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G = C4×D45order 360 = 23·32·5

Direct product of C4 and D45

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

Aliases: C4×D45, C202D9, C362D5, C1802C2, C60.5S3, C6.9D30, C2.1D90, D90.2C2, C12.5D15, C18.9D10, C10.9D18, C30.41D6, Dic455C2, C90.9C22, C53(C4×D9), C92(C4×D5), C457(C2×C4), C3.(C4×D15), C15.4(C4×S3), SmallGroup(360,26)

Series: Derived Chief Lower central Upper central

C1C45 — C4×D45
C1C3C15C45C90D90 — C4×D45
C45 — C4×D45
C1C4

Generators and relations for C4×D45
 G = < a,b,c | a4=b45=c2=1, ab=ba, ac=ca, cbc=b-1 >

45C2
45C2
45C4
45C22
15S3
15S3
9D5
9D5
45C2×C4
15Dic3
15D6
5D9
5D9
9Dic5
9D10
3D15
3D15
15C4×S3
5D18
5Dic9
9C4×D5
3D30
3Dic15
5C4×D9
3C4×D15

Smallest permutation representation of C4×D45
On 180 points
Generators in S180
(1 141 68 122)(2 142 69 123)(3 143 70 124)(4 144 71 125)(5 145 72 126)(6 146 73 127)(7 147 74 128)(8 148 75 129)(9 149 76 130)(10 150 77 131)(11 151 78 132)(12 152 79 133)(13 153 80 134)(14 154 81 135)(15 155 82 91)(16 156 83 92)(17 157 84 93)(18 158 85 94)(19 159 86 95)(20 160 87 96)(21 161 88 97)(22 162 89 98)(23 163 90 99)(24 164 46 100)(25 165 47 101)(26 166 48 102)(27 167 49 103)(28 168 50 104)(29 169 51 105)(30 170 52 106)(31 171 53 107)(32 172 54 108)(33 173 55 109)(34 174 56 110)(35 175 57 111)(36 176 58 112)(37 177 59 113)(38 178 60 114)(39 179 61 115)(40 180 62 116)(41 136 63 117)(42 137 64 118)(43 138 65 119)(44 139 66 120)(45 140 67 121)
(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)
(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 90)(47 89)(48 88)(49 87)(50 86)(51 85)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 100)(109 135)(110 134)(111 133)(112 132)(113 131)(114 130)(115 129)(116 128)(117 127)(118 126)(119 125)(120 124)(121 123)(136 146)(137 145)(138 144)(139 143)(140 142)(147 180)(148 179)(149 178)(150 177)(151 176)(152 175)(153 174)(154 173)(155 172)(156 171)(157 170)(158 169)(159 168)(160 167)(161 166)(162 165)(163 164)

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

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

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

96 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B 6 9A9B9C10A10B12A12B15A15B15C15D18A18B18C20A20B20C20D30A30B30C30D36A···36F45A···45L60A···60H90A···90L180A···180X
order1222344445569991010121215151515181818202020203030303036···3645···4560···6090···90180···180
size114545211454522222222222222222222222222···22···22···22···22···2

96 irreducible representations

dim11111222222222222222
type++++++++++++++
imageC1C2C2C2C4S3D5D6D9D10C4×S3D15D18C4×D5D30C4×D9D45C4×D15D90C4×D45
kernelC4×D45Dic45C180D90D45C60C36C30C20C18C15C12C10C9C6C5C4C3C2C1
# reps11114121322434461281224

Matrix representation of C4×D45 in GL2(𝔽181) generated by

190
019
,
76149
32108
,
11944
10662
G:=sub<GL(2,GF(181))| [19,0,0,19],[76,32,149,108],[119,106,44,62] >;

C4×D45 in GAP, Magma, Sage, TeX

C_4\times D_{45}
% in TeX

G:=Group("C4xD45");
// GroupNames label

G:=SmallGroup(360,26);
// by ID

G=gap.SmallGroup(360,26);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,31,3267,741,2884,8645]);
// Polycyclic

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

Export

Subgroup lattice of C4×D45 in TeX

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