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## G = C40⋊D5order 400 = 24·52

### 4th semidirect product of C40 and D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C40⋊D5
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C4×C5⋊D5 — C40⋊D5
 Lower central C52 — C5×C10 — C40⋊D5
 Upper central C1 — C4 — C8

Generators and relations for C40⋊D5
G = < a,b,c | a40=b5=c2=1, ab=ba, cac=a29, cbc=b-1 >

Subgroups: 424 in 80 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D5, C10, M4(2), Dic5, C20, D10, C52, C52C8, C40, C4×D5, C5⋊D5, C5×C10, C8⋊D5, C526C4, C5×C20, C2×C5⋊D5, C527C8, C5×C40, C4×C5⋊D5, C40⋊D5
Quotients: C1, C2, C4, C22, C2×C4, D5, M4(2), D10, C4×D5, C5⋊D5, C8⋊D5, C2×C5⋊D5, C4×C5⋊D5, C40⋊D5

Smallest permutation representation of C40⋊D5
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)
(1 111 167 150 78)(2 112 168 151 79)(3 113 169 152 80)(4 114 170 153 41)(5 115 171 154 42)(6 116 172 155 43)(7 117 173 156 44)(8 118 174 157 45)(9 119 175 158 46)(10 120 176 159 47)(11 81 177 160 48)(12 82 178 121 49)(13 83 179 122 50)(14 84 180 123 51)(15 85 181 124 52)(16 86 182 125 53)(17 87 183 126 54)(18 88 184 127 55)(19 89 185 128 56)(20 90 186 129 57)(21 91 187 130 58)(22 92 188 131 59)(23 93 189 132 60)(24 94 190 133 61)(25 95 191 134 62)(26 96 192 135 63)(27 97 193 136 64)(28 98 194 137 65)(29 99 195 138 66)(30 100 196 139 67)(31 101 197 140 68)(32 102 198 141 69)(33 103 199 142 70)(34 104 200 143 71)(35 105 161 144 72)(36 106 162 145 73)(37 107 163 146 74)(38 108 164 147 75)(39 109 165 148 76)(40 110 166 149 77)
(1 78)(2 67)(3 56)(4 45)(5 74)(6 63)(7 52)(8 41)(9 70)(10 59)(11 48)(12 77)(13 66)(14 55)(15 44)(16 73)(17 62)(18 51)(19 80)(20 69)(21 58)(22 47)(23 76)(24 65)(25 54)(26 43)(27 72)(28 61)(29 50)(30 79)(31 68)(32 57)(33 46)(34 75)(35 64)(36 53)(37 42)(38 71)(39 60)(40 49)(81 160)(82 149)(83 138)(84 127)(85 156)(86 145)(87 134)(88 123)(89 152)(90 141)(91 130)(92 159)(93 148)(94 137)(95 126)(96 155)(97 144)(98 133)(99 122)(100 151)(101 140)(102 129)(103 158)(104 147)(105 136)(106 125)(107 154)(108 143)(109 132)(110 121)(111 150)(112 139)(113 128)(114 157)(115 146)(116 135)(117 124)(118 153)(119 142)(120 131)(161 193)(162 182)(163 171)(164 200)(165 189)(166 178)(168 196)(169 185)(170 174)(172 192)(173 181)(175 199)(176 188)(179 195)(180 184)(183 191)(186 198)(190 194)```

