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G = D9×Dic5order 360 = 23·32·5

Direct product of D9 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C45 — D9×Dic5
 Chief series C1 — C3 — C15 — C45 — C90 — C9×Dic5 — D9×Dic5
 Lower central C45 — D9×Dic5
 Upper central C1 — C2

Generators and relations for D9×Dic5
G = < a,b,c,d | a9=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of D9×Dic5
On 180 points
Generators in S180
(1 141 28 37 108 64 89 76 96)(2 142 29 38 109 65 90 77 97)(3 143 30 39 110 66 81 78 98)(4 144 21 40 101 67 82 79 99)(5 145 22 31 102 68 83 80 100)(6 146 23 32 103 69 84 71 91)(7 147 24 33 104 70 85 72 92)(8 148 25 34 105 61 86 73 93)(9 149 26 35 106 62 87 74 94)(10 150 27 36 107 63 88 75 95)(11 111 46 155 123 60 165 172 138)(12 112 47 156 124 51 166 173 139)(13 113 48 157 125 52 167 174 140)(14 114 49 158 126 53 168 175 131)(15 115 50 159 127 54 169 176 132)(16 116 41 160 128 55 170 177 133)(17 117 42 151 129 56 161 178 134)(18 118 43 152 130 57 162 179 135)(19 119 44 153 121 58 163 180 136)(20 120 45 154 122 59 164 171 137)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 116)(12 117)(13 118)(14 119)(15 120)(16 111)(17 112)(18 113)(19 114)(20 115)(21 87)(22 88)(23 89)(24 90)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 61)(40 62)(41 138)(42 139)(43 140)(44 131)(45 132)(46 133)(47 134)(48 135)(49 136)(50 137)(51 56)(52 57)(53 58)(54 59)(55 60)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)(101 106)(102 107)(103 108)(104 109)(105 110)(121 168)(122 169)(123 170)(124 161)(125 162)(126 163)(127 164)(128 165)(129 166)(130 167)(151 173)(152 174)(153 175)(154 176)(155 177)(156 178)(157 179)(158 180)(159 171)(160 172)
(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)
(1 115 6 120)(2 114 7 119)(3 113 8 118)(4 112 9 117)(5 111 10 116)(11 95 16 100)(12 94 17 99)(13 93 18 98)(14 92 19 97)(15 91 20 96)(21 156 26 151)(22 155 27 160)(23 154 28 159)(24 153 29 158)(25 152 30 157)(31 123 36 128)(32 122 37 127)(33 121 38 126)(34 130 39 125)(35 129 40 124)(41 145 46 150)(42 144 47 149)(43 143 48 148)(44 142 49 147)(45 141 50 146)(51 106 56 101)(52 105 57 110)(53 104 58 109)(54 103 59 108)(55 102 60 107)(61 162 66 167)(62 161 67 166)(63 170 68 165)(64 169 69 164)(65 168 70 163)(71 137 76 132)(72 136 77 131)(73 135 78 140)(74 134 79 139)(75 133 80 138)(81 174 86 179)(82 173 87 178)(83 172 88 177)(84 171 89 176)(85 180 90 175)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6 9A 9B 9C 10A 10B 10C 10D 10E 10F 12A 12B 15A 15B 18A 18B 18C 30A 30B 36A ··· 36F 45A ··· 45F 90A ··· 90F order 1 2 2 2 3 4 4 4 4 5 5 6 9 9 9 10 10 10 10 10 10 12 12 15 15 18 18 18 30 30 36 ··· 36 45 ··· 45 90 ··· 90 size 1 1 9 9 2 5 5 45 45 2 2 2 2 2 2 2 2 18 18 18 18 10 10 4 4 2 2 2 4 4 10 ··· 10 4 ··· 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + + - + - image C1 C2 C2 C2 C4 S3 D5 D6 D9 Dic5 D10 C4×S3 D18 C4×D9 S3×D5 S3×Dic5 D5×D9 D9×Dic5 kernel D9×Dic5 C9×Dic5 Dic45 C10×D9 C5×D9 C3×Dic5 D18 C30 Dic5 D9 C18 C15 C10 C5 C6 C3 C2 C1 # reps 1 1 1 1 4 1 2 1 3 4 2 2 3 6 2 2 6 6

Matrix representation of D9×Dic5 in GL5(𝔽181)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 50 177 0 0 0 4 54
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 50 54 0 0 0 4 131
,
 180 0 0 0 0 0 28 3 0 0 0 101 166 0 0 0 0 0 1 0 0 0 0 0 1
,
 162 0 0 0 0 0 154 68 0 0 0 149 27 0 0 0 0 0 180 0 0 0 0 0 180

G:=sub<GL(5,GF(181))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,50,4,0,0,0,177,54],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,50,4,0,0,0,54,131],[180,0,0,0,0,0,28,101,0,0,0,3,166,0,0,0,0,0,1,0,0,0,0,0,1],[162,0,0,0,0,0,154,149,0,0,0,68,27,0,0,0,0,0,180,0,0,0,0,0,180] >;

D9×Dic5 in GAP, Magma, Sage, TeX

D_9\times {\rm Dic}_5
% in TeX

G:=Group("D9xDic5");
// GroupNames label

G:=SmallGroup(360,8);
// by ID

G=gap.SmallGroup(360,8);
# by ID

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

G:=Group<a,b,c,d|a^9=b^2=c^10=1,d^2=c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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