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G = C5×D36order 360 = 23·32·5

Direct product of C5 and D36

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

Aliases: C5×D36, C455D4, C203D9, C361C10, C1803C2, C60.9S3, D181C10, C30.58D6, C15.3D12, C10.15D18, C90.15C22, C4⋊(C5×D9), C91(C5×D4), C3.(C5×D12), (C10×D9)⋊4C2, C6.8(S3×C10), C12.2(C5×S3), C2.4(C10×D9), C18.3(C2×C10), SmallGroup(360,22)

Series: Derived Chief Lower central Upper central

C1C18 — C5×D36
C1C3C9C18C90C10×D9 — C5×D36
C9C18 — C5×D36
C1C10C20

Generators and relations for C5×D36
 G = < a,b,c | a5=b36=c2=1, ab=ba, ac=ca, cbc=b-1 >

18C2
18C2
9C22
9C22
6S3
6S3
18C10
18C10
9D4
3D6
3D6
2D9
2D9
9C2×C10
9C2×C10
6C5×S3
6C5×S3
3D12
9C5×D4
3S3×C10
3S3×C10
2C5×D9
2C5×D9
3C5×D12

Smallest permutation representation of C5×D36
On 180 points
Generators in S180
(1 91 172 137 38)(2 92 173 138 39)(3 93 174 139 40)(4 94 175 140 41)(5 95 176 141 42)(6 96 177 142 43)(7 97 178 143 44)(8 98 179 144 45)(9 99 180 109 46)(10 100 145 110 47)(11 101 146 111 48)(12 102 147 112 49)(13 103 148 113 50)(14 104 149 114 51)(15 105 150 115 52)(16 106 151 116 53)(17 107 152 117 54)(18 108 153 118 55)(19 73 154 119 56)(20 74 155 120 57)(21 75 156 121 58)(22 76 157 122 59)(23 77 158 123 60)(24 78 159 124 61)(25 79 160 125 62)(26 80 161 126 63)(27 81 162 127 64)(28 82 163 128 65)(29 83 164 129 66)(30 84 165 130 67)(31 85 166 131 68)(32 86 167 132 69)(33 87 168 133 70)(34 88 169 134 71)(35 89 170 135 72)(36 90 171 136 37)
(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)
(1 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(37 38)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(109 128)(110 127)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 120)(118 119)(129 144)(130 143)(131 142)(132 141)(133 140)(134 139)(135 138)(136 137)(145 162)(146 161)(147 160)(148 159)(149 158)(150 157)(151 156)(152 155)(153 154)(163 180)(164 179)(165 178)(166 177)(167 176)(168 175)(169 174)(170 173)(171 172)

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

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

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

105 conjugacy classes

class 1 2A2B2C 3  4 5A5B5C5D 6 9A9B9C10A10B10C10D10E···10L12A12B15A15B15C15D18A18B18C20A20B20C20D30A30B30C30D36A···36F45A···45L60A···60H90A···90L180A···180X
order122234555569991010101010···10121215151515181818202020203030303036···3645···4560···6090···90180···180
size1118182211112222111118···18222222222222222222···22···22···22···22···2

105 irreducible representations

dim11111122222222222222
type++++++++++
imageC1C2C2C5C10C10S3D4D6D9D12C5×S3D18C5×D4S3×C10D36C5×D9C5×D12C10×D9C5×D36
kernelC5×D36C180C10×D9D36C36D18C60C45C30C20C15C12C10C9C6C5C4C3C2C1
# reps11244811132434461281224

Matrix representation of C5×D36 in GL2(𝔽181) generated by

420
042
,
37123
5895
,
159170
1122
G:=sub<GL(2,GF(181))| [42,0,0,42],[37,58,123,95],[159,11,170,22] >;

C5×D36 in GAP, Magma, Sage, TeX

C_5\times D_{36}
% in TeX

G:=Group("C5xD36");
// GroupNames label

G:=SmallGroup(360,22);
// by ID

G=gap.SmallGroup(360,22);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,265,127,6004,208,8645]);
// Polycyclic

G:=Group<a,b,c|a^5=b^36=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D36 in TeX

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