metabelian, supersoluble, monomial
Aliases: C40⋊2D5, C10.7D20, C52⋊6SD16, C20.46D10, (C5×C40)⋊3C2, C8⋊2(C5⋊D5), C5⋊1(C40⋊C2), (C5×C10).22D4, C52⋊4Q8⋊1C2, C20⋊D5.1C2, C2.3(C20⋊D5), (C5×C20).32C22, C4.8(C2×C5⋊D5), SmallGroup(400,94)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊2D5
G = < a,b,c | a40=b5=c2=1, ab=ba, cac=a19, cbc=b-1 >
Subgroups: 648 in 80 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, D4, Q8, D5, C10, SD16, Dic5, C20, D10, C52, C40, Dic10, D20, C5⋊D5, C5×C10, C40⋊C2, C52⋊6C4, C5×C20, C2×C5⋊D5, C5×C40, C52⋊4Q8, C20⋊D5, C40⋊2D5
Quotients: C1, C2, C22, D4, D5, SD16, D10, D20, C5⋊D5, C40⋊C2, C2×C5⋊D5, C20⋊D5, C40⋊2D5
►On 200 points
Generators in S200
(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)
(1 146 59 101 179)(2 147 60 102 180)(3 148 61 103 181)(4 149 62 104 182)(5 150 63 105 183)(6 151 64 106 184)(7 152 65 107 185)(8 153 66 108 186)(9 154 67 109 187)(10 155 68 110 188)(11 156 69 111 189)(12 157 70 112 190)(13 158 71 113 191)(14 159 72 114 192)(15 160 73 115 193)(16 121 74 116 194)(17 122 75 117 195)(18 123 76 118 196)(19 124 77 119 197)(20 125 78 120 198)(21 126 79 81 199)(22 127 80 82 200)(23 128 41 83 161)(24 129 42 84 162)(25 130 43 85 163)(26 131 44 86 164)(27 132 45 87 165)(28 133 46 88 166)(29 134 47 89 167)(30 135 48 90 168)(31 136 49 91 169)(32 137 50 92 170)(33 138 51 93 171)(34 139 52 94 172)(35 140 53 95 173)(36 141 54 96 174)(37 142 55 97 175)(38 143 56 98 176)(39 144 57 99 177)(40 145 58 100 178)
(1 179)(2 198)(3 177)(4 196)(5 175)(6 194)(7 173)(8 192)(9 171)(10 190)(11 169)(12 188)(13 167)(14 186)(15 165)(16 184)(17 163)(18 182)(19 161)(20 180)(21 199)(22 178)(23 197)(24 176)(25 195)(26 174)(27 193)(28 172)(29 191)(30 170)(31 189)(32 168)(33 187)(34 166)(35 185)(36 164)(37 183)(38 162)(39 181)(40 200)(41 77)(42 56)(43 75)(44 54)(45 73)(46 52)(47 71)(48 50)(49 69)(51 67)(53 65)(55 63)(57 61)(58 80)(60 78)(62 76)(64 74)(66 72)(68 70)(81 126)(82 145)(83 124)(84 143)(85 122)(86 141)(87 160)(88 139)(89 158)(90 137)(91 156)(92 135)(93 154)(94 133)(95 152)(96 131)(97 150)(98 129)(99 148)(100 127)(101 146)(102 125)(103 144)(104 123)(105 142)(106 121)(107 140)(108 159)(109 138)(110 157)(111 136)(112 155)(113 134)(114 153)(115 132)(116 151)(117 130)(118 149)(119 128)(120 147)
G:=sub<Sym(200)| (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), (1,146,59,101,179)(2,147,60,102,180)(3,148,61,103,181)(4,149,62,104,182)(5,150,63,105,183)(6,151,64,106,184)(7,152,65,107,185)(8,153,66,108,186)(9,154,67,109,187)(10,155,68,110,188)(11,156,69,111,189)(12,157,70,112,190)(13,158,71,113,191)(14,159,72,114,192)(15,160,73,115,193)(16,121,74,116,194)(17,122,75,117,195)(18,123,76,118,196)(19,124,77,119,197)(20,125,78,120,198)(21,126,79,81,199)(22,127,80,82,200)(23,128,41,83,161)(24,129,42,84,162)(25,130,43,85,163)(26,131,44,86,164)(27,132,45,87,165)(28,133