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G = D4×C45order 360 = 23·32·5

Direct product of C45 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C45, C4⋊C90, C203C18, C363C10, C1807C2, C60.11C6, C12.3C30, C222C90, C90.23C22, C3.(D4×C15), (C2×C90)⋊1C2, (C3×D4).C15, (D4×C15).C3, (C2×C18)⋊1C10, (C2×C10)⋊3C18, C6.6(C2×C30), (C2×C30).3C6, (C2×C6).2C30, C2.1(C2×C90), C15.3(C3×D4), C30.29(C2×C6), C18.6(C2×C10), C10.6(C2×C18), SmallGroup(360,31)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C45
C1C3C6C30C90C2×C90 — D4×C45
C1C2 — D4×C45
C1C90 — D4×C45

Generators and relations for D4×C45
 G = < a,b,c | a45=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C6
2C6
2C10
2C10
2C18
2C18
2C30
2C30
2C90
2C90

Smallest permutation representation of D4×C45
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 46 98 170)(2 47 99 171)(3 48 100 172)(4 49 101 173)(5 50 102 174)(6 51 103 175)(7 52 104 176)(8 53 105 177)(9 54 106 178)(10 55 107 179)(11 56 108 180)(12 57 109 136)(13 58 110 137)(14 59 111 138)(15 60 112 139)(16 61 113 140)(17 62 114 141)(18 63 115 142)(19 64 116 143)(20 65 117 144)(21 66 118 145)(22 67 119 146)(23 68 120 147)(24 69 121 148)(25 70 122 149)(26 71 123 150)(27 72 124 151)(28 73 125 152)(29 74 126 153)(30 75 127 154)(31 76 128 155)(32 77 129 156)(33 78 130 157)(34 79 131 158)(35 80 132 159)(36 81 133 160)(37 82 134 161)(38 83 135 162)(39 84 91 163)(40 85 92 164)(41 86 93 165)(42 87 94 166)(43 88 95 167)(44 89 96 168)(45 90 97 169)
(46 170)(47 171)(48 172)(49 173)(50 174)(51 175)(52 176)(53 177)(54 178)(55 179)(56 180)(57 136)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(65 144)(66 145)(67 146)(68 147)(69 148)(70 149)(71 150)(72 151)(73 152)(74 153)(75 154)(76 155)(77 156)(78 157)(79 158)(80 159)(81 160)(82 161)(83 162)(84 163)(85 164)(86 165)(87 166)(88 167)(89 168)(90 169)

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

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

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

225 conjugacy classes

class 1 2A2B2C3A3B 4 5A5B5C5D6A6B6C6D6E6F9A···9F10A10B10C10D10E···10L12A12B15A···15H18A···18F18G···18R20A20B20C20D30A···30H30I···30X36A···36F45A···45X60A···60H90A···90X90Y···90BT180A···180X
order122233455556666669···91010101010···10121215···1518···1818···182020202030···3030···3036···3645···4560···6090···9090···90180···180
size112211211111122221···111112···2221···11···12···222221···12···22···21···12···21···12···22···2

225 irreducible representations

dim111111111111111111222222
type++++
imageC1C2C2C3C5C6C6C9C10C10C15C18C18C30C30C45C90C90D4C3×D4C5×D4D4×C9D4×C15D4×C45
kernelD4×C45C180C2×C90D4×C15D4×C9C60C2×C30C5×D4C36C2×C18C3×D4C20C2×C10C12C2×C6D4C4C22C45C15C9C5C3C1
# reps112242464886128162424481246824

Matrix representation of D4×C45 in GL2(𝔽181) generated by

1690
0169
,
1802
1801
,
10
1180
G:=sub<GL(2,GF(181))| [169,0,0,169],[180,180,2,1],[1,1,0,180] >;

D4×C45 in GAP, Magma, Sage, TeX

D_4\times C_{45}
% in TeX

G:=Group("D4xC45");
// GroupNames label

G:=SmallGroup(360,31);
// by ID

G=gap.SmallGroup(360,31);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-3,745,554]);
// Polycyclic

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

Export

Subgroup lattice of D4×C45 in TeX

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