direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C44, C20⋊2C22, C220⋊6C2, Dic5⋊2C22, D10.2C22, C22.14D10, C110.19C22, C55⋊9(C2×C4), C5⋊2(C2×C44), C2.1(D5×C22), C10.2(C2×C22), (D5×C22).4C2, (C11×Dic5)⋊5C2, SmallGroup(440,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C44 |
Generators and relations for D5×C44
G = < a,b,c | a44=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C44
►On 220 points
Generators in S220
(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)(177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220)
(1 87 188 170 119)(2 88 189 171 120)(3 45 190 172 121)(4 46 191 173 122)(5 47 192 174 123)(6 48 193 175 124)(7 49 194 176 125)(8 50 195 133 126)(9 51 196 134 127)(10 52 197 135 128)(11 53 198 136 129)(12 54 199 137 130)(13 55 200 138 131)(14 56 201 139 132)(15 57 202 140 89)(16 58 203 141 90)(17 59 204 142 91)(18 60 205 143 92)(19 61 206 144 93)(20 62 207 145 94)(21 63 208 146 95)(22 64 209 147 96)(23 65 210 148 97)(24 66 211 149 98)(25 67 212 150 99)(26 68 213 151 100)(27 69 214 152 101)(28 70 215 153 102)(29 71 216 154 103)(30 72 217 155 104)(31 73 218 156 105)(32 74 219 157 106)(33 75 220 158 107)(34 76 177 159 108)(35 77 178 160 109)(36 78 179 161 110)(37 79 180 162 111)(38 80 181 163 112)(39 81 182 164 113)(40 82 183 165 114)(41 83 184 166 115)(42 84 185 167 116)(43 85 186 168 117)(44 86 187 169 118)
(1 119)(2 120)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 101)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 172)(46 173)(47 174)(48 175)(49 176)(50 133)(51 134)(52 135)(53 136)(54 137)(55 138)(56 139)(57 140)(58 141)(59 142)(60 143)(61 144)(62 145)(63 146)(64 147)(65 148)(66 149)(67 150)(68 151)(69 152)(70 153)(71 154)(72 155)(73 156)(74 157)(75 158)(76 159)(77 160)(78 161)(79 162)(80 163)(81 164)(82 165)(83 166)(84 167)(85 168)(86 169)(87 170)(88 171)
G:=sub<Sym(220)| (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)(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220), (1,87,188,170,119)(2,88,189,171,120)(3,45,190,172,121)(4,46,191,173,122)(5,47,192,174,123)(6,48,193,175,124)(7,49,194,176,125)(8,50,195,133,126)(9,51,196,134,127)(10,52,197,135,128)(11,53,198,136,129)(12,54,199,137,130)(13,55,200,138,131)(14,56,201,139,132)(15,57,202,140,89)(16,58,203,141,90)(17,59,204,142,91)(18,60,205,143,92)(19,61,206,144,93)(20,62,207,145,94)(21,63,208,146,95)(22,64,209,147,96)(23,65,210,148,97)(24,66,211,149,98)(25,67,212,150,99)(26,68,213,151,100)(27,69,214,152,101)(28,70,215,153,102)(29,71,216,154,103)(30,72,217,155,104)(31,73,218,156,105)(32,74,219,157,106)(33,75,220,158,107)(34,76,177,159,108)(35,77,178,160,109)(36,78,179,161,110)(37,79,180,162,111)(38,80,181,163,112)(39,81,182,164,113)(40,82,183,165,114)(41,83,184,166,115)(42,84,185,167,116)(43,85,186,168,117)(44,86,187,169,118), (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,172)(46,173)(47,174)(48,175)(49,176)(50,133)(51,134)(52,135)(53,136)(54,137)(55,138)(56,139)(57,140)(58,141)(59,142)(60,143)(61,144)(62,145)(63,146)(64,147)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,159)(77,160)(78,161)(79,162)(80,163)(81,164)(82,165)(83,166)(84,167)(85,168)(86,169)(87,170)(88,171)>;
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)(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220), (1,87,188,170,119)(2,88,189,171,120)(3,45,190,172,121)(4,46,191,173,122)(5,47,192,174,123)(6,48,193,175,124)(7,49,194,176,125)(8,50,195,133,126)(9,51,196,134,127)(10,52,197,135,128)(11,53,198,136,129)(12,54,199,137,130)(13,55,200,138,131)(14,56,201,139,132)(15,57,202,140,89)(16,58,203,141,90)(17,59,204,142,91)(18,60,205,143,92)(19,61,206,144,93)(20,62,207,145,94)(21,63,208,146,95)(22,64,209,147,96)(23,65,210,148,97)(24,66,211,149,98)(25,67,212,150,99)(26,68,213,151,100)(27,69,214,152,101)(28,70,215,153,102)(29,71,216,154,103)(30,72,217,155,104)(31,73,218,156,105)(32,74,219,157,106)(33,75,220,158,107)(34,76,177,159,108)(35,77,178,160,109)(36,78,179,161,110)(37,79,180,162,111)(38,80,181,163,112)(39,81,182,164,113)(40,82,183,165,114)(41,83,184,166,115)(42,84,185,167,116)(43,85,186,168,117)(44,86,187,169,118), (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,101)(28,102)(29,103)(30,104)(31,105)(32,106)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,172)(46,173)(47,174)(48,175)(49,176)(50,133)(51,134)(52,135)(53,136)(54,137)(55,138)(56,139)(57,140)(58,141)(59,142)(60,143)(61,144)(62,145)(63,146)(64,147)(65,148)(66,149)(67,150)(68,151)(69,152)(70,153)(71,154)(72,155)(73,156)(74,157)(75,158)(76,159)(77,160)(78,161)(79,162)(80,163)(81,164)(82,165)(83,166)(84,167)(85,168)(86,169)(87,170)(88,171) );
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),(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)], [(1,87,188,170,119),(2,88,189,171,120),(3,45,190,172,121),(4,46,191,173,122),(5,47,192,174,123),(6,48,193,175,124),(7,49,194,176,125),(8,50,195,133,126),(9,51,196,134,127),(10,52,197,135,128),(11,53,198,136,129),(12,54,199,137,130),(13,55,200,138,131),(14,56,201,139,132),(15,57,202,140,89),(16,58,203,141,90),(17,59,204,142,91),(18,60,205,143,92),(19,61,206,144,93),(20,62,207,145,94),(21,63,208,146,95),(22,64,209,147,96),(23,65,210,148,97),(24,66,211,149,98),(25,67,212,150,99),(26,68,213,151,100),(27,69,214,152,101),(28,70,215,153,102),(29,71,216,154,103),(30,72,217,155,104),(31,73,218,156,105),(32,74,219,157,106),(33,75,220,158,107),(34,76,177,159,108),(35,77,178,160,109),(36,78,179,161,110),(37,79,180,162,111),(38,80,181,163,112),(39,81,182,164,113),(40,82,183,165,114),(41,83,184,166,115),(42,84,185,167,116),(43,85,186,168,117),(44,86,187,169,118)], [(1,119),(2,120),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,101),(28,102),(29,103),(30,104),(31,105),(32,106),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,172),(46,173),(47,174),(48,175),(49,176),(50,133),(51,134),(52,135),(53,136),(54,137),(55,138),(56,139),(57,140),(58,141),(59,142),(60,143),(61,144),(62,145),(63,146),(64,147),(65,148),(66,149),(67,150),(68,151),(69,152),(70,153),(71,154),(72,155),(73,156),(74,157),(75,158),(76,159),(77,160),(78,161),(79,162),(80,163),(81,164),(82,165),(83,166),(84,167),(85,168),(86,169),(87,170),(88,171)]])
176 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 10A | 10B | 11A | ··· | 11J | 20A | 20B | 20C | 20D | 22A | ··· | 22J | 22K | ··· | 22AD | 44A | ··· | 44T | 44U | ··· | 44AN | 55A | ··· | 55T | 110A | ··· | 110T | 220A | ··· | 220AN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | 10 | 11 | ··· | 11 | 20 | 20 | 20 | 20 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 220 | ··· | 220 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
176 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C11 | C22 | C22 | C22 | C44 | D5 | D10 | C4×D5 | D5×C11 | D5×C22 | D5×C44 |
| kernel | D5×C44 | C11×Dic5 | C220 | D5×C22 | D5×C11 | C4×D5 | Dic5 | C20 | D10 | D5 | C44 | C22 | C11 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 40 | 2 | 2 | 4 | 20 | 20 | 40 |
Matrix representation of D5×C44 ►in GL2(𝔽661) generated by
| 379 | 0 |
| 0 | 379 |
| 660 | 1 |
| 56 | 604 |
| 660 | 0 |
| 56 | 1 |
G:=sub<GL(2,GF(661))| [379,0,0,379],[660,56,1,604],[660,56,0,1] >;
D5×C44 in GAP, Magma, Sage, TeX
D_5\times C_{44} % in TeX
G:=Group("D5xC44"); // GroupNames label
G:=SmallGroup(440,30);
// by ID
G=gap.SmallGroup(440,30);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-5,226,8804]);
// Polycyclic
G:=Group<a,b,c|a^44=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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