,46,88,166)(29,134,47,89,167)(30,135,48,90,168)(31,136,49,91,169)(32,137,50,92,170)(33,138,51,93,171)(34,139,52,94,172)(35,140,53,95,173)(36,141,54,96,174)(37,142,55,97,175)(38,143,56,98,176)(39,144,57,99,177)(40,145,58,100,178), (1,179)(2,198)(3,177)(4,196)(5,175)(6,194)(7,173)(8,192)(9,171)(10,190)(11,169)(12,188)(13,167)(14,186)(15,165)(16,184)(17,163)(18,182)(19,161)(20,180)(21,199)(22,178)(23,197)(24,176)(25,195)(26,174)(27,193)(28,172)(29,191)(30,170)(31,189)(32,168)(33,187)(34,166)(35,185)(36,164)(37,183)(38,162)(39,181)(40,200)(41,77)(42,56)(43,75)(44,54)(45,73)(46,52)(47,71)(48,50)(49,69)(51,67)(53,65)(55,63)(57,61)(58,80)(60,78)(62,76)(64,74)(66,72)(68,70)(81,126)(82,145)(83,124)(84,143)(85,122)(86,141)(87,160)(88,139)(89,158)(90,137)(91,156)(92,135)(93,154)(94,133)(95,152)(96,131)(97,150)(98,129)(99,148)(100,127)(101,146)(102,125)(103,144)(104,123)(105,142)(106,121)(107,140)(108,159)(109,138)(110,157)(111,136)(112,155)(113,134)(114,153)(115,132)(116,151)(117,130)(118,149)(119,128)(120,147)>;
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), (1,146,59,101,179)(2,147,60,102,180)(3,148,61,103,181)(4,149,62,104,182)(5,150,63,105,183)(6,151,64,106,184)(7,152,65,107,185)(8,153,66,108,186)(9,154,67,109,187)(10,155,68,110,188)(11,156,69,111,189)(12,157,70,112,190)(13,158,71,113,191)(14,159,72,114,192)(15,160,73,115,193)(16,121,74,116,194)(17,122,75,117,195)(18,123,76,118,196)(19,124,77,119,197)(20,125,78,120,198)(21,126,79,81,199)(22,127,80,82,200)(23,128,41,83,161)(24,129,42,84,162)(25,130,43,85,163)(26,131,44,86,164)(27,132,45,87,165)(28,133,46,88,166)(29,134,47,89,167)(30,135,48,90,168)(31,136,49,91,169)(32,137,50,92,170)(33,138,51,93,171)(34,139,52,94,172)(35,140,53,95,173)(36,141,54,96,174)(37,142,55,97,175)(38,143,56,98,176)(39,144,57,99,177)(40,145,58,100,178), (1,179)(2,198)(3,177)(4,196)(5,175)(6,194)(7,173)(8,192)(9,171)(10,190)(11,169)(12,188)(13,167)(14,186)(15,165)(16,184)(17,163)(18,182)(19,161)(20,180)(21,199)(22,178)(23,197)(24,176)(25,195)(26,174)(27,193)(28,172)(29,191)(30,170)(31,189)(32,168)(33,187)(34,166)(35,185)(36,164)(37,183)(38,162)(39,181)(40,200)(41,77)(42,56)(43,75)(44,54)(45,73)(46,52)(47,71)(48,50)(49,69)(51,67)(53,65)(55,63)(57,61)(58,80)(60,78)(62,76)(64,74)(66,72)(68,70)(81,126)(82,145)(83,124)(84,143)(85,122)(86,141)(87,160)(88,139)(89,158)(90,137)(91,156)(92,135)(93,154)(94,133)(95,152)(96,131)(97,150)(98,129)(99,148)(100,127)(101,146)(102,125)(103,144)(104,123)(105,142)(106,121)(107,140)(108,159)(109,138)(110,157)(111,136)(112,155)(113,134)(114,153)(115,132)(116,151)(117,130)(118,149)(119,128)(120,147) );
