direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D55, C44⋊2D5, C220⋊2C2, C20⋊2D11, C10.9D22, C2.1D110, C22.9D10, Dic55⋊5C2, D110.2C2, C110.9C22, C55⋊7(C2×C4), C5⋊3(C4×D11), C11⋊2(C4×D5), SmallGroup(440,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C55 — C4×D55 |
Generators and relations for C4×D55
G = < a,b,c | a4=b55=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D55
►On 220 points
Generators in S220
(1 209 106 112)(2 210 107 113)(3 211 108 114)(4 212 109 115)(5 213 110 116)(6 214 56 117)(7 215 57 118)(8 216 58 119)(9 217 59 120)(10 218 60 121)(11 219 61 122)(12 220 62 123)(13 166 63 124)(14 167 64 125)(15 168 65 126)(16 169 66 127)(17 170 67 128)(18 171 68 129)(19 172 69 130)(20 173 70 131)(21 174 71 132)(22 175 72 133)(23 176 73 134)(24 177 74 135)(25 178 75 136)(26 179 76 137)(27 180 77 138)(28 181 78 139)(29 182 79 140)(30 183 80 141)(31 184 81 142)(32 185 82 143)(33 186 83 144)(34 187 84 145)(35 188 85 146)(36 189 86 147)(37 190 87 148)(38 191 88 149)(39 192 89 150)(40 193 90 151)(41 194 91 152)(42 195 92 153)(43 196 93 154)(44 197 94 155)(45 198 95 156)(46 199 96 157)(47 200 97 158)(48 201 98 159)(49 202 99 160)(50 203 100 161)(51 204 101 162)(52 205 102 163)(53 206 103 164)(54 207 104 165)(55 208 105 111)
(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)
(1 105)(2 104)(3 103)(4 102)(5 101)(6 100)(7 99)(8 98)(9 97)(10 96)(11 95)(12 94)(13 93)(14 92)(15 91)(16 90)(17 89)(18 88)(19 87)(20 86)(21 85)(22 84)(23 83)(24 82)(25 81)(26 80)(27 79)(28 78)(29 77)(30 76)(31 75)(32 74)(33 73)(34 72)(35 71)(36 70)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 60)(47 59)(48 58)(49 57)(50 56)(51 110)(52 109)(53 108)(54 107)(55 106)(111 209)(112 208)(113 207)(114 206)(115 205)(116 204)(117 203)(118 202)(119 201)(120 200)(121 199)(122 198)(123 197)(124 196)(125 195)(126 194)(127 193)(128 192)(129 191)(130 190)(131 189)(132 188)(133 187)(134 186)(135 185)(136 184)(137 183)(138 182)(139 181)(140 180)(141 179)(142 178)(143 177)(144 176)(145 175)(146 174)(147 173)(148 172)(149 171)(150 170)(151 169)(152 168)(153 167)(154 166)(155 220)(156 219)(157 218)(158 217)(159 216)(160 215)(161 214)(162 213)(163 212)(164 211)(165 210)
G:=sub<Sym(220)| (1,209,106,112)(2,210,107,113)(3,211,108,114)(4,212,109,115)(5,213,110,116)(6,214,56,117)(7,215,57,118)(8,216,58,119)(9,217,59,120)(10,218,60,121)(11,219,61,122)(12,220,62,123)(13,166,63,124)(14,167,64,125)(15,168,65,126)(16,169,66,127)(17,170,67,128)(18,171,68,129)(19,172,69,130)(20,173,70,131)(21,174,71,132)(22,175,72,133)(23,176,73,134)(24,177,74,135)(25,178,75,136)(26,179,76,137)(27,180,77,138)(28,181,78,139)(29,182,79,140)(30,183,80,141)(31,184,81,142)(32,185,82,143)(33,186,83,144)(34,187,84,145)(35,188,85,146)(36,189,86,147)(37,190,87,148)(38,191,88,149)(39,192,89,150)(40,193,90,151)(41,194,91,152)(42,195,92,153)(43,196,93,154)(44,197,94,155)(45,198,95,156)(46,199,96,157)(47,200,97,158)(48,201,98,159)(49,202,99,160)(50,203,100,161)(51,204,101,162)(52,205,102,163)(53,206,103,164)(54,207,104,165)(55,208,105,111), (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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,84)(23,83)(24,82)(25,81)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,72)(35,71)(36,70)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,110)(52,109)(53,108)(54,107)(55,106)(111,209)(112,208)(113,207)(114,206)(115,205)(116,204)(117,203)(118,202)(119,201)(120,200)(121,199)(122,198)(123,197)(124,196)(125,195)(126,194)(127,193)(128,192)(129,191)(130,190)(131,189)(132,188)(133,187)(134,186)(135,185)(136,184)(137,183)(138,182)(139,181)(140,180)(141,179)(142,178)(143,177)(144,176)(145,175)(146,174)(147,173)(148,172)(149,171)(150,170)(151,169)(152,168)(153,167)(154,166)(155,220)(156,219)(157,218)(158,217)(159,216)(160,215)(161,214)(162,213)(163,212)(164,211)(165,210)>;
