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G = C457D4order 360 = 23·32·5

1st semidirect product of C45 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C457D4, D902C2, C2.5D90, C6.12D30, C30.44D6, C222D45, Dic451C2, C18.12D10, C10.12D18, C90.12C22, (C2×C90)⋊2C2, (C2×C18)⋊2D5, (C2×C10)⋊4D9, C93(C5⋊D4), C53(C9⋊D4), (C2×C30).4S3, C3.(C157D4), (C2×C6).3D15, C15.3(C3⋊D4), SmallGroup(360,29)

Series: Derived Chief Lower central Upper central

C1C90 — C457D4
C1C3C15C45C90D90 — C457D4
C45C90 — C457D4
C1C2C22

Generators and relations for C457D4
 G = < a,b,c | a45=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
90C2
45C4
45C22
2C6
30S3
2C10
18D5
45D4
15D6
15Dic3
2C18
10D9
9D10
9Dic5
2C30
6D15
15C3⋊D4
5D18
5Dic9
9C5⋊D4
3Dic15
3D30
2D45
2C90
5C9⋊D4
3C157D4

Smallest permutation representation of C457D4
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)
(1 113 78 180)(2 112 79 179)(3 111 80 178)(4 110 81 177)(5 109 82 176)(6 108 83 175)(7 107 84 174)(8 106 85 173)(9 105 86 172)(10 104 87 171)(11 103 88 170)(12 102 89 169)(13 101 90 168)(14 100 46 167)(15 99 47 166)(16 98 48 165)(17 97 49 164)(18 96 50 163)(19 95 51 162)(20 94 52 161)(21 93 53 160)(22 92 54 159)(23 91 55 158)(24 135 56 157)(25 134 57 156)(26 133 58 155)(27 132 59 154)(28 131 60 153)(29 130 61 152)(30 129 62 151)(31 128 63 150)(32 127 64 149)(33 126 65 148)(34 125 66 147)(35 124 67 146)(36 123 68 145)(37 122 69 144)(38 121 70 143)(39 120 71 142)(40 119 72 141)(41 118 73 140)(42 117 74 139)(43 116 75 138)(44 115 76 137)(45 114 77 136)
(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 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(91 157)(92 156)(93 155)(94 154)(95 153)(96 152)(97 151)(98 150)(99 149)(100 148)(101 147)(102 146)(103 145)(104 144)(105 143)(106 142)(107 141)(108 140)(109 139)(110 138)(111 137)(112 136)(113 180)(114 179)(115 178)(116 177)(117 176)(118 175)(119 174)(120 173)(121 172)(122 171)(123 170)(124 169)(125 168)(126 167)(127 166)(128 165)(129 164)(130 163)(131 162)(132 161)(133 160)(134 159)(135 158)

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

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

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

93 conjugacy classes

class 1 2A2B2C 3  4 5A5B6A6B6C9A9B9C10A···10F15A15B15C15D18A···18I30A···30L45A···45L90A···90AJ
order1222345566699910···101515151518···1830···3045···4590···90
size11290290222222222···222222···22···22···22···2

93 irreducible representations

dim11112222222222222222
type+++++++++++++++
imageC1C2C2C2S3D4D5D6D9D10C3⋊D4D15D18C5⋊D4D30C9⋊D4D45C157D4D90C457D4
kernelC457D4Dic45D90C2×C90C2×C30C45C2×C18C30C2×C10C18C15C2×C6C10C9C6C5C22C3C2C1
# reps11111121322434461281224

Matrix representation of C457D4 in GL2(𝔽181) generated by

13539
14296
,
153125
15328
,
10
180180
G:=sub<GL(2,GF(181))| [135,142,39,96],[153,153,125,28],[1,180,0,180] >;

C457D4 in GAP, Magma, Sage, TeX

C_{45}\rtimes_7D_4
% in TeX

G:=Group("C45:7D4");
// GroupNames label

G:=SmallGroup(360,29);
// by ID

G=gap.SmallGroup(360,29);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,73,3267,741,2884,8645]);
// Polycyclic

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

Export

Subgroup lattice of C457D4 in TeX

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