direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C9⋊D4, C45⋊8D4, Dic9⋊C10, D18⋊2C10, C30.60D6, C10.17D18, C90.17C22, C9⋊2(C5×D4), (C2×C90)⋊4C2, (C2×C10)⋊3D9, (C2×C18)⋊2C10, (C10×D9)⋊5C2, (C2×C30).9S3, C2.5(C10×D9), C22⋊2(C5×D9), C6.10(S3×C10), C18.5(C2×C10), (C5×Dic9)⋊4C2, C15.4(C3⋊D4), C3.(C5×C3⋊D4), (C2×C6).3(C5×S3), SmallGroup(360,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C9⋊D4
G = < a,b,c,d | a5=b9=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C5×C9⋊D4
►On 180 points
Generators in S180
(1 79 61 43 25)(2 80 62 44 26)(3 81 63 45 27)(4 73 55 37 19)(5 74 56 38 20)(6 75 57 39 21)(7 76 58 40 22)(8 77 59 41 23)(9 78 60 42 24)(10 82 64 46 28)(11 83 65 47 29)(12 84 66 48 30)(13 85 67 49 31)(14 86 68 50 32)(15 87 69 51 33)(16 88 70 52 34)(17 89 71 53 35)(18 90 72 54 36)(91 163 145 127 109)(92 164 146 128 110)(93 165 147 129 111)(94 166 148 130 112)(95 167 149 131 113)(96 168 150 132 114)(97 169 151 133 115)(98 170 152 134 116)(99 171 153 135 117)(100 172 154 136 118)(101 173 155 137 119)(102 174 156 138 120)(103 175 157 139 121)(104 176 158 140 122)(105 177 159 141 123)(106 178 160 142 124)(107 179 161 143 125)(108 180 162 144 126)
(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 97 16 106)(2 96 17 105)(3 95 18 104)(4 94 10 103)(5 93 11 102)(6 92 12 101)(7 91 13 100)(8 99 14 108)(9 98 15 107)(19 112 28 121)(20 111 29 120)(21 110 30 119)(22 109 31 118)(23 117 32 126)(24 116 33 125)(25 115 34 124)(26 114 35 123)(27 113 36 122)(37 130 46 139)(38 129 47 138)(39 128 48 137)(40 127 49 136)(41 135 50 144)(42 134 51 143)(43 133 52 142)(44 132 53 141)(45 131 54 140)(55 148 64 157)(56 147 65 156)(57 146 66 155)(58 145 67 154)(59 153 68 162)(60 152 69 161)(61 151 70 160)(62 150 71 159)(63 149 72 158)(73 166 82 175)(74 165 83 174)(75 164 84 173)(76 163 85 172)(77 171 86 180)(78 170 87 179)(79 169 88 178)(80 168 89 177)(81 167 90 176)
(2 9)(3 8)(4 7)(5 6)(10 13)(11 12)(14 18)(15 17)(19 22)(20 21)(23 27)(24 26)(28 31)(29 30)(32 36)(33 35)(37 40)(38 39)(41 45)(42 44)(46 49)(47 48)(50 54)(51 53)(55 58)(56 57)(59 63)(60 62)(64 67)(65 66)(68 72)(69 71)(73 76)(74 75)(77 81)(78 80)(82 85)(83 84)(86 90)(87 89)(91 103)(92 102)(93 101)(94 100)(95 108)(96 107)(97 106)(98 105)(99 104)(109 121)(110 120)(111 119)(112 118)(113 126)(114 125)(115 124)(116 123)(117 122)(127 139)(128 138)(129 137)(130 136)(131 144)(132 143)(133 142)(134 141)(135 140)(145 157)(146 156)(147 155)(148 154)(149 162)(150 161)(151 160)(152 159)(153 158)(163 175)(164 174)(165 173)(166 172)(167 180)(168 179)(169 178)(170 177)(171 176)
G:=sub<Sym(180)| (1,79,61,43,25)(2,80,62,44,26)(3,81,63,45,27)(4,73,55,37,19)(5,74,56,38,20)(6,75,57,39,21)(7,76,58,40,22)(8,77,59,41,23)(9,78,60,42,24)(10,82,64,46,28)(11,83,65,47,29)(12,84,66,48,30)(13,85,67,49,31)(14,86,68,50,32)(15,87,69,51,33)(16,88,70,52,34)(17,89,71,53,35)(18,90,72,54,36)(91,163,145,127,109)(92,164,146,128,110)(93,165,147,129,111)(94,166,148,130,112)(95,167,149,131,113)(96,168,150,132,114)(97,169,151,133,115)(98,170,152,134,116)(99,171,153,135,117)(100,172,154,136,118)(101,173,155,137,119)(102,174,156,138,120)(103,175,157,139,121)(104,176,158,140,122)(105,177,159,141,123)(106,178,160,142,124)(107,179,161,143,125)(108,180,162,144,126), (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), (1,97,16,106)(2,96,17,105)(3,95,18,104)(4,94,10,103)(5,93,11,102)(6,92,12,101)(7,91,13,100)(8,99,14,108)(9,98,15,107)(19,112,28,121)(20,111,29,120)(21,110,30,119)(22,109