G:=Group( (1,209,106,112)(2,210,107,113)(3,211,108,114)(4,212,109,115)(5,213,110,116)(6,214,56,117)(7,215,57,118)(8,216,58,119)(9,217,59,120)(10,218,60,121)(11,219,61,122)(12,220,62,123)(13,166,63,124)(14,167,64,125)(15,168,65,126)(16,169,66,127)(17,170,67,128)(18,171,68,129)(19,172,69,130)(20,173,70,131)(21,174,71,132)(22,175,72,133)(23,176,73,134)(24,177,74,135)(25,178,75,136)(26,179,76,137)(27,180,77,138)(28,181,78,139)(29,182,79,140)(30,183,80,141)(31,184,81,142)(32,185,82,143)(33,186,83,144)(34,187,84,145)(35,188,85,146)(36,189,86,147)(37,190,87,148)(38,191,88,149)(39,192,89,150)(40,193,90,151)(41,194,91,152)(42,195,92,153)(43,196,93,154)(44,197,94,155)(45,198,95,156)(46,199,96,157)(47,200,97,158)(48,201,98,159)(49,202,99,160)(50,203,100,161)(51,204,101,162)(52,205,102,163)(53,206,103,164)(54,207,104,165)(55,208,105,111), (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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,84)(23,83)(24,82)(25,81)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,72)(35,71)(36,70)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,60)(47,59)(48,58)(49,57)(50,56)(51,110)(52,109)(53,108)(54,107)(55,106)(111,209)(112,208)(113,207)(114,206)(115,205)(116,204)(117,203)(118,202)(119,201)(120,200)(121,199)(122,198)(123,197)(124,196)(125,195)(126,194)(127,193)(128,192)(129,191)(130,190)(131,189)(132,188)(133,187)(134,186)(135,185)(136,184)(137,183)(138,182)(139,181)(140,180)(141,179)(142,178)(143,177)(144,176)(145,175)(146,174)(147,173)(148,172)(149,171)(150,170)(151,169)(152,168)(153,167)(154,166)(155,220)(156,219)(157,218)(158,217)(159,216)(160,215)(161,214)(162,213)(163,212)(164,211)(165,210) );
G=PermutationGroup([[(1,209,106,112),(2,210,107,113),(3,211,108,114),(4,212,109,115),(5,213,110,116),(6,214,56,117),(7,215,57,118),(8,216,58,119),(9,217,59,120),(10,218,60,121),(11,219,61,122),(12,220,62,123),(13,166,63,124),(14,167,64,125),(15,168,65,126),(16,169,66,127),(17,170,67,128),(18,171,68,129),(19,172,69,130),(20,173,70,131),(21,174,71,132),(22,175,72,133),(23,176,73,134),(24,177,74,135),(25,178,75,136),(26,179,76,137),(27,180,77,138),(28,181,78,139),(29,182,79,140),(30,183,80,141),(31,184,81,142),(32,185,82,143),(33,186,83,144),(34,187,84,145),(35,188,85,146),(36,189,86,147),(37,190,87,148),(38,191,88,149),(39,192,89,150),(40,193,90,151),(41,194,91,152),(42,195,92,153),(43,196,93,154),(44,197,94,155),(45,198,95,156),(46,199,96,157),(47,200,97,158),(48,201,98,159),(49,202,99,160),(50,203,100,161),(51,204,101,162),(52,205,102,163),(53,206,103,164),(54,207,104,165),(55,208,105,111)], [(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)], [(1,105),(2,104),(3,103),(4,102),(5,101),(6,100),(7,99),(8,98),(9,97),(10,96),(11,95),(12,94),(13,93),(14,92),(15,91),(16,90),(17,89),(18,88),(19,87),(20,86),(21,85),(22,84),(23,83),(24,82),(25,81),(26,80),(27,79),(28,78),(29,77),(30,76),(31,75),(32,74),(33,73),(34,72),(35,71),(36,70),(37,69),(38,68),(39,67),(40,66),(41,65),(42,64),(43,63),(44,62),(45,61),(46,60),(47,59),(48,58),(49,57),(50,56),(51,110),(52,109),(53,108),(54,107),(55,106),(111,209),(112,208),(113,207),(114,206),(115,205),(116,204),(117,203),(118,202),(119,201),(120,200),(121,199),(122,198),(123,197),(124,196),(125,195),(126,194),(127,193),(128,192),(129,191),(130,190),(131,189),(132,188),(133,187),(134,186),(135,185),(136,184),(137,183),(138,182),(139,181),(140,180),(141,179),(142,178),(143,177),(144,176),(145,175),(146,174),(147,173),(148,172),(149,171),(150,170),(151,169),(152,168),(153,167),(154,166),(155,220),(156,219),(157,218),(158,217),(159,216),(160,215),(161,214),(162,213),(163,212),(164,211),(165,210)]])
116 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 10A | 10B | 11A | ··· | 11E | 20A | 20B | 20C | 20D | 22A | ··· | 22E | 44A | ··· | 44J | 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 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 220 | ··· | 220 |
| size | 1 | 1 | 55 | 55 | 1 | 1 | 55 | 55 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | D5 | D10 | D11 | C4×D5 | D22 | C4×D11 | D55 | D110 | C4×D55 |
| kernel | C4×D55 | Dic55 | C220 | D110 | D55 | C44 | C22 | C20 | C11 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 5 | 4 | 5 | 10 | 20 | 20 | 40 |
Matrix representation of C4×D55 ►in GL2(𝔽661) generated by
| 106 | 0 |
| 0 | 106 |
| 50 | 482 |
| 179 | 549 |
| 611 | 179 |
| 311 | 50 |
G:=sub<GL(2,GF(661))| [106,0,0,106],[50,179,482,549],[611,311,179,50] >;
C4×D55 in GAP, Magma, Sage, TeX
C_4\times D_{55} % in TeX
G:=Group("C4xD55"); // GroupNames label
G:=SmallGroup(440,35);
// by ID
G=gap.SmallGroup(440,35);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,26,643,10004]);
// Polycyclic
G:=Group<a,b,c|a^4=b^55=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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