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G = C14×D13order 364 = 22·7·13

Direct product of C14 and D13

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

Aliases: C14×D13, C26⋊C14, C1822C2, C913C22, C13⋊(C2×C14), SmallGroup(364,8)

Series: Derived Chief Lower central Upper central

C1C13 — C14×D13
C1C13C91C7×D13 — C14×D13
C13 — C14×D13
C1C14

Generators and relations for C14×D13
 G = < a,b,c | a14=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

13C2
13C2
13C22
13C14
13C14
13C2×C14

Smallest permutation representation of C14×D13
On 182 points
Generators in S182
(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)
(1 39 76 69 51 97 15 160 102 175 130 150 116)(2 40 77 70 52 98 16 161 103 176 131 151 117)(3 41 78 57 53 85 17 162 104 177 132 152 118)(4 42 79 58 54 86 18 163 105 178 133 153 119)(5 29 80 59 55 87 19 164 106 179 134 154 120)(6 30 81 60 56 88 20 165 107 180 135 141 121)(7 31 82 61 43 89 21 166 108 181 136 142 122)(8 32 83 62 44 90 22 167 109 182 137 143 123)(9 33 84 63 45 91 23 168 110 169 138 144 124)(10 34 71 64 46 92 24 155 111 170 139 145 125)(11 35 72 65 47 93 25 156 112 171 140 146 126)(12 36 73 66 48 94 26 157 99 172 127 147 113)(13 37 74 67 49 95 27 158 100 173 128 148 114)(14 38 75 68 50 96 28 159 101 174 129 149 115)
(1 123)(2 124)(3 125)(4 126)(5 113)(6 114)(7 115)(8 116)(9 117)(10 118)(11 119)(12 120)(13 121)(14 122)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 147)(30 148)(31 149)(32 150)(33 151)(34 152)(35 153)(36 154)(37 141)(38 142)(39 143)(40 144)(41 145)(42 146)(43 101)(44 102)(45 103)(46 104)(47 105)(48 106)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 99)(56 100)(57 170)(58 171)(59 172)(60 173)(61 174)(62 175)(63 176)(64 177)(65 178)(66 179)(67 180)(68 181)(69 182)(70 169)(71 132)(72 133)(73 134)(74 135)(75 136)(76 137)(77 138)(78 139)(79 140)(80 127)(81 128)(82 129)(83 130)(84 131)(85 155)(86 156)(87 157)(88 158)(89 159)(90 160)(91 161)(92 162)(93 163)(94 164)(95 165)(96 166)(97 167)(98 168)

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

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

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

112 conjugacy classes

class 1 2A2B2C7A···7F13A···13F14A···14F14G···14R26A···26F91A···91AJ182A···182AJ
order12227···713···1314···1414···1426···2691···91182···182
size1113131···12···21···113···132···22···22···2

112 irreducible representations

dim1111112222
type+++++
imageC1C2C2C7C14C14D13D26C7×D13C14×D13
kernelC14×D13C7×D13C182D26D13C26C14C7C2C1
# reps1216126663636

Matrix representation of C14×D13 in GL2(𝔽547) generated by

4660
0466
,
01
54664
,
0546
5460
G:=sub<GL(2,GF(547))| [466,0,0,466],[0,546,1,64],[0,546,546,0] >;

C14×D13 in GAP, Magma, Sage, TeX

C_{14}\times D_{13}
% in TeX

G:=Group("C14xD13");
// GroupNames label

G:=SmallGroup(364,8);
// by ID

G=gap.SmallGroup(364,8);
# by ID

G:=PCGroup([4,-2,-2,-7,-13,5379]);
// Polycyclic

G:=Group<a,b,c|a^14=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C14×D13 in TeX

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