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G = D5×Dic9order 360 = 23·32·5

Direct product of D5 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C45 — D5×Dic9
 Chief series C1 — C3 — C15 — C45 — C90 — D5×C18 — D5×Dic9
 Lower central C45 — D5×Dic9
 Upper central C1 — C2

Generators and relations for D5×Dic9
G = < a,b,c,d | a5=b2=c18=1, d2=c9, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of D5×Dic9
On 180 points
Generators in S180
(1 89 173 121 106)(2 90 174 122 107)(3 73 175 123 108)(4 74 176 124 91)(5 75 177 125 92)(6 76 178 126 93)(7 77 179 109 94)(8 78 180 110 95)(9 79 163 111 96)(10 80 164 112 97)(11 81 165 113 98)(12 82 166 114 99)(13 83 167 115 100)(14 84 168 116 101)(15 85 169 117 102)(16 86 170 118 103)(17 87 171 119 104)(18 88 172 120 105)(19 51 131 148 68)(20 52 132 149 69)(21 53 133 150 70)(22 54 134 151 71)(23 37 135 152 72)(24 38 136 153 55)(25 39 137 154 56)(26 40 138 155 57)(27 41 139 156 58)(28 42 140 157 59)(29 43 141 158 60)(30 44 142 159 61)(31 45 143 160 62)(32 46 144 161 63)(33 47 127 162 64)(34 48 128 145 65)(35 49 129 146 66)(36 50 130 147 67)
(1 106)(2 107)(3 108)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 51)(20 52)(21 53)(22 54)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 123)(74 124)(75 125)(76 126)(77 109)(78 110)(79 111)(80 112)(81 113)(82 114)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 121)(90 122)
(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 54 10 45)(2 53 11 44)(3 52 12 43)(4 51 13 42)(5 50 14 41)(6 49 15 40)(7 48 16 39)(8 47 17 38)(9 46 18 37)(19 100 28 91)(20 99 29 108)(21 98 30 107)(22 97 31 106)(23 96 32 105)(24 95 33 104)(25 94 34 103)(26 93 35 102)(27 92 36 101)(55 110 64 119)(56 109 65 118)(57 126 66 117)(58 125 67 116)(59 124 68 115)(60 123 69 114)(61 122 70 113)(62 121 71 112)(63 120 72 111)(73 132 82 141)(74 131 83 140)(75 130 84 139)(76 129 85 138)(77 128 86 137)(78 127 87 136)(79 144 88 135)(80 143 89 134)(81 142 90 133)(145 170 154 179)(146 169 155 178)(147 168 156 177)(148 167 157 176)(149 166 158 175)(150 165 159 174)(151 164 160 173)(152 163 161 172)(153 180 162 171)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 9A 9B 9C 10A 10B 15A 15B 18A 18B 18C 18D ··· 18I 20A 20B 20C 20D 30A 30B 45A ··· 45F 90A ··· 90F order 1 2 2 2 3 4 4 4 4 5 5 6 6 6 9 9 9 10 10 15 15 18 18 18 18 ··· 18 20 20 20 20 30 30 45 ··· 45 90 ··· 90 size 1 1 5 5 2 9 9 45 45 2 2 2 10 10 2 2 2 2 2 4 4 2 2 2 10 ··· 10 18 18 18 18 4 4 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 Dic3 D6 D9 D10 Dic9 D18 C4×D5 S3×D5 D5×Dic3 D5×D9 D5×Dic9 kernel D5×Dic9 C5×Dic9 Dic45 D5×C18 C9×D5 C6×D5 Dic9 C3×D5 C30 D10 C18 D5 C10 C9 C6 C3 C2 C1 # reps 1 1 1 1 4 1 2 2 1 3 2 6 3 4 2 2 6 6

Matrix representation of D5×Dic9 in GL4(𝔽181) generated by

 166 3 0 0 55 1 0 0 0 0 1 0 0 0 0 1
,
 180 0 0 0 55 1 0 0 0 0 1 0 0 0 0 1
,
 180 0 0 0 0 180 0 0 0 0 131 54 0 0 127 4
,
 19 0 0 0 0 19 0 0 0 0 159 11 0 0 170 22
G:=sub<GL(4,GF(181))| [166,55,0,0,3,1,0,0,0,0,1,0,0,0,0,1],[180,55,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[180,0,0,0,0,180,0,0,0,0,131,127,0,0,54,4],[19,0,0,0,0,19,0,0,0,0,159,170,0,0,11,22] >;

D5×Dic9 in GAP, Magma, Sage, TeX

D_5\times {\rm Dic}_9
% in TeX

G:=Group("D5xDic9");
// GroupNames label

G:=SmallGroup(360,11);
// by ID

G=gap.SmallGroup(360,11);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,31,1641,741,2884,4331]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^18=1,d^2=c^9,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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