metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D55⋊2C4, D110.C2, C10.3D22, C22.3D10, Dic11⋊2D5, Dic5⋊2D11, C110.3C22, C55⋊6(C2×C4), C11⋊1(C4×D5), C5⋊2(C4×D11), C2.3(D5×D11), (C5×Dic11)⋊2C2, (C11×Dic5)⋊2C2, SmallGroup(440,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C55 — D55⋊2C4 |
Generators and relations for D55⋊2C4
G = < a,b,c | a55=b2=c4=1, bab=a-1, cac-1=a34, cbc-1=a33b >
Smallest permutation representation of D55⋊2C4
►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 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 110)(11 109)(12 108)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 101)(20 100)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(111 207)(112 206)(113 205)(114 204)(115 203)(116 202)(117 201)(118 200)(119 199)(120 198)(121 197)(122 196)(123 195)(124 194)(125 193)(126 192)(127 191)(128 190)(129 189)(130 188)(131 187)(132 186)(133 185)(134 184)(135 183)(136 182)(137 181)(138 180)(139 179)(140 178)(141 177)(142 176)(143 175)(144 174)(145 173)(146 172)(147 171)(148 170)(149 169)(150 168)(151 167)(152 166)(153 220)(154 219)(155 218)(156 217)(157 216)(158 215)(159 214)(160 213)(161 212)(162 211)(163 210)(164 209)(165 208)
(1 202 65 117)(2 181 66 151)(3 215 67 130)(4 194 68 164)(5 173 69 143)(6 207 70 122)(7 186 71 156)(8 220 72 135)(9 199 73 114)(10 178 74 148)(11 212 75 127)(12 191 76 161)(13 170 77 140)(14 204 78 119)(15 183 79 153)(16 217 80 132)(17 196 81 111)(18 175 82 145)(19 209 83 124)(20 188 84 158)(21 167 85 137)(22 201 86 116)(23 180 87 150)(24 214 88 129)(25 193 89 163)(26 172 90 142)(27 206 91 121)(28 185 92 155)(29 219 93 134)(30 198 94 113)(31 177 95 147)(32 211 96 126)(33 190 97 160)(34 169 98 139)(35 203 99 118)(36 182 100 152)(37 216 101 131)(38 195 102 165)(39 174 103 144)(40 208 104 123)(41 187 105 157)(42 166 106 136)(43 200 107 115)(44 179 108 149)(45 213 109 128)(46 192 110 162)(47 171 56 141)(48 205 57 120)(49 184 58 154)(50 218 59 133)(51 197 60 112)(52 176 61 146)(53 210 62 125)(54 189 63 159)(55 168 64 138)
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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(111,207)(112,206)(113,205)(114,204)(115,203)(116,202)(117,201)(118,200)(119,199)(120,198)(121,197)(122,196)(123,195)(124,194)(125,193)(126,192)(127,191)(128,190)(129,189)(130,188)(131,187)(132,186)(133,185)(134,184)(135,183)(136,182)(137,181)(138,180)(139,179)(140,178)(141,177)(142,176)(143,175)(144,174)(145,173)(146,172)(147,171)(148,170)(149,169)(150,168)(151,167)(152,166)(153,220)(154,219)(155,218)(156,217)(157,216)(158,215)(159,214)(160,213)(161,212)(162,211)(163,210)(164,209)(165,208), (1,202,65,117)(2,181,66,151)(3,215,67,130)(4,194,68,164)(5,173,69,143)(6,207,70,122)(7,186,71,156)(8,220,72,135)(9,199,73,114)(10,178,74,148)(11,212,75,127)(12,191,76,161)(13,170,77,140)(14,204,78,119)(15,183,79,153)(16,217,80,132)(17,196,81,111)(18,175,82,145)(19,209,83,124)(20,188,84,158)(21,167,85,137)(22,201,86,116)(23,180,87,150)(24,214,88,129)(25,193,89,163)(26,172,90,142)(27,206,91,121)(28,185,92,155)(29,219,93,134)(30,198,94,113)(31,177,95,147)(32,211,96,126)(33,190,97,160)(34,169,98,139)(35,203,99,118)(36,182,100,152)(37,216,101,131)(38,195,102,165)(39,174,103,144)(40,208,104,123)(41,187,105,157)(42,166,106,136)(43,200,107,115)(44,179,108,149)(45,213,109,128)(46,192,110,162)(47,171,56,141)(48,205,57,120)(49,184,58,154)(50,218,59,133)(51,197,60,112)(52,176,61,146)(53,210,62,125)(54,189,63,159)(55,168,64,138)>;
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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,110)(11,109)(12,108)(13,107)(14,106)(15,105)(16,104)(17,103)(18,102)(19,101)(20,100)(21,99)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(111,207)(112,206)(113,205)(114,204)(115,203)(116,202)(117,201)(118,200)(119,199)(120,198)(121,197)(122,196)(123,195)(124,194)(125,193)(126,192)(127,191)(128,190)(129,189)(130,188)(131,187)(132,186)(133,185)(134,184)(135,183)(136,182)(137,181)(138,180)(139,179)(140,178)(141,177)(142,176)(143,175)(144,174)(145,173)(146,172)(147,171)(148,170)(149,169)(150,168)(151,167)(152,166)(153,220)(154,219)(155,218)(156,217)(157,216)(158,215)(159,214)(160,213)(161,212)(