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G = C5⋊D36order 360 = 23·32·5

The semidirect product of C5 and D36 acting via D36/D18=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52D36, C451D4, Dic5⋊D9, C15.D12, D181D5, D903C2, C30.4D6, C10.4D18, C18.4D10, C90.4C22, C91(C5⋊D4), C2.5(D5×D9), (C10×D9)⋊1C2, C3.(C5⋊D12), C6.11(S3×D5), (C9×Dic5)⋊3C2, (C3×Dic5).4S3, SmallGroup(360,10)

Series: Derived Chief Lower central Upper central

C1C90 — C5⋊D36
C1C3C15C45C90C9×Dic5 — C5⋊D36
C45C90 — C5⋊D36
C1C2

Generators and relations for C5⋊D36
 G = < a,b,c | a45=b4=c2=1, bab-1=a19, cac=a-1, cbc=b-1 >

18C2
90C2
5C4
9C22
45C22
6S3
30S3
18D5
18C10
45D4
3D6
5C12
15D6
2D9
10D9
9C2×C10
9D10
6C5×S3
6D15
15D12
5C36
5D18
9C5⋊D4
3S3×C10
3D30
2D45
2C5×D9
5D36
3C5⋊D12

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

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

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

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

45 conjugacy classes

class 1 2A2B2C 3  4 5A5B 6 9A9B9C10A10B10C10D10E10F12A12B15A15B18A18B18C30A30B36A···36F45A···45F90A···90F
order12223455699910101010101012121515181818303036···3645···4590···90
size11189021022222222181818181010442224410···104···44···4

45 irreducible representations

dim111122222222224444
type+++++++++++++++++
imageC1C2C2C2S3D4D5D6D9D10D12D18C5⋊D4D36S3×D5C5⋊D12D5×D9C5⋊D36
kernelC5⋊D36C9×Dic5C10×D9D90C3×Dic5C45D18C30Dic5C18C15C10C9C5C6C3C2C1
# reps111111213223462266

Matrix representation of C5⋊D36 in GL4(𝔽181) generated by

1318000
1000
00177131
0050127
,
7410000
15710700
001800
000180
,
1000
1318000
0054177
0050127
G:=sub<GL(4,GF(181))| [13,1,0,0,180,0,0,0,0,0,177,50,0,0,131,127],[74,157,0,0,100,107,0,0,0,0,180,0,0,0,0,180],[1,13,0,0,0,180,0,0,0,0,54,50,0,0,177,127] >;

C5⋊D36 in GAP, Magma, Sage, TeX

C_5\rtimes D_{36}
% in TeX

G:=Group("C5:D36");
// GroupNames label

G:=SmallGroup(360,10);
// by ID

G=gap.SmallGroup(360,10);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,24,73,1641,741,2884,4331]);
// Polycyclic

G:=Group<a,b,c|a^45=b^4=c^2=1,b*a*b^-1=a^19,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5⋊D36 in TeX

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