direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×D44, C55⋊5D4, C220⋊3C2, C44⋊5C10, C20⋊3D11, D22⋊4C10, C10.15D22, C110.15C22, C4⋊(C5×D11), C11⋊4(C5×D4), (C10×D11)⋊4C2, C2.4(C10×D11), C22.11(C2×C10), SmallGroup(440,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D44
G = < a,b,c | a5=b44=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D44
►On 220 points
Generators in S220
(1 188 167 114 61)(2 189 168 115 62)(3 190 169 116 63)(4 191 170 117 64)(5 192 171 118 65)(6 193 172 119 66)(7 194 173 120 67)(8 195 174 121 68)(9 196 175 122 69)(10 197 176 123 70)(11 198 133 124 71)(12 199 134 125 72)(13 200 135 126 73)(14 201 136 127 74)(15 202 137 128 75)(16 203 138 129 76)(17 204 139 130 77)(18 205 140 131 78)(19 206 141 132 79)(20 207 142 89 80)(21 208 143 90 81)(22 209 144 91 82)(23 210 145 92 83)(24 211 146 93 84)(25 212 147 94 85)(26 213 148 95 86)(27 214 149 96 87)(28 215 150 97 88)(29 216 151 98 45)(30 217 152 99 46)(31 218 153 100 47)(32 219 154 101 48)(33 220 155 102 49)(34 177 156 103 50)(35 178 157 104 51)(36 179 158 105 52)(37 180 159 106 53)(38 181 160 107 54)(39 182 161 108 55)(40 183 162 109 56)(41 184 163 110 57)(42 185 164 111 58)(43 186 165 112 59)(44 187 166 113 60)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(45 87)(46 86)(47 85)(48 84)(49 83)(50 82)(51 81)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)(89 105)(90 104)(91 103)(92 102)(93 101)(94 100)(95 99)(96 98)(106 132)(107 131)(108 130)(109 129)(110 128)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 120)(133 167)(134 166)(135 165)(136 164)(137 163)(138 162)(139 161)(140 160)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)(168 176)(169 175)(170 174)(171 173)(177 209)(178 208)(179 207)(180 206)(181 205)(182 204)(183 203)(184 202)(185 201)(186 200)(187 199)(188 198)(189 197)(190 196)(191 195)(192 194)(210 220)(211 219)(212 218)(213 217)(214 216)
G:=sub<Sym(220)| (1,188,167,114,61)(2,189,168,115,62)(3,190,169,116,63)(4,191,170,117,64)(5,192,171,118,65)(6,193,172,119,66)(7,194,173,120,67)(8,195,174,121,68)(9,196,175,122,69)(10,197,176,123,70)(11,198,133,124,71)(12,199,134,125,72)(13,200,135,126,73)(14,201,136,127,74)(15,202,137,128,75)(16,203,138,129,76)(17,204,139,130,77)(18,205,140,131,78)(19,206,141,132,79)(20,207,142,89,80)(21,208,143,90,81)(22,209,144,91,82)(23,210,145,92,83)(24,211,146,93,84)(25,212,147,94,85)(26,213,148,95,86)(27,214,149,96,87)(28,215,150,97,88)(29,216,151,98,45)(30,217,152,99,46)(31,218,153,100,47)(32,219,154,101,48)(33,220,155,102,49)(34,177,156,103,50)(35,178,157,104,51)(36,179,158,105,52)(37,180,159,106,53)(38,181,160,107,54)(39,182,161,108,55)(40,183,162,109,56)(41,184,163,110,57)(42,185,164,111,58)(43,186,165,112,59)(44,187,166,113,60), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(106,132)(107,131)(108,130)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(133,167)(134,166)(135,165)(136,164)(137,163)(138,162)(139,161)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(168,176)(169,175)(170,174)(171,173)(177,209)(178,208)(179,207)(180,206)(181,205)(182,204)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,196)(191,195)(192,194)(210,220)(211,219)(212,218)(213,217)(214,216)>;
G:=Group( (1,188,167,114,61)(2,189,168,115,62)(3,190,169,116,63)(4,191,170,117,64)(5,192,171,118,65)(6,193,172,119,66)(7,194,173,120,67)(8,195,174,121,68)(9,196,175,122,69)(10,197,176,123,70)(11,198,133,124,71)(12,199,134,125,72)(13,200,135,126,73)(14,201,136,127,74)(15,202,137,128,75)(16,203,138,129,76)(17,204,139,130,77)(18,205,140,131,78)(19,206,141,132,79)(20,207,142,89,80)(21,208,143,90,81)(22,209,144,91,82)(23,210,145,92,83)(24,211,146,93,84)(25,212,147,94,85)(26,213,148,95,86)(27,214,149,96,87)(28,215,150,97,88)(29,216,151,98,45)(30,217,152,99,46)(31,218,153,100,47)(32,219,154,101,48)(33,220,155,102,49)(34,177,156,103,50)(35,178,157,104,