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G = C3×D67order 402 = 2·3·67

Direct product of C3 and D67

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

Aliases: C3×D67, C673C6, C2012C2, SmallGroup(402,4)

Series: Derived Chief Lower central Upper central

C1C67 — C3×D67
C1C67C201 — C3×D67
C67 — C3×D67
C1C3

Generators and relations for C3×D67
 G = < a,b,c | a3=b67=c2=1, ab=ba, ac=ca, cbc=b-1 >

67C2
67C6

Smallest permutation representation of C3×D67
On 201 points
Generators in S201
(1 180 79)(2 181 80)(3 182 81)(4 183 82)(5 184 83)(6 185 84)(7 186 85)(8 187 86)(9 188 87)(10 189 88)(11 190 89)(12 191 90)(13 192 91)(14 193 92)(15 194 93)(16 195 94)(17 196 95)(18 197 96)(19 198 97)(20 199 98)(21 200 99)(22 201 100)(23 135 101)(24 136 102)(25 137 103)(26 138 104)(27 139 105)(28 140 106)(29 141 107)(30 142 108)(31 143 109)(32 144 110)(33 145 111)(34 146 112)(35 147 113)(36 148 114)(37 149 115)(38 150 116)(39 151 117)(40 152 118)(41 153 119)(42 154 120)(43 155 121)(44 156 122)(45 157 123)(46 158 124)(47 159 125)(48 160 126)(49 161 127)(50 162 128)(51 163 129)(52 164 130)(53 165 131)(54 166 132)(55 167 133)(56 168 134)(57 169 68)(58 170 69)(59 171 70)(60 172 71)(61 173 72)(62 174 73)(63 175 74)(64 176 75)(65 177 76)(66 178 77)(67 179 78)
(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 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 60)(9 59)(10 58)(11 57)(12 56)(13 55)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(90 134)(91 133)(92 132)(93 131)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)(101 123)(102 122)(103 121)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(135 157)(136 156)(137 155)(138 154)(139 153)(140 152)(141 151)(142 150)(143 149)(144 148)(145 147)(158 201)(159 200)(160 199)(161 198)(162 197)(163 196)(164 195)(165 194)(166 193)(167 192)(168 191)(169 190)(170 189)(171 188)(172 187)(173 186)(174 185)(175 184)(176 183)(177 182)(178 181)(179 180)

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

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

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

105 conjugacy classes

class 1  2 3A3B6A6B67A···67AG201A···201BN
order12336667···67201···201
size1671167672···22···2

105 irreducible representations

dim111122
type+++
imageC1C2C3C6D67C3×D67
kernelC3×D67C201D67C67C3C1
# reps11223366

Matrix representation of C3×D67 in GL2(𝔽1609) generated by

2500
0250
,
4841042
1608932
,
484950
16081125
G:=sub<GL(2,GF(1609))| [250,0,0,250],[484,1608,1042,932],[484,1608,950,1125] >;

C3×D67 in GAP, Magma, Sage, TeX

C_3\times D_{67}
% in TeX

G:=Group("C3xD67");
// GroupNames label

G:=SmallGroup(402,4);
// by ID

G=gap.SmallGroup(402,4);
# by ID

G:=PCGroup([3,-2,-3,-67,3566]);
// Polycyclic

G:=Group<a,b,c|a^3=b^67=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D67 in TeX

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