162,211)(163,210)(164,209)(165,208), (1,202,65,117)(2,181,66,151)(3,215,67,130)(4,194,68,164)(5,173,69,143)(6,207,70,122)(7,186,71,156)(8,220,72,135)(9,199,73,114)(10,178,74,148)(11,212,75,127)(12,191,76,161)(13,170,77,140)(14,204,78,119)(15,183,79,153)(16,217,80,132)(17,196,81,111)(18,175,82,145)(19,209,83,124)(20,188,84,158)(21,167,85,137)(22,201,86,116)(23,180,87,150)(24,214,88,129)(25,193,89,163)(26,172,90,142)(27,206,91,121)(28,185,92,155)(29,219,93,134)(30,198,94,113)(31,177,95,147)(32,211,96,126)(33,190,97,160)(34,169,98,139)(35,203,99,118)(36,182,100,152)(37,216,101,131)(38,195,102,165)(39,174,103,144)(40,208,104,123)(41,187,105,157)(42,166,106,136)(43,200,107,115)(44,179,108,149)(45,213,109,128)(46,192,110,162)(47,171,56,141)(48,205,57,120)(49,184,58,154)(50,218,59,133)(51,197,60,112)(52,176,61,146)(53,210,62,125)(54,189,63,159)(55,168,64,138) );
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)], [(1,64),(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,56),(10,110),(11,109),(12,108),(13,107),(14,106),(15,105),(16,104),(17,103),(18,102),(19,101),(20,100),(21,99),(22,98),(23,97),(24,96),(25,95),(26,94),(27,93),(28,92),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,84),(37,83),(38,82),(39,81),(40,80),(41,79),(42,78),(43,77),(44,76),(45,75),(46,74),(47,73),(48,72),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(111,207),(112,206),(113,205),(114,204),(115,203),(116,202),(117,201),(118,200),(119,199),(120,198),(121,197),(122,196),(123,195),(124,194),(125,193),(126,192),(127,191),(128,190),(129,189),(130,188),(131,187),(132,186),(133,185),(134,184),(135,183),(136,182),(137,181),(138,180),(139,179),(140,178),(141,177),(142,176),(143,175),(144,174),(145,173),(146,172),(147,171),(148,170),(149,169),(150,168),(151,167),(152,166),(153,220),(154,219),(155,218),(156,217),(157,216),(158,215),(159,214),(160,213),(161,212),(162,211),(163,210),(164,209),(165,208)], [(1,202,65,117),(2,181,66,151),(3,215,67,130),(4,194,68,164),(5,173,69,143),(6,207,70,122),(7,186,71,156),(8,220,72,135),(9,199,73,114),(10,178,74,148),(11,212,75,127),(12,191,76,161),(13,170,77,140),(14,204,78,119),(15,183,79,153),(16,217,80,132),(17,196,81,111),(18,175,82,145),(19,209,83,124),(20,188,84,158),(21,167,85,137),(22,201,86,116),(23,180,87,150),(24,214,88,129),(25,193,89,163),(26,172,90,142),(27,206,91,121),(28,185,92,155),(29,219,93,134),(30,198,94,113),(31,177,95,147),(32,211,96,126),(33,190,97,160),(34,169,98,139),(35,203,99,118),(36,182,100,152),(37,216,101,131),(38,195,102,165),(39,174,103,144),(40,208,104,123),(41,187,105,157),(42,166,106,136),(43,200,107,115),(44,179,108,149),(45,213,109,128),(46,192,110,162),(47,171,56,141),(48,205,57,120),(49,184,58,154),(50,218,59,133),(51,197,60,112),(52,176,61,146),(53,210,62,125),(54,189,63,159),(55,168,64,138)]])
56 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 | ··· | 55J | 110A | ··· | 110J |
| 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 |
| size | 1 | 1 | 55 | 55 | 5 | 5 | 11 | 11 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D5 | D10 | D11 | C4×D5 | D22 | C4×D11 | D5×D11 | D55⋊2C4 |
| kernel | D55⋊2C4 | C11×Dic5 | C5×Dic11 | D110 | D55 | Dic11 | C22 | Dic5 | C11 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 5 | 4 | 5 | 10 | 10 | 10 |
Matrix representation of D55⋊2C4 ►in GL4(𝔽661) generated by
| 218 | 175 | 0 | 0 |
| 486 | 660 | 0 | 0 |
| 0 | 0 | 1 | 177 |
| 0 | 0 | 459 | 602 |
| 218 | 175 | 0 | 0 |
| 363 | 443 | 0 | 0 |
| 0 | 0 | 660 | 484 |
| 0 | 0 | 0 | 1 |
| 660 | 0 | 0 | 0 |
| 0 | 660 | 0 | 0 |
| 0 | 0 | 555 | 0 |
| 0 | 0 | 260 | 106 |
G:=sub<GL(4,GF(661))| [218,486,0,0,175,660,0,0,0,0,1,459,0,0,177,602],[218,363,0,0,175,443,0,0,0,0,660,0,0,0,484,1],[660,0,0,0,0,660,0,0,0,0,555,260,0,0,0,106] >;
D55⋊2C4 in GAP, Magma, Sage, TeX
D_{55}\rtimes_2C_4 % in TeX
G:=Group("D55:2C4"); // GroupNames label
G:=SmallGroup(440,19);
// by ID
G=gap.SmallGroup(440,19);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,20,26,328,10004]);
// Polycyclic
G:=Group<a,b,c|a^55=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^34,c*b*c^-1=a^33*b>;
// generators/relations
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