direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C45, C4⋊C90, C20⋊3C18, C36⋊3C10, C180⋊7C2, C60.11C6, C12.3C30, C22⋊2C90, 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
Generators and relations for D4×C45
G = < a,b,c | a45=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
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 71 143 103)(2 72 144 104)(3 73 145 105)(4 74 146 106)(5 75 147 107)(6 76 148 108)(7 77 149 109)(8 78 150 110)(9 79 151 111)(10 80 152 112)(11 81 153 113)(12 82 154 114)(13 83 155 115)(14 84 156 116)(15 85 157 117)(16 86 158 118)(17 87 159 119)(18 88 160 120)(19 89 161 121)(20 90 162 122)(21 46 163 123)(22 47 164 124)(23 48 165 125)(24 49 166 126)(25 50 167 127)(26 51 168 128)(27 52 169 129)(28 53 170 130)(29 54 171 131)(30 55 172 132)(31 56 173 133)(32 57 174 134)(33 58 175 135)(34 59 176 91)(35 60 177 92)(36 61 178 93)(37 62 179 94)(38 63 180 95)(39 64 136 96)(40 65 137 97)(41 66 138 98)(42 67 139 99)(43 68 140 100)(44 69 141 101)(45 70 142 102)
(46 123)(47 124)(48 125)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 132)(56 133)(57 134)(58 135)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(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)
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), (1,71,143,103)(2,72,144,104)(3,73,145,105)(4,74,146,106)(5,75,147,107)(6,76,148,108)(7,77,149,109)(8,78,150,110)(9,79,151,111)(10,80,152,112)(11,81,153,113)(12,82,154,114)(13,83,155,115)(14,84,156,116)(15,85,157,117)(16,86,158,118)(17,87,159,119)(18,88,160,120)(19,89,161,121)(20,90,162,122)(21,46,163,123)(22,47,164,124)(23,48,165,125)(24,49,166,126)(25,50,167,127)(26,51,168,128)(27,52,169,129)(28,53,170,130)(29,54,171,131)(30,55,172,132)(31,56,173,133)(32,57,174,134)(33,58,175,135)(34,59,176,91)(35,60,177,92)(36,61,178,93)(37,62,179,94)(38,63,180,95)(39,64,136,96)(40,65,137,97)(41,66,138,98)(42,67,139,99)(43,68,140,100)(44,69,141,101)(45,70,142,102), (46,123)(47,124)(48,125)(49,126)(50,127)(51,128)(52,129)(53,130)(54,131)(55,132)(56,133)(57,134)(58,135)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(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)>;
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), (1,71,143,103)(2,72,144,104)(3,73,145,105)(4,74,146,106)(5,75,147,107)(6,76,148,108)(7,77,149,109)(8,78,150,110)(9,79,151,111)(10,80,152,112)(11,81,153,113)(12,82,154,114)(13,83,155,115)(14,84,156,116)(15,85,157,117)(16,86,158,118)(17,87,159,119)(18,88,160,120)(19,89,161,121)(20,90,162,122)(21,46,163,123)(22,47,164,124)(23,48,165,125)(24,49,166,126)(25,50,167,127)(26,51,168,128)(27,52,169,129)(28,53,170,130)(29,54,171,131)(30,55,172,132)(31,56,173,133)(32,57,174,134)(33,58,175,135)(34,59,176,91)(35,60,177,92)(36,61,178,93)(37,62,179,94)(38,63,180,95)(39,64,136,96)(40,65,137,97)(41,66,138,98)(42,67,139,99)(43,68,140,100)(44,69,141,101)(45,70,142,102), (46,123)(47,124)(48,125)(49,126)(50,127)(51,128)(52,129)(53,130)(54,131)(55,132)(56,133)(57,134)(58,135)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(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) );
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)], [(1,71,143,103),(2,72,144,104),(3,73,145,105),(4,74,146,106),(5,75,147,107),(6,76,148,108),(7,77,149,109),(8,78,150,110),(9,79,151,111),(10,80,152,112),(11,81,153,113),(12,82,154,114),(13,83,155,115),(14,84,156,116),(15,85,157,117),(16,86,158,118),(17,87,159,119),(18,88,160,120),(19,89,161,121),(20,90,162,122),(21,46,163,123),(22,47,164,124),(23,48,165,125),(24,49,166,126),(25,50,167,127),(26,51,168,128),(27,52,169,129),(28,53,170,130),(29,54,171,131),(30,55,172,132),(31,56,173,133),(32,57,174,134),(33,58,175,135),(34,59,176,91),(35,60,177,92),(36,61,178,93),(37,62,179,94),(38,63,180,95),(39,64,136,96),(40,65,137,97),(41,66,138,98),(42,67,139,99),(43,68,140,100),(44,69,141,101),(45,70,142,102)], [(46,123),(47,124),(48,125),(49,126),(50,127),(51,128),(52,129),(53,130),(54,131),(55,132),(56,133),(57,134),(58,135),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,99),(68,100),(69,101),(70,102),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(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)]])
225 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 12A | 12B | 15A | ··· | 15H | 18A | ··· | 18F | 18G | ··· | 18R | 20A | 20B | 20C | 20D | 30A | ··· | 30H | 30I | ··· | 30X | 36A | ··· | 36F | 45A | ··· | 45X | 60A | ··· | 60H | 90A | ··· | 90X | 90Y | ··· | 90BT | 180A | ··· | 180X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 15 | ··· | 15 | 18 | ··· | 18 | 18 | ··· | 18 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 36 | ··· | 36 | 45 | ··· | 45 | 60 | ··· | 60 | 90 | ··· | 90 | 90 | ··· | 90 | 180 | ··· | 180 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
225 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C9 | C10 | C10 | C15 | C18 | C18 | C30 | C30 | C45 | C90 | C90 | D4 | C3×D4 | C5×D4 | D4×C9 | D4×C15 | D4×C45 |
| kernel | D4×C45 | C180 | C2×C90 | D4×C15 | D4×C9 | C60 | C2×C30 | C5×D4 | C36 | C2×C18 | C3×D4 | C20 | C2×C10 | C12 | C2×C6 | D4 | C4 | C22 | C45 | C15 | C9 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 6 | 4 | 8 | 8 | 6 | 12 | 8 | 16 | 24 | 24 | 48 | 1 | 2 | 4 | 6 | 8 | 24 |
Matrix representation of D4×C45 ►in GL2(𝔽181) generated by
| 169 | 0 |
| 0 | 169 |
| 180 | 2 |
| 180 | 1 |
| 1 | 0 |
| 1 | 180 |
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
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