metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C55⋊2D4, C5⋊2D44, D22⋊2D5, Dic5⋊D11, D110⋊3C2, C22.5D10, C10.5D22, C110.5C22, C11⋊1(C5⋊D4), C2.5(D5×D11), (C10×D11)⋊2C2, (C11×Dic5)⋊3C2, SmallGroup(440,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊D44
G = < a,b,c | a5=b44=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C5⋊D44
►On 220 points
Generators in S220
(1 138 129 75 195)(2 196 76 130 139)(3 140 131 77 197)(4 198 78 132 141)(5 142 89 79 199)(6 200 80 90 143)(7 144 91 81 201)(8 202 82 92 145)(9 146 93 83 203)(10 204 84 94 147)(11 148 95 85 205)(12 206 86 96 149)(13 150 97 87 207)(14 208 88 98 151)(15 152 99 45 209)(16 210 46 100 153)(17 154 101 47 211)(18 212 48 102 155)(19 156 103 49 213)(20 214 50 104 157)(21 158 105 51 215)(22 216 52 106 159)(23 160 107 53 217)(24 218 54 108 161)(25 162 109 55 219)(26 220 56 110 163)(27 164 111 57 177)(28 178 58 112 165)(29 166 113 59 179)(30 180 60 114 167)(31 168 115 61 181)(32 182 62 116 169)(33 170 117 63 183)(34 184 64 118 171)(35 172 119 65 185)(36 186 66 120 173)(37 174 121 67 187)(38 188 68 122 175)(39 176 123 69 189)(40 190 70 124 133)(41 134 125 71 191)(42 192 72 126 135)(43 136 127 73 193)(44 194 74 128 137)
(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)(45 125)(46 124)(47 123)(48 122)(49 121)(50 120)(51 119)(52 118)(53 117)(54 116)(55 115)(56 114)(57 113)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 132)(83 131)(84 130)(85 129)(86 128)(87 127)(88 126)(133 210)(134 209)(135 208)(136 207)(137 206)(138 205)(139 204)(140 203)(141 202)(142 201)(143 200)(144 199)(145 198)(146 197)(147 196)(148 195)(149 194)(150 193)(151 192)(152 191)(153 190)(154 189)(155 188)(156 187)(157 186)(158 185)(159 184)(160 183)(161 182)(162 181)(163 180)(164 179)(165 178)(166 177)(167 220)(168 219)(169 218)(170 217)(171 216)(172 215)(173 214)(174 213)(175 212)(176 211)
G:=sub<Sym(220)| (1,138,129,75,195)(2,196,76,130,139)(3,140,131,77,197)(4,198,78,132,141)(5,142,89,79,199)(6,200,80,90,143)(7,144,91,81,201)(8,202,82,92,145)(9,146,93,83,203)(10,204,84,94,147)(11,148,95,85,205)(12,206,86,96,149)(13,150,97,87,207)(14,208,88,98,151)(15,152,99,45,209)(16,210,46,100,153)(17,154,101,47,211)(18,212,48,102,155)(19,156,103,49,213)(20,214,50,104,157)(21,158,105,51,215)(22,216,52,106,159)(23,160,107,53,217)(24,218,54,108,161)(25,162,109,55,219)(26,220,56,110,163)(27,164,111,57,177)(28,178,58,112,165)(29,166,113,59,179)(30,180,60,114,167)(31,168,115,61,181)(32,182,62,116,169)(33,170,117,63,183)(34,184,64,118,171)(35,172,119,65,185)(36,186,66,120,173)(37,174,121,67,187)(38,188,68,122,175)(39,176,123,69,189)(40,190,70,124,133)(41,134,125,71,191)(42,192,72,126,135)(43,136,127,73,193)(44,194,74,128,137), (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)(45,125)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,132)(83,131)(84,130)(85,129)(86,128)(87,127)(88,126)(133,210)(134,209)(135,208)(136,207)(137,206)(138,205)(139,204)(140,203)(141,202)(142,201)(143,200)(144,199)(145,198)(146,197)(147,196)(148,195)(149,194)(150,193)(151,192)(152,191)(153,190)(154,189)(155,188)(156,187)(157,186)(158,185)(159,184)(160,183)(161,182)(162,181)(163,180)(164,179)(165,178)(166,177)(167,220)(168,219)(169,218)(170,217)(171,216)(172,215)(173,214)(174,213)(175,212)(176,211)>;
