direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C11×C5⋊D4, C55⋊9D4, Dic5⋊C22, D10⋊2C22, C22.17D10, C110.22C22, C5⋊2(D4×C11), (C2×C22)⋊1D5, C22⋊(D5×C11), (C2×C110)⋊6C2, (C2×C10)⋊2C22, (D5×C22)⋊5C2, C2.5(D5×C22), C10.5(C2×C22), (C11×Dic5)⋊4C2, SmallGroup(440,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C5⋊D4
G = < a,b,c,d | a11=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C11×C5⋊D4
►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)
(1 162 173 113 185)(2 163 174 114 186)(3 164 175 115 187)(4 165 176 116 177)(5 155 166 117 178)(6 156 167 118 179)(7 157 168 119 180)(8 158 169 120 181)(9 159 170 121 182)(10 160 171 111 183)(11 161 172 112 184)(12 218 50 61 72)(13 219 51 62 73)(14 220 52 63 74)(15 210 53 64 75)(16 211 54 65 76)(17 212 55 66 77)(18 213 45 56 67)(19 214 46 57 68)(20 215 47 58 69)(21 216 48 59 70)(22 217 49 60 71)(23 95 35 107 83)(24 96 36 108 84)(25 97 37 109 85)(26 98 38 110 86)(27 99 39 100 87)(28 89 40 101 88)(29 90 41 102 78)(30 91 42 103 79)(31 92 43 104 80)(32 93 44 105 81)(33 94 34 106 82)(122 194 134 206 146)(123 195 135 207 147)(124 196 136 208 148)(125 197 137 209 149)(126 198 138 199 150)(127 188 139 200 151)(128 189 140 201 152)(129 190 141 202 153)(130 191 142 203 154)(131 192 143 204 144)(132 193 133 205 145)
(1 210 125 87)(2 211 126 88)(3 212 127 78)(4 213 128 79)(5 214 129 80)(6 215 130 81)(7 216 131 82)(8 217 132 83)(9 218 122 84)(10 219 123 85)(11 220 124 86)(12 194 108 159)(13 195 109 160)(14 196 110 161)(15 197 100 162)(16 198 101 163)(17 188 102 164)(18 189 103 165)(19 190 104 155)(20 191 105 156)(21 192 106 157)(22 193 107 158)(23 181 49 145)(24 182 50 146)(25 183 51 147)(26 184 52 148)(27 185 53 149)(28 186 54 150)(29 187 55 151)(30 177 45 152)(31 178 46 153)(32 179 47 154)(33 180 48 144)(34 168 70 143)(35 169 71 133)(36 170 72 134)(37 171 73 135)(38 172 74 136)(39 173 75 137)(40 174 76 138)(41 175 77 139)(42 176 67 140)(43 166 68 141)(44 167 69 142)(56 201 91 116)(57 202 92 117)(58 203 93 118)(59 204 94 119)(60 205 95 120)(61 206 96 121)(62 207 97 111)(63 208 98 112)(64 209 99 113)(65 199 89 114)(66 200 90 115)
(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 23)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 65)(41 66)(42 56)(43 57)(44 58)(45 103)(46 104)(47 105)(48 106)(49 107)(50 108)(51 109)(52 110)(53 100)(54 101)(55 102)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(73 97)(74 98)(75 99)(76 89)(77 90)(78 212)(79 213)(80 214)(81 215)(82 216)(83 217)(84 218)(85 219)(86 220)(87 210)(88 211)(111 171)(112 172)(113 173)(114 174)(115 175)(116 176)(117 166)(118 167)(119 168)(120 169)(121 170)(133 205)(134 206)(135 207)(136 208)(137 209)(138 199)(139 200)(140 201)(141 202)(142 203)(143 204)(144 192)(145 193)(146 194)(147 195)(148 196)(149 197)(150 198)(151 188)(152 189)(153 190)(154 191)(155 178)(156 179)(157 180)(158 181)(159 182)(160 183)(161 184)(162 185)(163 186)(164 187)(165 177)