,31,118)(23,117,32,126)(24,116,33,125)(25,115,34,124)(26,114,35,123)(27,113,36,122)(37,130,46,139)(38,129,47,138)(39,128,48,137)(40,127,49,136)(41,135,50,144)(42,134,51,143)(43,133,52,142)(44,132,53,141)(45,131,54,140)(55,148,64,157)(56,147,65,156)(57,146,66,155)(58,145,67,154)(59,153,68,162)(60,152,69,161)(61,151,70,160)(62,150,71,159)(63,149,72,158)(73,166,82,175)(74,165,83,174)(75,164,84,173)(76,163,85,172)(77,171,86,180)(78,170,87,179)(79,169,88,178)(80,168,89,177)(81,167,90,176), (2,9)(3,8)(4,7)(5,6)(10,13)(11,12)(14,18)(15,17)(19,22)(20,21)(23,27)(24,26)(28,31)(29,30)(32,36)(33,35)(37,40)(38,39)(41,45)(42,44)(46,49)(47,48)(50,54)(51,53)(55,58)(56,57)(59,63)(60,62)(64,67)(65,66)(68,72)(69,71)(73,76)(74,75)(77,81)(78,80)(82,85)(83,84)(86,90)(87,89)(91,103)(92,102)(93,101)(94,100)(95,108)(96,107)(97,106)(98,105)(99,104)(109,121)(110,120)(111,119)(112,118)(113,126)(114,125)(115,124)(116,123)(117,122)(127,139)(128,138)(129,137)(130,136)(131,144)(132,143)(133,142)(134,141)(135,140)(145,157)(146,156)(147,155)(148,154)(149,162)(150,161)(151,160)(152,159)(153,158)(163,175)(164,174)(165,173)(166,172)(167,180)(168,179)(169,178)(170,177)(171,176)>;
G:=Group( (1,79,61,43,25)(2,80,62,44,26)(3,81,63,45,27)(4,73,55,37,19)(5,74,56,38,20)(6,75,57,39,21)(7,76,58,40,22)(8,77,59,41,23)(9,78,60,42,24)(10,82,64,46,28)(11,83,65,47,29)(12,84,66,48,30)(13,85,67,49,31)(14,86,68,50,32)(15,87,69,51,33)(16,88,70,52,34)(17,89,71,53,35)(18,90,72,54,36)(91,163,145,127,109)(92,164,146,128,110)(93,165,147,129,111)(94,166,148,130,112)(95,167,149,131,113)(96,168,150,132,114)(97,169,151,133,115)(98,170,152,134,116)(99,171,153,135,117)(100,172,154,136,118)(101,173,155,137,119)(102,174,156,138,120)(103,175,157,139,121)(104,176,158,140,122)(105,177,159,141,123)(106,178,160,142,124)(107,179,161,143,125)(108,180,162,144,126), (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), (1,97,16,106)(2,96,17,105)(3,95,18,104)(4,94,10,103)(5,93,11,102)(6,92,12,101)(7,91,13,100)(8,99,14,108)(9,98,15,107)(19,112,28,121)(20,111,29,120)(21,110,30,119)(22,109,31,118)(23,117,32,126)(24,116,33,125)(25,115,34,124)(26,114,35,123)(27,113,36,122)(37,130,46,139)(38,129,47,138)(39,128,48,137)(40,127,49,136)(41,135,50,144)(42,134,51,143)(43,133,52,142)(44,132,53,141)(45,131,54,140)(55,148,64,157)(56,147,65,156)(57,146,66,155)(58,145,67,154)(59,153,68,162)(60,152,69,161)(61,151,70,160)(62,150,71,159)(63,149,72,158)(73,166,82,175)(74,165,83,174)(75,164,84,173)(76,163,85,172)(77,171,86,180)(78,170,87,179)(79,169,88,178)(80,168,89,177)(81,167,90,176), (2,9)(3,8)(4,7)(5,6)(10,13)(11,12)(14,18)(15,17)(19,22)(20,21)(23,27)(24,26)(28,31)(29,30)(32,36)(33,35)(37,40)(38,39)(41,45)(42,44)(46,49)(47,48)(50,54)(51,53)(55,58)(56,57)(59,63)(60,62)(64,67)(65,66)(68,72)(69,71)(73,76)(74,75)(77,81)(78,80)(82,85)(83,84)(86,90)(87,89)(91,103)(92,102)(93,101)(94,100)(95,108)(96,107)(97,106)(98,105)(99,104)(109,121)(110,120)(111,119)(112,118)(113,126)(114,125)(115,124)(116,123)(117,122)(127,139)(128,138)(129,137)(130,136)(131,144)(132,143)(133,142)(134,141)(135,140)(145,157)(146,156)(147,155)(148,154)(149,162)(150,161)(151,160)(152,159)(153,158)(163,175)(164,174)(165,173)(166,172)(167,180)(168,179)(169,178)(170,177)(171,176) );