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)], [(1,146,59,101,179),(2,147,60,102,180),(3,148,61,103,181),(4,149,62,104,182),(5,150,63,105,183),(6,151,64,106,184),(7,152,65,107,185),(8,153,66,108,186),(9,154,67,109,187),(10,155,68,110,188),(11,156,69,111,189),(12,157,70,112,190),(13,158,71,113,191),(14,159,72,114,192),(15,160,73,115,193),(16,121,74,116,194),(17,122,75,117,195),(18,123,76,118,196),(19,124,77,119,197),(20,125,78,120,198),(21,126,79,81,199),(22,127,80,82,200),(23,128,41,83,161),(24,129,42,84,162),(25,130,43,85,163),(26,131,44,86,164),(27,132,45,87,165),(28,133,46,88,166),(29,134,47,89,167),(30,135,48,90,168),(31,136,49,91,169),(32,137,50,92,170),(33,138,51,93,171),(34,139,52,94,172),(35,140,53,95,173),(36,141,54,96,174),(37,142,55,97,175),(38,143,56,98,176),(39,144,57,99,177),(40,145,58,100,178)], [(1,179),(2,198),(3,177),(4,196),(5,175),(6,194),(7,173),(8,192),(9,171),(10,190),(11,169),(12,188),(13,167),(14,186),(15,165),(16,184),(17,163),(18,182),(19,161),(20,180),(21,199),(22,178),(23,197),(24,176),(25,195),(26,174),(27,193),(28,172),(29,191),(30,170),(31,189),(32,168),(33,187),(34,166),(35,185),(36,164),(37,183),(38,162),(39,181),(40,200),(41,77),(42,56),(43,75),(44,54),(45,73),(46,52),(47,71),(48,50),(49,69),(51,67),(53,65),(55,63),(57,61),(58,80),(60,78),(62,76),(64,74),(66,72),(68,70),(81,126),(82,145),(83,124),(84,143),(85,122),(86,141),(87,160),(88,139),(89,158),(90,137),(91,156),(92,135),(93,154),(94,133),(95,152),(96,131),(97,150),(98,129),(99,148),(100,127),(101,146),(102,125),(103,144),(104,123),(105,142),(106,121),(107,140),(108,159),(109,138),(110,157),(111,136),(112,155),(113,134),(114,153),(115,132),(116,151),(117,130),(118,149),(119,128),(120,147)]])
103 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 5A | ··· | 5L | 8A | 8B | 10A | ··· | 10L | 20A | ··· | 20X | 40A | ··· | 40AV |
| order | 1 | 2 | 2 | 4 | 4 | 5 | ··· | 5 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 100 | 2 | 100 | 2 | ··· | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
103 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D5 | SD16 | D10 | D20 | C40⋊C2 |
| kernel | C40⋊2D5 | C5×C40 | C52⋊4Q8 | C20⋊D5 | C5×C10 | C40 | C52 | C20 | C10 | C5 |
| # reps | 1 | 1 | 1 | 1 | 1 | 12 | 2 | 12 | 24 | 48 |
Matrix representation of C40⋊2D5 ►in GL4(𝔽41) generated by
| 25 | 23 | 0 | 0 |
| 18 | 28 | 0 | 0 |
| 0 | 0 | 16 | 18 |
| 0 | 0 | 23 | 13 |
| 0 | 1 | 0 | 0 |
| 40 | 34 | 0 | 0 |
| 0 | 0 | 40 | 34 |
| 0 | 0 | 7 | 7 |
| 34 | 34 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 40 | 34 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [25,18,0,0,23,28,0,0,0,0,16,23,0,0,18,13],[0,40,0,0,1,34,0,0,0,0,40,7,0,0,34,7],[34,1,0,0,34,7,0,0,0,0,40,0,0,0,34,1] >;
C40⋊2D5 in GAP, Magma, Sage, TeX
C_{40}\rtimes_2D_5 % in TeX
G:=Group("C40:2D5"); // GroupNames label
G:=SmallGroup(400,94);
// by ID
G=gap.SmallGroup(400,94);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,73,31,218,50,1924,11525]);
// Polycyclic
G:=Group<a,b,c|a^40=b^5=c^2=1,a*b=b*a,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations