direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C55, C4⋊C110, C220⋊7C2, C20⋊3C22, C44⋊7C10, C22⋊C110, C110.23C22, (C2×C22)⋊5C10, (C2×C110)⋊1C2, (C2×C10)⋊1C22, C10.6(C2×C22), C2.1(C2×C110), C22.14(C2×C10), SmallGroup(440,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C55
G = < a,b,c | a55=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C55
►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 96 197 134)(2 97 198 135)(3 98 199 136)(4 99 200 137)(5 100 201 138)(6 101 202 139)(7 102 203 140)(8 103 204 141)(9 104 205 142)(10 105 206 143)(11 106 207 144)(12 107 208 145)(13 108 209 146)(14 109 210 147)(15 110 211 148)(16 56 212 149)(17 57 213 150)(18 58 214 151)(19 59 215 152)(20 60 216 153)(21 61 217 154)(22 62 218 155)(23 63 219 156)(24 64 220 157)(25 65 166 158)(26 66 167 159)(27 67 168 160)(28 68 169 161)(29 69 170 162)(30 70 171 163)(31 71 172 164)(32 72 173 165)(33 73 174 111)(34 74 175 112)(35 75 176 113)(36 76 177 114)(37 77 178 115)(38 78 179 116)(39 79 180 117)(40 80 181 118)(41 81 182 119)(42 82 183 120)(43 83 184 121)(44 84 185 122)(45 85 186 123)(46 86 187 124)(47 87 188 125)(48 88 189 126)(49 89 190 127)(50 90 191 128)(51 91 192 129)(52 92 193 130)(53 93 194 131)(54 94 195 132)(55 95 196 133)
(56 149)(57 150)(58 151)(59 152)(60 153)(61 154)(62 155)(63 156)(64 157)(65 158)(66 159)(67 160)(68 161)(69 162)(70 163)(71 164)(72 165)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)(81 119)(82 120)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 145)(108 146)(109 147)(110 148)
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), (1,96,197,134)(2,97,198,135)(3,98,199,136)(4,99,200,137)(5,100,201,138)(6,101,202,139)(7,102,203,140)(8,103,204,141)(9,104,205,142)(10,105,206,143)(11,106,207,144)(12,107,208,145)(13,108,209,146)(14,109,210,147)(15,110,211,148)(16,56,212,149)(17,57,213,150)(18,58,214,151)(19,59,215,152)(20,60,216,153)(21,61,217,154)(22,62,218,155)(23,63,219,156)(24,64,220,157)(25,65,166,158)(26,66,167,159)(27,67,168,160)(28,68,169,161)(29,69,170,162)(30,70,171,163)(31,71,172,164)(32,72,173,165)(33,73,174,111)(34,74,175,112)(35,75,176,113)(36,76,177,114)(37,77,178,115)(38,78,179,116)(39,79,180,117)(40,80,181,118)(41,81,182,119)(42,82,183,120)(43,83,184,121)(44,84,185,122)(45,85,186,123)(46,86,187,124)(47,87,188,125)(48,88,189,126)(49,89,190,127)(50,90,191,128)(51,91,192,129)(52,92,193,130)(53,93,194,131)(54,94,195,132)(55,95,196,133), (56,149)(57,150)(58,151)(59,152)(60,153)(61,154)(62,155)(63,156)(64,157)(65,158)(66,159)(67,160)(68,161)(69,162)(70,163)(71,164)(72,165)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148)>;
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), (1,96,197,134)(2,97,198,135)(3,98,199,136)(4,99,200,137)(5,100,201,138)(6,101,202,139)(7,102,203,140)(8,103,204,141)(9,104,205,142)(10,105,206,143)(11,106,207,144)(12,107,208,145)(13,108,209,146)(14,109,210,147)(15,110,211,148)(16,56,212,149)(17,57,213,150)(18,58,214,151)(19,59,215,152)(20,60,216,153)(21,61,217,154)(22,62,218,155)(23,63,219,156)(24,64,220,157)(25,65,166,158)(26,66,167,159)(27,67,168,160)(28,68,169,161)(29,69,170,162)(30,70,171,163)(31,71,172,164)(32,72,173,165)(33,73,174,111)(34,74,175,112)(35,75,176,113)(36,76,177,114)(37,77,178,115)(38,78,179,116)(39,79,180,117)(40,80,181,118)(41,81,182,119)(42,82,183,120)(43,83,184,121)(44,84,185,122)(45,85,186,123)(46,86,187,124)(47,87,188,125)(48,88,189,126)(49,89,190,127)(50,90,191,128)(51,91,192,129)(52,92,193,130)(53,93,194,131)(54,94,195,132)(55,95,196,133), (56,149)(57,150)(58,151)(59,152)(60,153)(61,154)(62,155)(63,156)(64,157)(65,158)(66,159)(67,160)(68,161)(69,162)(70,163)(71,164)(72,165)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148) );
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)], [(1,96,197,134),(2,97,198,135),(3,98,199,136),(4,99,200,137),(5,100,201,138),(6,101,202,139),(7,102,203,140),(8,103,204,141),(9,104,205,142),(10,105,206,143),(11,106,207,144),(12,107,208,145),(13,108,209,146),(14,109,210,147),(15,110,211,148),(16,56,212,149),(17,57,213,150),(18,58,214,151),(19,59,215,152),(20,60,216,153),(21,61,217,154),(22,62,218,155),(23,63,219,156),(24,64,220,157),(25,65,166,158),(26,66,167,159),(27,67,168,160),(28,68,169,161),(29,69,170,162),(30,70,171,163),(31,71,172,164),(32,72,173,165),(33,73,174,111),(34,74,175,112),(35,75,176,113),(36,76,177,114),(37,77,178,115),(38,78,179,116),(39,79,180,117),(40,80,181,118),(41,81,182,119),(42,82,183,120),(43,83,184,121),(44,84,185,122),(45,85,186,123),(46,86,187,124),(47,87,188,125),(48,88,189,126),(49,89,190,127),(50,90,191,128),(51,91,192,129),(52,92,193,130),(53,93,194,131),(54,94,195,132),(55,95,196,133)], [(56,149),(57,150),(58,151),(59,152),(60,153),(61,154),(62,155),(63,156),(64,157),(65,158),(66,159),(67,160),(68,161),(69,162),(70,163),(71,164),(72,165),(73,111),(74,112),(75,113),(76,114),(77,115),(78,116),(79,117),(80,118),(81,119),(82,120),(83,121),(84,122),(85,123),(86,124),(87,125),(88,126),(89,127),(90,128),(91,129),(92,130),(93,131),(94,132),(95,133),(96,134),(97,135),(98,136),(99,137),(100,138),(101,139),(102,140),(103,141),(104,142),(105,143),(106,144),(107,145),(108,146),(109,147),(110,148)]])
275 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 11A | ··· | 11J | 20A | 20B | 20C | 20D | 22A | ··· | 22J | 22K | ··· | 22AD | 44A | ··· | 44J | 55A | ··· | 55AN | 110A | ··· | 110AN | 110AO | ··· | 110DP | 220A | ··· | 220AN |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 11 | ··· | 11 | 20 | 20 | 20 | 20 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 110 | ··· | 110 | 220 | ··· | 220 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
275 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | C11 | C22 | C22 | C55 | C110 | C110 | D4 | C5×D4 | D4×C11 | D4×C55 |
| kernel | D4×C55 | C220 | C2×C110 | D4×C11 | C44 | C2×C22 | C5×D4 | C20 | C2×C10 | D4 | C4 | C22 | C55 | C11 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 40 | 40 | 80 | 1 | 4 | 10 | 40 |
Matrix representation of D4×C55 ►in GL3(𝔽661) generated by
| 406 | 0 | 0 |
| 0 | 634 | 0 |
| 0 | 0 | 634 |
| 660 | 0 | 0 |
| 0 | 460 | 2 |
| 0 | 290 | 201 |
| 660 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 201 | 660 |
G:=sub<GL(3,GF(661))| [406,0,0,0,634,0,0,0,634],[660,0,0,0,460,290,0,2,201],[660,0,0,0,1,201,0,0,660] >;
D4×C55 in GAP, Magma, Sage, TeX
D_4\times C_{55} % in TeX
G:=Group("D4xC55"); // GroupNames label
G:=SmallGroup(440,40);
// by ID
G=gap.SmallGroup(440,40);
# by ID
G:=PCGroup([5,-2,-2,-5,-11,-2,2221]);
// Polycyclic
G:=Group<a,b,c|a^55=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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