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), (1,162,173,113,185)(2,163,174,114,186)(3,164,175,115,187)(4,165,176,116,177)(5,155,166,117,178)(6,156,167,118,179)(7,157,168,119,180)(8,158,169,120,181)(9,159,170,121,182)(10,160,171,111,183)(11,161,172,112,184)(12,218,50,61,72)(13,219,51,62,73)(14,220,52,63,74)(15,210,53,64,75)(16,211,54,65,76)(17,212,55,66,77)(18,213,45,56,67)(19,214,46,57,68)(20,215,47,58,69)(21,216,48,59,70)(22,217,49,60,71)(23,95,35,107,83)(24,96,36,108,84)(25,97,37,109,85)(26,98,38,110,86)(27,99,39,100,87)(28,89,40,101,88)(29,90,41,102,78)(30,91,42,103,79)(31,92,43,104,80)(32,93,44,105,81)(33,94,34,106,82)(122,194,134,206,146)(123,195,135,207,147)(124,196,136,208,148)(125,197,137,209,149)(126,198,138,199,150)(127,188,139,200,151)(128,189,140,201,152)(129,190,141,202,153)(130,191,142,203,154)(131,192,143,204,144)(132,193,133,205,145), (1,210,125,87)(2,211,126,88)(3,212,127,78)(4,213,128,79)(5,214,129,80)(6,215,130,81)(7,216,131,82)(8,217,132,83)(9,218,122,84)(10,219,123,85)(11,220,124,86)(12,194,108,159)(13,195,109,160)(14,196,110,161)(15,197,100,162)(16,198,101,163)(17,188,102,164)(18,189,103,165)(19,190,104,155)(20,191,105,156)(21,192,106,157)(22,193,107,158)(23,181,49,145)(24,182,50,146)(25,183,51,147)(26,184,52,148)(27,185,53,149)(28,186,54,150)(29,187,55,151)(30,177,45,152)(31,178,46,153)(32,179,47,154)(33,180,48,144)(34,168,70,143)(35,169,71,133)(36,170,72,134)(37,171,73,135)(38,172,74,136)(39,173,75,137)(40,174,76,138)(41,175,77,139)(42,176,67,140)(43,166,68,141)(44,167,69,142)(56,201,91,116)(57,202,92,117)(58,203,93,118)(59,204,94,119)(60,205,95,120)(61,206,96,121)(62,207,97,111)(63,208,98,112)(64,209,99,113)(65,199,89,114)(66,200,90,115), (12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,23)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,56)(43,57)(44,58)(45,103)(46,104)(47,105)(48,106)(49,107)(50,108)(51,109)(52,110)(53,100)(54,101)(55,102)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,89)(77,90)(78,212)(79,213)(80,214)(81,215)(82,216)(83,217)(84,218)(85,219)(86,220)(87,210)(88,211)(111,171)(112,172)(113,173)(114,174)(115,175)(116,176)(117,166)(118,167)(119,168)(120,169)(121,170)(133,205)(134,206)(135,207)(136,208)(137,209)(138,199)(139,200)(140,201)(141,202)(142,203)(143,204)(144,192)(145,193)(146,194)(147,195)(148,196)(149,197)(150,198)(151,188)(152,189)(153,190)(154,191)(155,178)(156,179)(157,180)(158,181)(159,182)(160,183)(161,184)(162,185)(163,186)(164,187)(165,177)>;
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), (1,162,173,113,185)(2,163,174,114,186)(3,164,175,115,187)(4,165,176,116,177)(5,155,166,117,178)(6,156,167,118,179)(7,157,168,119,180)(8,158,169,120,181)(9,159,170,121,182)(10,160,171,111,183)(11,161,172,112,184)(12,218,50,61,72)(13,219,51,62,73)(14,220,52,63,74)(15,210,53,64,75)(16,211,54,65,76)(17,212,55,66,77)(18,213,45,56,67)(19,214,46,57,68)(20,215,47,58,69)(21,216,48,59,70)(22,217,49,60,71)(23,95,35,107,83)(24,96,36,108,84)(25,