metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C45⋊7D4, D90⋊2C2, C2.5D90, C6.12D30, C30.44D6, C22⋊2D45, Dic45⋊1C2, C18.12D10, C10.12D18, C90.12C22, (C2×C90)⋊2C2, (C2×C18)⋊2D5, (C2×C10)⋊4D9, C9⋊3(C5⋊D4), C5⋊3(C9⋊D4), (C2×C30).4S3, C3.(C15⋊7D4), (C2×C6).3D15, C15.3(C3⋊D4), SmallGroup(360,29)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C45⋊7D4
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
3D30
2D45
2C90
Smallest permutation representation of C45⋊7D4
►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)
(1 100 69 147)(2 99 70 146)(3 98 71 145)(4 97 72 144)(5 96 73 143)(6 95 74 142)(7 94 75 141)(8 93 76 140)(9 92 77 139)(10 91 78 138)(11 135 79 137)(12 134 80 136)(13 133 81 180)(14 132 82 179)(15 131 83 178)(16 130 84 177)(17 129 85 176)(18 128 86 175)(19 127 87 174)(20 126 88 173)(21 125 89 172)(22 124 90 171)(23 123 46 170)(24 122 47 169)(25 121 48 168)(26 120 49 167)(27 119 50 166)(28 118 51 165)(29 117 52 164)(30 116 53 163)(31 115 54 162)(32 114 55 161)(33 113 56 160)(34 112 57 159)(35 111 58 158)(36 110 59 157)(37 109 60 156)(38 108 61 155)(39 107 62 154)(40 106 63 153)(41 105 64 152)(42 104 65 151)(43 103 66 150)(44 102 67 149)(45 101 68 148)
(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 47)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(91 156)(92 155)(93 154)(94 153)(95 152)(96 151)(97 150)(98 149)(99 148)(100 147)(101 146)(102 145)(103 144)(104 143)(105 142)(106 141)(107 140)(108 139)(109 138)(110 137)(111 136)(112 180)(113 179)(114 178)(115 177)(116 176)(117 175)(118 174)(119 173)(120 172)(121 171)(122 170)(123 169)(124 168)(125 167)(126 166)(127 165)(128 164)(129 163)(130 162)(131 161)(132 160)(133 159)(134 158)(135 157)
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), (1,100,69,147)(2,99,70,146)(3,98,71,145)(4,97,72,144)(5,96,73,143)(6,95,74,142)(7,94,75,141)(8,93,76,140)(9,92,77,139)(10,91,78,138)(11,135,79,137)(12,134,80,136)(13,133,81,180)(14,132,82,179)(15,131,83,178)(16,130,84,177)(17,129,85,176)(18,128,86,175)(19,127,87,174)(20,126,88,173)(21,125,89,172)(22,124,90,171)(23,123,46,170)(24,122,47,169)(25,121,48,168)(26,120,49,167)(27,119,50,166)(28,118,51,165)(29,117,52,164)(30,116,53,163)(31,115,54,162)(32,114,55,161)(33,113,56,160)(34,112,57,159)(35,111,58,158)(36,110,59,157)(37,109,60,156)(38,108,61,155)(39,107,62,154)(40,106,63,153)(41,105,64,152)(42,104,65,151)(43,103,66,150)(44,102,67,149)(45,101,68,148), (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)(46,47)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(91,156)(92,155)(93,154)(94,153)(95,152)(96,151)(97,150)(98,149)(99,148)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,174)(119,173)(120,172)(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)(127,165)(128,164)(129,163)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)>;