`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), (1,111,167,150,78)(2,112,168,151,79)(3,113,169,152,80)(4,114,170,153,41)(5,115,171,154,42)(6,116,172,155,43)(7,117,173,156,44)(8,118,174,157,45)(9,119,175,158,46)(10,120,176,159,47)(11,81,177,160,48)(12,82,178,121,49)(13,83,179,122,50)(14,84,180,123,51)(15,85,181,124,52)(16,86,182,125,53)(17,87,183,126,54)(18,88,184,127,55)(19,89,185,128,56)(20,90,186,129,57)(21,91,187,130,58)(22,92,188,131,59)(23,93,189,132,60)(24,94,190,133,61)(25,95,191,134,62)(26,96,192,135,63)(27,97,193,136,64)(28,98,194,137,65)(29,99,195,138,66)(30,100,196,139,67)(31,101,197,140,68)(32,102,198,141,69)(33,103,199,142,70)(34,104,200,143,71)(35,105,161,144,72)(36,106,162,145,73)(37,107,163,146,74)(38,108,164,147,75)(39,109,165,148,76)(40,110,166,149,77), (1,78)(2,67)(3,56)(4,45)(5,74)(6,63)(7,52)(8,41)(9,70)(10,59)(11,48)(12,77)(13,66)(14,55)(15,44)(16,73)(17,62)(18,51)(19,80)(20,69)(21,58)(22,47)(23,76)(24,65)(25,54)(26,43)(27,72)(28,61)(29,50)(30,79)(31,68)(32,57)(33,46)(34,75)(35,64)(36,53)(37,42)(38,71)(39,60)(40,49)(81,160)(82,149)(83,138)(84,127)(85,156)(86,145)(87,134)(88,123)(89,152)(90,141)(91,130)(92,159)(93,148)(94,137)(95,126)(96,155)(97,144)(98,133)(99,122)(100,151)(101,140)(102,129)(103,158)(104,147)(105,136)(106,125)(107,154)(108,143)(109,132)(110,121)(111,150)(112,139)(113,128)(114,157)(115,146)(116,135)(117,124)(118,153)(119,142)(120,131)(161,193)(162,182)(163,171)(164,200)(165,189)(166,178)(168,196)(169,185)(170,174)(172,192)(173,181)(175,199)(176,188)(179,195)(180,184)(183,191)(186,198)(190,194)>;`

`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), (1,111,167,150,78)(2,112,168,151,79)(3,113,169,152,80)(4,114,170,153,41)(5,115,171,154,42)(6,116,172,155,43)(7,117,173,156,44)(8,118,174,157,45)(9,119,175,158,46)(10,120,176,159,47)(11,81,177,160,48)(12,82,178,121,49)(13,83,179,122,50)(14,84,180,123,51)(15,85,181,124,52)(16,86,182,125,53)(17,87,183,126,54)(18,88,184,127,55)(19,89,185,128,56)(20,90,186,129,57)(21,91,187,130,58)(22,92,188,131,59)(23,93,189,132,60)(24,94,190,133,61)(25,95,191,134,62)(26,96,192,135,63)(27,97,193,136,64)(28,98,194,137,65)(29,99,195,138,66)(30,100,196,139,67)(31,101,197,140,68)(32,102,198,141,69)(33,103,199,142,70)(34,104,200,143,71)(35,105,161,144,72)(36,106,162,145,73)(37,107,163,146,74)(38,108,164,147,75)(39,109,165,148,76)(40,110,166,149,77), (1,78)(2,67)(3,56)(4,45)(5,74)(6,63)(7,52)(8,41)(9,70)(10,59)(11,48)(12,77)(13,66)(14,55)(15,44)(16,73)(17,62)(18,51)(19,80)(20,69)(21,58)(22,47)(23,76)(24,65)(25,54)(26,43)(27,72)(28,61)(29,50)(30,79)(31,68)(32,57)(33,46)(34,75)(35,64)(36,53)(37,42)(38,71)(39,60)(40,49)(81,160)(82,149)(83,138)(84,127)(85,156)(86,145)(87,134)(88,123)(89,152)(90,141)(91,130)(92,159)(93,148)(94,137)(95,126)(96,155)(97,144)(98,133)(99,122)(100,151)(101,140)(102,129)(103,158)(104,147)(105,136)(106,125)(107,154)(108,143)(109,132)(110,121)(111,150)(112,139)(113,128)(114,157)(115,146)(116,135)(117,124)(118,153)(119,142)(120,131)(161,193)(162,182)(163,171)(164,200)(165,189)(166,178)(168,196)(169,185)(170,174)(172,192)(173,181)(175,199)(176,188)(179,195)(180,184)(183,191)(186,198)(190,194) );`