97,37,109,85)(26,98,38,110,86)(27,99,39,100,87)(28,89,40,101,88)(29,90,41,102,78)(30,91,42,103,79)(31,92,43,104,80)(32,93,44,105,81)(33,94,34,106,82)(122,194,134,206,146)(123,195,135,207,147)(124,196,136,208,148)(125,197,137,209,149)(126,198,138,199,150)(127,188,139,200,151)(128,189,140,201,152)(129,190,141,202,153)(130,191,142,203,154)(131,192,143,204,144)(132,193,133,205,145), (1,210,125,87)(2,211,126,88)(3,212,127,78)(4,213,128,79)(5,214,129,80)(6,215,130,81)(7,216,131,82)(8,217,132,83)(9,218,122,84)(10,219,123,85)(11,220,124,86)(12,194,108,159)(13,195,109,160)(14,196,110,161)(15,197,100,162)(16,198,101,163)(17,188,102,164)(18,189,103,165)(19,190,104,155)(20,191,105,156)(21,192,106,157)(22,193,107,158)(23,181,49,145)(24,182,50,146)(25,183,51,147)(26,184,52,148)(27,185,53,149)(28,186,54,150)(29,187,55,151)(30,177,45,152)(31,178,46,153)(32,179,47,154)(33,180,48,144)(34,168,70,143)(35,169,71,133)(36,170,72,134)(37,171,73,135)(38,172,74,136)(39,173,75,137)(40,174,76,138)(41,175,77,139)(42,176,67,140)(43,166,68,141)(44,167,69,142)(56,201,91,116)(57,202,92,117)(58,203,93,118)(59,204,94,119)(60,205,95,120)(61,206,96,121)(62,207,97,111)(63,208,98,112)(64,209,99,113)(65,199,89,114)(66,200,90,115), (12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,23)(34,59)(35,60)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,56)(43,57)(44,58)(45,103)(46,104)(47,105)(48,106)(49,107)(50,108)(51,109)(52,110)(53,100)(54,101)(55,102)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,89)(77,90)(78,212)(79,213)(80,214)(81,215)(82,216)(83,217)(84,218)(85,219)(86,220)(87,210)(88,211)(111,171)(112,172)(113,173)(114,174)(115,175)(116,176)(117,166)(118,167)(119,168)(120,169)(121,170)(133,205)(134,206)(135,207)(136,208)(137,209)(138,199)(139,200)(140,201)(141,202)(142,203)(143,204)(144,192)(145,193)(146,194)(147,195)(148,196)(149,197)(150,198)(151,188)(152,189)(153,190)(154,191)(155,178)(156,179)(157,180)(158,181)(159,182)(160,183)(161,184)(162,185)(163,186)(164,187)(165,177) );
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)], [(1,162,173,113,185),(2,163,174,114,186),(3,164,175,115,187),(4,165,176,116,177),(5,155,166,117,178),(6,156,167,118,179),(7,157,168,119,180),(8,158,169,120,181),(9,159,170,121,182),(10,160,171,111,183),(11,161,172,112,184),(12,218,50,61,72),(13,219,51,62,73),(14,220,52,63,74),(15,210,53,64,75),(16,211,54,65,76),(17,212,55,66,77),(18,213,45,56,67),(19,214,46,57,68),(20,215,47,58,69),(21,216,48,59,70),(22,217,49,60,71),(23,95,35,107,83),(24,96,36,108,84),(25,97,37,109,85),(26,98,38,110,86),(27,99,39,100,87),(28,89,40,101,88),(29,90,41,102,78),(30,91,42,103,79),(31,92,43,104,80),(32,93,44,105,81),(33,94,34,106,82),(122,194,134,206,146),(123,195,135,207,147),(124,196,136,208,148),(125,197,137,209,149),(126,198,138,199,150),(127,188,139,200,151),(128,189,140,201,152),(129,190,141,202,153),(130,191,142,203,154),(131,192,143,204,144),(132,193,133,205,145)], [(1,210,125,87),(2,211,