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## G = C4×D45order 360 = 23·32·5

### Direct product of C4 and D45

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

 Derived series C1 — C45 — C4×D45
 Chief series C1 — C3 — C15 — C45 — C90 — D90 — C4×D45
 Lower central C45 — C4×D45
 Upper central C1 — C4

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

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 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6 9A 9B 9C 10A 10B 12A 12B 15A 15B 15C 15D 18A 18B 18C 20A 20B 20C 20D 30A 30B 30C 30D 36A ··· 36F 45A ··· 45L 60A ··· 60H 90A ··· 90L 180A ··· 180X order 1 2 2 2 3 4 4 4 4 5 5 6 9 9 9 10 10 12 12 15 15 15 15 18 18 18 20 20 20 20 30 30 30 30 36 ··· 36 45 ··· 45 60 ··· 60 90 ··· 90 180 ··· 180 size 1 1 45 45 2 1 1 45 45 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D5 D6 D9 D10 C4×S3 D15 D18 C4×D5 D30 C4×D9 D45 C4×D15 D90 C4×D45 kernel C4×D45 Dic45 C180 D90 D45 C60 C36 C30 C20 C18 C15 C12 C10 C9 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 4 1 2 1 3 2 2 4 3 4 4 6 12 8 12 24

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

 19 0 0 19
,
 76 149 32 108
,
 119 44 106 62
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

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