G=PermutationGroup([[(1,79,61,43,25),(2,80,62,44,26),(3,81,63,45,27),(4,73,55,37,19),(5,74,56,38,20),(6,75,57,39,21),(7,76,58,40,22),(8,77,59,41,23),(9,78,60,42,24),(10,82,64,46,28),(11,83,65,47,29),(12,84,66,48,30),(13,85,67,49,31),(14,86,68,50,32),(15,87,69,51,33),(16,88,70,52,34),(17,89,71,53,35),(18,90,72,54,36),(91,163,145,127,109),(92,164,146,128,110),(93,165,147,129,111),(94,166,148,130,112),(95,167,149,131,113),(96,168,150,132,114),(97,169,151,133,115),(98,170,152,134,116),(99,171,153,135,117),(100,172,154,136,118),(101,173,155,137,119),(102,174,156,138,120),(103,175,157,139,121),(104,176,158,140,122),(105,177,159,141,123),(106,178,160,142,124),(107,179,161,143,125),(108,180,162,144,126)], [(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)], [(1,97,16,106),(2,96,17,105),(3,95,18,104),(4,94,10,103),(5,93,11,102),(6,92,12,101),(7,91,13,100),(8,99,14,108),(9,98,15,107),(19,112,28,121),(20,111,29,120),(21,110,30,119),(22,109,31,118),(23,117,32,126),(24,116,33,125),(25,115,34,124),(26,114,35,123),(27,113,36,122),(37,130,46,139),(38,129,47,138),(39,128,48,137),(40,127,49,136),(41,135,50,144),(42,134,51,143),(43,133,52,142),(44,132,53,141),(45,131,54,140),(55,148,64,157),(56,147,65,156),(57,146,66,155),(58,145,67,154),(59,153,68,162),(60,152,69,161),(61,151,70,160),(62,150,71,159),(63,149,72,158),(73,166,82,175),(74,165,83,174),(75,164,84,173),(76,163,85,172),(77,171,86,180),(78,170,87,179),(79,169,88,178),(80,168,89,177),(81,167,90,176)], [(2,9),(3,8),(4,7),(5,6),(10,13),(11,12),(14,18),(15,17),(19,22),(20,21),(23,27),(24,26),(28,31),(29,30),(32,36),(33,35),(37,40),(38,39),(41,45),(42,44),(46,49),(47,48),(50,54),(51,53),(55,58),(56,57),(59,63),(60,62),(64,67),(65,66),(68,72),(69,71),(73,76),(74,75),(77,81),(78,80),(82,85),(83,84),(86,90),(87,89),(91,103),(92,102),(93,101),(94,100),(95,108),(96,107),(97,106),(98,105),(99,104),(109,121),(110,120),(111,119),(112,118),(113,126),(114,125),(115,124),(116,123),(117,122),(127,139),(128,138),(129,137),(130,136),(131,144),(132,143),(133,142),(134,141),(135,140),(145,157),(146,156),(147,155),(148,154),(149,162),(150,161),(151,160),(152,159),(153,158),(163,175),(164,174),(165,173),(166,172),(167,180),(168,179),(169,178),(170,177),(171,176)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 9A | 9B | 9C | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 15A | 15B | 15C | 15D | 18A | ··· | 18I | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 45A | ··· | 45L | 90A | ··· | 90AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 9 | 9 | 9 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 18 | ··· | 18 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 2 | 18 | 2 | 18 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D6 | D9 | C3⋊D4 | C5×S3 | D18 | C5×D4 | S3×C10 | C9⋊D4 | C5×D9 | C5×C3⋊D4 | C10×D9 | C5×C9⋊D4 |
| kernel | C5×C9⋊D4 | C5×Dic9 | C10×D9 | C2×C90 | C9⋊D4 | Dic9 | D18 | C2×C18 | C2×C30 | C45 | C30 | C2×C10 | C15 | C2×C6 | C10 | C9 | C6 | C5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 3 | 2 | 4 | 3 | 4 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C5×C9⋊D4 ►in GL2(𝔽181) generated by
| 59 | 0 |
| 0 | 59 |
| 4 | 127 |
| 54 | 131 |
| 153 | 153 |
| 125 | 28 |
| 1 | 180 |
| 0 | 180 |
G:=sub<GL(2,GF(181))| [59,0,0,59],[4,54,127,131],[153,125,153,28],[1,0,180,180] >;
C5×C9⋊D4 in GAP, Magma, Sage, TeX
C_5\times C_9\rtimes D_4
% in TeX
G:=Group("C5xC9:D4"); // GroupNames label
G:=SmallGroup(360,24);
// by ID
G=gap.SmallGroup(360,24);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-3,-3,265,6004,208,8645]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^9=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export