126,88),(3,212,127,78),(4,213,128,79),(5,214,129,80),(6,215,130,81),(7,216,131,82),(8,217,132,83),(9,218,122,84),(10,219,123,85),(11,220,124,86),(12,194,108,159),(13,195,109,160),(14,196,110,161),(15,197,100,162),(16,198,101,163),(17,188,102,164),(18,189,103,165),(19,190,104,155),(20,191,105,156),(21,192,106,157),(22,193,107,158),(23,181,49,145),(24,182,50,146),(25,183,51,147),(26,184,52,148),(27,185,53,149),(28,186,54,150),(29,187,55,151),(30,177,45,152),(31,178,46,153),(32,179,47,154),(33,180,48,144),(34,168,70,143),(35,169,71,133),(36,170,72,134),(37,171,73,135),(38,172,74,136),(39,173,75,137),(40,174,76,138),(41,175,77,139),(42,176,67,140),(43,166,68,141),(44,167,69,142),(56,201,91,116),(57,202,92,117),(58,203,93,118),(59,204,94,119),(60,205,95,120),(61,206,96,121),(62,207,97,111),(63,208,98,112),(64,209,99,113),(65,199,89,114),(66,200,90,115)], [(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,23),(34,59),(35,60),(36,61),(37,62),(38,63),(39,64),(40,65),(41,66),(42,56),(43,57),(44,58),(45,103),(46,104),(47,105),(48,106),(49,107),(50,108),(51,109),(52,110),(53,100),(54,101),(55,102),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96),(73,97),(74,98),(75,99),(76,89),(77,90),(78,212),(79,213),(80,214),(81,215),(82,216),(83,217),(84,218),(85,219),(86,220),(87,210),(88,211),(111,171),(112,172),(113,173),(114,174),(115,175),(116,176),(117,166),(118,167),(119,168),(120,169),(121,170),(133,205),(134,206),(135,207),(136,208),(137,209),(138,199),(139,200),(140,201),(141,202),(142,203),(143,204),(144,192),(145,193),(146,194),(147,195),(148,196),(149,197),(150,198),(151,188),(152,189),(153,190),(154,191),(155,178),(156,179),(157,180),(158,181),(159,182),(160,183),(161,184),(162,185),(163,186),(164,187),(165,177)]])
143 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 10A | ··· | 10F | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22T | 22U | ··· | 22AD | 44A | ··· | 44J | 55A | ··· | 55T | 110A | ··· | 110BH |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 10 | ··· | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | D4 | D5 | D10 | C5⋊D4 | D4×C11 | D5×C11 | D5×C22 | C11×C5⋊D4 |
| kernel | C11×C5⋊D4 | C11×Dic5 | D5×C22 | C2×C110 | C5⋊D4 | Dic5 | D10 | C2×C10 | C55 | C2×C22 | C22 | C11 | C5 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 1 | 2 | 2 | 4 | 10 | 20 | 20 | 40 |
Matrix representation of C11×C5⋊D4 ►in GL2(𝔽661) generated by
| 418 | 0 |
| 0 | 418 |
| 603 | 660 |
| 1 | 0 |
| 44 | 389 |
| 481 | 617 |
| 1 | 0 |
| 603 | 660 |
G:=sub<GL(2,GF(661))| [418,0,0,418],[603,1,660,0],[44,481,389,617],[1,603,0,660] >;
C11×C5⋊D4 in GAP, Magma, Sage, TeX
C_{11}\times C_5\rtimes D_4 % in TeX
G:=Group("C11xC5:D4"); // GroupNames label
G:=SmallGroup(440,33);
// by ID
G=gap.SmallGroup(440,33);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-5,461,8804]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^5=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export