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), (1,100,69,147)(2,99,70,146)(3,98,71,145)(4,97,72,144)(5,96,73,143)(6,95,74,142)(7,94,75,141)(8,93,76,140)(9,92,77,139)(10,91,78,138)(11,135,79,137)(12,134,80,136)(13,133,81,180)(14,132,82,179)(15,131,83,178)(16,130,84,177)(17,129,85,176)(18,128,86,175)(19,127,87,174)(20,126,88,173)(21,125,89,172)(22,124,90,171)(23,123,46,170)(24,122,47,169)(25,121,48,168)(26,120,49,167)(27,119,50,166)(28,118,51,165)(29,117,52,164)(30,116,53,163)(31,115,54,162)(32,114,55,161)(33,113,56,160)(34,112,57,159)(35,111,58,158)(36,110,59,157)(37,109,60,156)(38,108,61,155)(39,107,62,154)(40,106,63,153)(41,105,64,152)(42,104,65,151)(43,103,66,150)(44,102,67,149)(45,101,68,148), (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)(46,47)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(91,156)(92,155)(93,154)(94,153)(95,152)(96,151)(97,150)(98,149)(99,148)(100,147)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,174)(119,173)(120,172)(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)(127,165)(128,164)(129,163)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157) );
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)], [(1,100,69,147),(2,99,70,146),(3,98,71,145),(4,97,72,144),(5,96,73,143),(6,95,74,142),(7,94,75,141),(8,93,76,140),(9,92,77,139),(10,91,78,138),(11,135,79,137),(12,134,80,136),(13,133,81,180),(14,132,82,179),(15,131,83,178),(16,130,84,177),(17,129,85,176),(18,128,86,175),(19,127,87,174),(20,126,88,173),(21,125,89,172),(22,124,90,171),(23,123,46,170),(24,122,47,169),(25,121,48,168),(26,120,49,167),(27,119,50,166),(28,118,51,165),(29,117,52,164),(30,116,53,163),(31,115,54,162),(32,114,55,161),(33,113,56,160),(34,112,57,159),(35,111,58,158),(36,110,59,157),(37,109,60,156),(38,108,61,155),(39,107,62,154),(40,106,63,153),(41,105,64,152),(42,104,65,151),(43,103,66,150),(44,102,67,149),(45,101,68,148)], [(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),(46,47),(48,90),(49,89),(50,88),(51,87),(52,86),(53,85),(54,84),(55,83),(56,82),(57,81),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(91,156),(92,155),(93,154),(94,153),(95,152),(96,151),(97,150),(98,149),(99,148),(100,147),(101,146),(102,145),(103,144),(104,143),(105,142),(106,141),(107,140),(108,139),(109,138),(110,137),(111,136),(112,180),(113,179),(114,178),(115,177),(116,176),(117,175),(118,174),(119,173),(120,172),(121,171),(122,170),(123,169),(124,168),(125,167),(126,166),(127,165),(128,164),(129,163),(130,162),(131,161),(132,160),(133,159),(134,158),(135,157)]])
93 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6A | 6B | 6C | 9A | 9B | 9C | 10A | ··· | 10F | 15A | 15B | 15C | 15D | 18A | ··· | 18I | 30A | ··· | 30L | 45A | ··· | 45L | 90A | ··· | 90AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 6 | 6 | 9 | 9 | 9 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 18 | ··· | 18 | 30 | ··· | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 2 | 90 | 2 | 90 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D9 | D10 | C3⋊D4 | D15 | D18 | C5⋊D4 | D30 | C9⋊D4 | D45 | C15⋊7D4 | D90 | C45⋊7D4 |
| kernel | C45⋊7D4 | Dic45 | D90 | C2×C90 | C2×C30 | C45 | C2×C18 | C30 | C2×C10 | C18 | C15 | C2×C6 | C10 | C9 | C6 | C5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 4 | 3 | 4 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C45⋊7D4 ►in GL2(𝔽181) generated by
| 135 | 39 |
| 142 | 96 |
| 153 | 125 |
| 153 | 28 |
| 1 | 0 |
| 180 | 180 |
G:=sub<GL(2,GF(181))| [135,142,39,96],[153,153,125,28],[1,180,0,180] >;
C45⋊7D4 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