");
// GroupNames label

G:=SmallGroup(444,15);
// by ID

G=gap.SmallGroup(444,15);
# by ID

G:=PCGroup([4,-2,-2,-3,-37,6915]);
// Polycyclic

G:=Group<a,b,c|a^6=b^37=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D37 in TeX

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