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G = C3×D71order 426 = 2·3·71

Direct product of C3 and D71

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

Aliases: C3×D71, C71⋊C6, C2132C2, SmallGroup(426,2)

Series: Derived Chief Lower central Upper central

C1C71 — C3×D71
C1C71C213 — C3×D71
C71 — C3×D71
C1C3

Generators and relations for C3×D71
 G = < a,b,c | a3=b71=c2=1, ab=ba, ac=ca, cbc=b-1 >

71C2
71C6

Smallest permutation representation of C3×D71
On 213 points
Generators in S213
(1 168 131)(2 169 132)(3 170 133)(4 171 134)(5 172 135)(6 173 136)(7 174 137)(8 175 138)(9 176 139)(10 177 140)(11 178 141)(12 179 142)(13 180 72)(14 181 73)(15 182 74)(16 183 75)(17 184 76)(18 185 77)(19 186 78)(20 187 79)(21 188 80)(22 189 81)(23 190 82)(24 191 83)(25 192 84)(26 193 85)(27 194 86)(28 195 87)(29 196 88)(30 197 89)(31 198 90)(32 199 91)(33 200 92)(34 201 93)(35 202 94)(36 203 95)(37 204 96)(38 205 97)(39 206 98)(40 207 99)(41 208 100)(42 209 101)(43 210 102)(44 211 103)(45 212 104)(46 213 105)(47 143 106)(48 144 107)(49 145 108)(50 146 109)(51 147 110)(52 148 111)(53 149 112)(54 150 113)(55 151 114)(56 152 115)(57 153 116)(58 154 117)(59 155 118)(60 156 119)(61 157 120)(62 158 121)(63 159 122)(64 160 123)(65 161 124)(66 162 125)(67 163 126)(68 164 127)(69 165 128)(70 166 129)(71 167 130)
(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)
(1 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 60)(13 59)(14 58)(15 57)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(72 118)(73 117)(74 116)(75 115)(76 114)(77 113)(78 112)(79 111)(80 110)(81 109)(82 108)(83 107)(84 106)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(119 142)(120 141)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(143 192)(144 191)(145 190)(146 189)(147 188)(148 187)(149 186)(150 185)(151 184)(152 183)(153 182)(154 181)(155 180)(156 179)(157 178)(158 177)(159 176)(160 175)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 168)(193 213)(194 212)(195 211)(196 210)(197 209)(198 208)(199 207)(200 206)(201 205)(202 204)

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

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

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

111 conjugacy classes

class 1  2 3A3B6A6B71A···71AI213A···213BR
order12336671···71213···213
size1711171712···22···2

111 irreducible representations

dim111122
type+++
imageC1C2C3C6D71C3×D71
kernelC3×D71C213D71C71C3C1
# reps11223570

Matrix representation of C3×D71 in GL2(𝔽853) generated by

6320
0632
,
01
852418
,
01
10
G:=sub<GL(2,GF(853))| [632,0,0,632],[0,852,1,418],[0,1,1,0] >;

C3×D71 in GAP, Magma, Sage, TeX

C_3\times D_{71}
% in TeX

G:=Group("C3xD71");
// GroupNames label

G:=SmallGroup(426,2);
// by ID

G=gap.SmallGroup(426,2);
# by ID

G:=PCGroup([3,-2,-3,-71,3782]);
// Polycyclic

G:=Group<a,b,c|a^3=b^71=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D71 in TeX

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