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G = C6×D29order 348 = 22·3·29

Direct product of C6 and D29

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

Aliases: C6×D29, C58⋊C6, C1742C2, C873C22, C29⋊(C2×C6), SmallGroup(348,9)

Series: Derived Chief Lower central Upper central

C1C29 — C6×D29
C1C29C87C3×D29 — C6×D29
C29 — C6×D29
C1C6

Generators and relations for C6×D29
 G = < a,b,c | a6=b29=c2=1, ab=ba, ac=ca, cbc=b-1 >

29C2
29C2
29C22
29C6
29C6
29C2×C6

Smallest permutation representation of C6×D29
On 174 points
Generators in S174
(1 117 65 95 44 168)(2 118 66 96 45 169)(3 119 67 97 46 170)(4 120 68 98 47 171)(5 121 69 99 48 172)(6 122 70 100 49 173)(7 123 71 101 50 174)(8 124 72 102 51 146)(9 125 73 103 52 147)(10 126 74 104 53 148)(11 127 75 105 54 149)(12 128 76 106 55 150)(13 129 77 107 56 151)(14 130 78 108 57 152)(15 131 79 109 58 153)(16 132 80 110 30 154)(17 133 81 111 31 155)(18 134 82 112 32 156)(19 135 83 113 33 157)(20 136 84 114 34 158)(21 137 85 115 35 159)(22 138 86 116 36 160)(23 139 87 88 37 161)(24 140 59 89 38 162)(25 141 60 90 39 163)(26 142 61 91 40 164)(27 143 62 92 41 165)(28 144 63 93 42 166)(29 145 64 94 43 167)
(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)
(1 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 116)(9 115)(10 114)(11 113)(12 112)(13 111)(14 110)(15 109)(16 108)(17 107)(18 106)(19 105)(20 104)(21 103)(22 102)(23 101)(24 100)(25 99)(26 98)(27 97)(28 96)(29 95)(30 130)(31 129)(32 128)(33 127)(34 126)(35 125)(36 124)(37 123)(38 122)(39 121)(40 120)(41 119)(42 118)(43 117)(44 145)(45 144)(46 143)(47 142)(48 141)(49 140)(50 139)(51 138)(52 137)(53 136)(54 135)(55 134)(56 133)(57 132)(58 131)(59 173)(60 172)(61 171)(62 170)(63 169)(64 168)(65 167)(66 166)(67 165)(68 164)(69 163)(70 162)(71 161)(72 160)(73 159)(74 158)(75 157)(76 156)(77 155)(78 154)(79 153)(80 152)(81 151)(82 150)(83 149)(84 148)(85 147)(86 146)(87 174)

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

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

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

96 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F29A···29N58A···58N87A···87AB174A···174AB
order12223366666629···2958···5887···87174···174
size1129291111292929292···22···22···22···2

96 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D29D58C3×D29C6×D29
kernelC6×D29C3×D29C174D58D29C58C6C3C2C1
# reps12124214142828

Matrix representation of C6×D29 in GL3(𝔽349) generated by

22600
03480
00348
,
100
001
0348322
,
34800
00348
03480
G:=sub<GL(3,GF(349))| [226,0,0,0,348,0,0,0,348],[1,0,0,0,0,348,0,1,322],[348,0,0,0,0,348,0,348,0] >;

C6×D29 in GAP, Magma, Sage, TeX

C_6\times D_{29}
% in TeX

G:=Group("C6xD29");
// GroupNames label

G:=SmallGroup(348,9);
// by ID

G=gap.SmallGroup(348,9);
# by ID

G:=PCGroup([4,-2,-2,-3,-29,5379]);
// Polycyclic

G:=Group<a,b,c|a^6=b^29=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D29 in TeX

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