51)(36,179,158,105,52)(37,180,159,106,53)(38,181,160,107,54)(39,182,161,108,55)(40,183,162,109,56)(41,184,163,110,57)(42,185,164,111,58)(43,186,165,112,59)(44,187,166,113,60), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(45,87)(46,86)(47,85)(48,84)(49,83)(50,82)(51,81)(52,80)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)(59,73)(60,72)(61,71)(62,70)(63,69)(64,68)(65,67)(89,105)(90,104)(91,103)(92,102)(93,101)(94,100)(95,99)(96,98)(106,132)(107,131)(108,130)(109,129)(110,128)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(133,167)(134,166)(135,165)(136,164)(137,163)(138,162)(139,161)(140,160)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)(168,176)(169,175)(170,174)(171,173)(177,209)(178,208)(179,207)(180,206)(181,205)(182,204)(183,203)(184,202)(185,201)(186,200)(187,199)(188,198)(189,197)(190,196)(191,195)(192,194)(210,220)(211,219)(212,218)(213,217)(214,216) );
G=PermutationGroup([[(1,188,167,114,61),(2,189,168,115,62),(3,190,169,116,63),(4,191,170,117,64),(5,192,171,118,65),(6,193,172,119,66),(7,194,173,120,67),(8,195,174,121,68),(9,196,175,122,69),(10,197,176,123,70),(11,198,133,124,71),(12,199,134,125,72),(13,200,135,126,73),(14,201,136,127,74),(15,202,137,128,75),(16,203,138,129,76),(17,204,139,130,77),(18,205,140,131,78),(19,206,141,132,79),(20,207,142,89,80),(21,208,143,90,81),(22,209,144,91,82),(23,210,145,92,83),(24,211,146,93,84),(25,212,147,94,85),(26,213,148,95,86),(27,214,149,96,87),(28,215,150,97,88),(29,216,151,98,45),(30,217,152,99,46),(31,218,153,100,47),(32,219,154,101,48),(33,220,155,102,49),(34,177,156,103,50),(35,178,157,104,51),(36,179,158,105,52),(37,180,159,106,53),(38,181,160,107,54),(39,182,161,108,55),(40,183,162,109,56),(41,184,163,110,57),(42,185,164,111,58),(43,186,165,112,59),(44,187,166,113,60)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(45,87),(46,86),(47,85),(48,84),(49,83),(50,82),(51,81),(52,80),(53,79),(54,78),(55,77),(56,76),(57,75),(58,74),(59,73),(60,72),(61,71),(62,70),(63,69),(64,68),(65,67),(89,105),(90,104),(91,103),(92,102),(93,101),(94,100),(95,99),(96,98),(106,132),(107,131),(108,130),(109,129),(110,128),(111,127),(112,126),(113,125),(114,124),(115,123),(116,122),(117,121),(118,120),(133,167),(134,166),(135,165),(136,164),(137,163),(138,162),(139,161),(140,160),(141,159),(142,158),(143,157),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151),(168,176),(169,175),(170,174),(171,173),(177,209),(178,208),(179,207),(180,206),(181,205),(182,204),(183,203),(184,202),(185,201),(186,200),(187,199),(188,198),(189,197),(190,196),(191,195),(192,194),(210,220),(211,219),(212,218),(213,217),(214,216)]])
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 11A | ··· | 11E | 20A | 20B | 20C | 20D | 22A | ··· | 22E | 44A | ··· | 44J | 55A | ··· | 55T | 110A | ··· | 110T | 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 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 220 | ··· | 220 |
| size | 1 | 1 | 22 | 22 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 22 | ··· | 22 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D4 | D11 | C5×D4 | D22 | D44 | C5×D11 | C10×D11 | C5×D44 |
| kernel | C5×D44 | C220 | C10×D11 | D44 | C44 | D22 | C55 | C20 | C11 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 5 | 4 | 5 | 10 | 20 | 20 | 40 |
Matrix representation of C5×D44 ►in GL2(𝔽661) generated by
| 197 | 0 |
| 0 | 197 |
| 409 | 377 |
| 284 | 564 |
| 0 | 660 |
| 660 | 0 |
G:=sub<GL(2,GF(661))| [197,0,0,197],[409,284,377,564],[0,660,660,0] >;
C5×D44 in GAP, Magma, Sage, TeX
C_5\times D_{44} % in TeX
G:=Group("C5xD44"); // GroupNames label
G:=SmallGroup(440,26);
// by ID
G=gap.SmallGroup(440,26);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-11,221,106,10004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^44=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export