`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)], [(1,111,167,150,78),(2,112,168,151,79),(3,113,169,152,80),(4,114,170,153,41),(5,115,171,154,42),(6,116,172,155,43),(7,117,173,156,44),(8,118,174,157,45),(9,119,175,158,46),(10,120,176,159,47),(11,81,177,160,48),(12,82,178,121,49),(13,83,179,122,50),(14,84,180,123,51),(15,85,181,124,52),(16,86,182,125,53),(17,87,183,126,54),(18,88,184,127,55),(19,89,185,128,56),(20,90,186,129,57),(21,91,187,130,58),(22,92,188,131,59),(23,93,189,132,60),(24,94,190,133,61),(25,95,191,134,62),(26,96,192,135,63),(27,97,193,136,64),(28,98,194,137,65),(29,99,195,138,66),(30,100,196,139,67),(31,101,197,140,68),(32,102,198,141,69),(33,103,199,142,70),(34,104,200,143,71),(35,105,161,144,72),(36,106,162,145,73),(37,107,163,146,74),(38,108,164,147,75),(39,109,165,148,76),(40,110,166,149,77)], [(1,78),(2,67),(3,56),(4,45),(5,74),(6,63),(7,52),(8,41),(9,70),(10,59),(11,48),(12,77),(13,66),(14,55),(15,44),(16,73),(17,62),(18,51),(19,80),(20,69),(21,58),(22,47),(23,76),(24,65),(25,54),(26,43),(27,72),(28,61),(29,50),(30,79),(31,68),(32,57),(33,46),(34,75),(35,64),(36,53),(37,42),(38,71),(39,60),(40,49),(81,160),(82,149),(83,138),(84,127),(85,156),(86,145),(87,134),(88,123),(89,152),(90,141),(91,130),(92,159),(93,148),(94,137),(95,126),(96,155),(97,144),(98,133),(99,122),(100,151),(101,140),(102,129),(103,158),(104,147),(105,136),(106,125),(107,154),(108,143),(109,132),(110,121),(111,150),(112,139),(113,128),(114,157),(115,146),(116,135),(117,124),(118,153),(119,142),(120,131),(161,193),(162,182),(163,171),(164,200),(165,189),(166,178),(168,196),(169,185),(170,174),(172,192),(173,181),(175,199),(176,188),(179,195),(180,184),(183,191),(186,198),(190,194)]])`

106 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A ··· 5L 8A 8B 8C 8D 10A ··· 10L 20A ··· 20X 40A ··· 40AV order 1 2 2 4 4 4 5 ··· 5 8 8 8 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 50 1 1 50 2 ··· 2 2 2 50 50 2 ··· 2 2 ··· 2 2 ··· 2

106 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 D5 M4(2) D10 C4×D5 C8⋊D5 kernel C40⋊D5 C52⋊7C8 C5×C40 C4×C5⋊D5 C52⋊6C4 C2×C5⋊D5 C40 C52 C20 C10 C5 # reps 1 1 1 1 2 2 12 2 12 24 48

Matrix representation of C40⋊D5 in GL4(𝔽41) generated by

 13 23 0 0 18 28 0 0 0 0 7 6 0 0 35 2
,
 0 1 0 0 40 6 0 0 0 0 0 1 0 0 40 6
,
 0 1 0 0 1 0 0 0 0 0 40 0 0 0 35 1
`G:=sub<GL(4,GF(41))| [13,18,0,0,23,28,0,0,0,0,7,35,0,0,6,2],[0,40,0,0,1,6,0,0,0,0,0,40,0,0,1,6],[0,1,0,0,1,0,0,0,0,0,40,35,0,0,0,1] >;`

C40⋊D5 in GAP, Magma, Sage, TeX

`C_{40}\rtimes D_5`
`% in TeX`

`G:=Group("C40:D5");`
`// GroupNames label`

`G:=SmallGroup(400,93);`
`// by ID`

`G=gap.SmallGroup(400,93);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,31,50,1924,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^5=c^2=1,a*b=b*a,c*a*c=a^29,c*b*c=b^-1>;`
`// generators/relations`

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