G:=Group( (1,138,129,75,195)(2,196,76,130,139)(3,140,131,77,197)(4,198,78,132,141)(5,142,89,79,199)(6,200,80,90,143)(7,144,91,81,201)(8,202,82,92,145)(9,146,93,83,203)(10,204,84,94,147)(11,148,95,85,205)(12,206,86,96,149)(13,150,97,87,207)(14,208,88,98,151)(15,152,99,45,209)(16,210,46,100,153)(17,154,101,47,211)(18,212,48,102,155)(19,156,103,49,213)(20,214,50,104,157)(21,158,105,51,215)(22,216,52,106,159)(23,160,107,53,217)(24,218,54,108,161)(25,162,109,55,219)(26,220,56,110,163)(27,164,111,57,177)(28,178,58,112,165)(29,166,113,59,179)(30,180,60,114,167)(31,168,115,61,181)(32,182,62,116,169)(33,170,117,63,183)(34,184,64,118,171)(35,172,119,65,185)(36,186,66,120,173)(37,174,121,67,187)(38,188,68,122,175)(39,176,123,69,189)(40,190,70,124,133)(41,134,125,71,191)(42,192,72,126,135)(43,136,127,73,193)(44,194,74,128,137), (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)(45,125)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(81,89)(82,132)(83,131)(84,130)(85,129)(86,128)(87,127)(88,126)(133,210)(134,209)(135,208)(136,207)(137,206)(138,205)(139,204)(140,203)(141,202)(142,201)(143,200)(144,199)(145,198)(146,197)(147,196)(148,195)(149,194)(150,193)(151,192)(152,191)(153,190)(154,189)(155,188)(156,187)(157,186)(158,185)(159,184)(160,183)(161,182)(162,181)(163,180)(164,179)(165,178)(166,177)(167,220)(168,219)(169,218)(170,217)(171,216)(172,215)(173,214)(174,213)(175,212)(176,211) );
G=PermutationGroup([[(1,138,129,75,195),(2,196,76,130,139),(3,140,131,77,197),(4,198,78,132,141),(5,142,89,79,199),(6,200,80,90,143),(7,144,91,81,201),(8,202,82,92,145),(9,146,93,83,203),(10,204,84,94,147),(11,148,95,85,205),(12,206,86,96,149),(13,150,97,87,207),(14,208,88,98,151),(15,152,99,45,209),(16,210,46,100,153),(17,154,101,47,211),(18,212,48,102,155),(19,156,103,49,213),(20,214,50,104,157),(21,158,105,51,215),(22,216,52,106,159),(23,160,107,53,217),(24,218,54,108,161),(25,162,109,55,219),(26,220,56,110,163),(27,164,111,57,177),(28,178,58,112,165),(29,166,113,59,179),(30,180,60,114,167),(31,168,115,61,181),(32,182,62,116,169),(33,170,117,63,183),(34,184,64,118,171),(35,172,119,65,185),(36,186,66,120,173),(37,174,121,67,187),(38,188,68,122,175),(39,176,123,69,189),(40,190,70,124,133),(41,134,125,71,191),(42,192,72,126,135),(43,136,127,73,193),(44,194,74,128,137)], [(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),(45,125),(46,124),(47,123),(48,122),(49,121),(50,120),(51,119),(52,118),(53,117),(54,116),(55,115),(56,114),(57,113),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(81,89),(82,132),(83,131),(84,130),(85,129),(86,128),(87,127),(88,126),(133,210),(134,209),(135,208),(136,207),(137,206),(138,205),(139,204),(140,203),(141,202),(142,201),(143,200),(144,199),(145,198),(146,197),(147,196),(148,195),(149,194),(150,193),(151,192),(152,191),(153,190),(154,189),(155,188),(156,187),(157,186),(158,185),(159,184),(160,183),(161,182),(162,181),(163,180),(164,179),(165,178),(166,177),(167,220),(168,219),(169,218),(170,217),(171,216),(172,215),(173,214),(174,213),(175,212),(176,211)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 10A | 10B | 10C | 10D | 10E | 10F | 11A | ··· | 11E | 22A | ··· | 22E | 44A | ··· | 44J | 55A | ··· | 55J | 110A | ··· | 110J |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 22 | 110 | 10 | 2 | 2 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D5 | D10 | D11 | C5⋊D4 | D22 | D44 | D5×D11 | C5⋊D44 |
| kernel | C5⋊D44 | C11×Dic5 | C10×D11 | D110 | C55 | D22 | C22 | Dic5 | C11 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 4 | 5 | 10 | 10 | 10 |
Matrix representation of C5⋊D44 ►in GL4(𝔽661) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 660 | 1 |
| 0 | 0 | 56 | 604 |
| 110 | 157 | 0 | 0 |
| 153 | 627 | 0 | 0 |
| 0 | 0 | 136 | 44 |
| 0 | 0 | 496 | 525 |
| 627 | 504 | 0 | 0 |
| 521 | 34 | 0 | 0 |
| 0 | 0 | 604 | 660 |
| 0 | 0 | 604 | 57 |
G:=sub<GL(4,GF(661))| [1,0,0,0,0,1,0,0,0,0,660,56,0,0,1,604],[110,153,0,0,157,627,0,0,0,0,136,496,0,0,44,525],[627,521,0,0,504,34,0,0,0,0,604,604,0,0,660,57] >;
C5⋊D44 in GAP, Magma, Sage, TeX
C_5\rtimes D_{44} % in TeX
G:=Group("C5:D44"); // GroupNames label
G:=SmallGroup(440,21);
// by ID
G=gap.SmallGroup(440,21);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,61,26,328,10004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^44=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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