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## G = C4⋊D40order 320 = 26·5

### The semidirect product of C4 and D40 acting via D40/D20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C4⋊D40
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4×D20 — C4⋊D40
 Lower central C5 — C10 — C2×C20 — C4⋊D40
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for C4⋊D40
G = < a,b,c | a4=b40=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 854 in 140 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, Dic5, C20, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4×D4, C41D4, C2×D8, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, C4⋊D8, D40, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C2×D20, C2×D20, D205C4, C5×C4⋊C8, C4×D20, C204D4, C2×D40, C4⋊D40
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C4⋊D4, C2×D8, C8⋊C22, D20, C22×D5, C4⋊D8, D40, C2×D20, D4×D5, Q82D5, C4⋊D20, C2×D40, C8⋊D10, C4⋊D40

Smallest permutation representation of C4⋊D40
On 160 points
Generators in S160
```(1 67 132 96)(2 97 133 68)(3 69 134 98)(4 99 135 70)(5 71 136 100)(6 101 137 72)(7 73 138 102)(8 103 139 74)(9 75 140 104)(10 105 141 76)(11 77 142 106)(12 107 143 78)(13 79 144 108)(14 109 145 80)(15 41 146 110)(16 111 147 42)(17 43 148 112)(18 113 149 44)(19 45 150 114)(20 115 151 46)(21 47 152 116)(22 117 153 48)(23 49 154 118)(24 119 155 50)(25 51 156 120)(26 81 157 52)(27 53 158 82)(28 83 159 54)(29 55 160 84)(30 85 121 56)(31 57 122 86)(32 87 123 58)(33 59 124 88)(34 89 125 60)(35 61 126 90)(36 91 127 62)(37 63 128 92)(38 93 129 64)(39 65 130 94)(40 95 131 66)
(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)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 120)(58 119)(59 118)(60 117)(61 116)(62 115)(63 114)(64 113)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(73 104)(74 103)(75 102)(76 101)(77 100)(78 99)(79 98)(80 97)(121 157)(122 156)(123 155)(124 154)(125 153)(126 152)(127 151)(128 150)(129 149)(130 148)(131 147)(132 146)(133 145)(134 144)(135 143)(136 142)(137 141)(138 140)(158 160)```

`G:=sub<Sym(160)| (1,67,132,96)(2,97,133,68)(3,69,134,98)(4,99,135,70)(5,71,136,100)(6,101,137,72)(7,73,138,102)(8,103,139,74)(9,75,140,104)(10,105,141,76)(11,77,142,106)(12,107,143,78)(13,79,144,108)(14,109,145,80)(15,41,146,110)(16,111,147,42)(17,43,148,112)(18,113,149,44)(19,45,150,114)(20,115,151,46)(21,47,152,116)(22,117,153,48)(23,49,154,118)(24,119,155,50)(25,51,156,120)(26,81,157,52)(27,53,158,82)(28,83,159,54)(29,55,160,84)(30,85,121,56)(31,57,122,86)(32,87,123,58)(33,59,124,88)(34,89,125,60)(35,61,126,90)(36,91,127,62)(37,63,128,92)(38,93,129,64)(39,65,130,94)(40,95,131,66), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140)(158,160)>;`

`G:=Group( (1,67,132,96)(2,97,133,68)(3,69,134,98)(4,99,135,70)(5,71,136,100)(6,101,137,72)(7,73,138,102)(8,103,139,74)(9,75,140,104)(10,105,141,76)(11,77,142,106)(12,107,143,78)(13,79,144,108)(14,109,145,80)(15,41,146,110)(16,111,147,42)(17,43,148,112)(18,113,149,44)(19,45,150,114)(20,115,151,46)(21,47,152,116)(22,117,153,48)(23,49,154,118)(24,119,155,50)(25,51,156,120)(26,81,157,52)(27,53,158,82)(28,83,159,54)(29,55,160,84)(30,85,121,56)(31,57,122,86)(32,87,123,58)(33,59,124,88)(34,89,125,60)(35,61,126,90)(36,91,127,62)(37,63,128,92)(38,93,129,64)(39,65,130,94)(40,95,131,66), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,140)(158,160) );`

`G=PermutationGroup([[(1,67,132,96),(2,97,133,68),(3,69,134,98),(4,99,135,70),(5,71,136,100),(6,101,137,72),(7,73,138,102),(8,103,139,74),(9,75,140,104),(10,105,141,76),(11,77,142,106),(12,107,143,78),(13,79,144,108),(14,109,145,80),(15,41,146,110),(16,111,147,42),(17,43,148,112),(18,113,149,44),(19,45,150,114),(20,115,151,46),(21,47,152,116),(22,117,153,48),(23,49,154,118),(24,119,155,50),(25,51,156,120),(26,81,157,52),(27,53,158,82),(28,83,159,54),(29,55,160,84),(30,85,121,56),(31,57,122,86),(32,87,123,58),(33,59,124,88),(34,89,125,60),(35,61,126,90),(36,91,127,62),(37,63,128,92),(38,93,129,64),(39,65,130,94),(40,95,131,66)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(41,96),(42,95),(43,94),(44,93),(45,92),(46,91),(47,90),(48,89),(49,88),(50,87),(51,86),(52,85),(53,84),(54,83),(55,82),(56,81),(57,120),(58,119),(59,118),(60,117),(61,116),(62,115),(63,114),(64,113),(65,112),(66,111),(67,110),(68,109),(69,108),(70,107),(71,106),(72,105),(73,104),(74,103),(75,102),(76,101),(77,100),(78,99),(79,98),(80,97),(121,157),(122,156),(123,155),(124,154),(125,153),(126,152),(127,151),(128,150),(129,149),(130,148),(131,147),(132,146),(133,145),(134,144),(135,143),(136,142),(137,141),(138,140),(158,160)]])`

59 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 20 20 40 40 2 2 2 2 4 20 20 2 2 4 4 4 4 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D8 C4○D4 D10 D10 D20 D40 C8⋊C22 D4×D5 Q8⋊2D5 C8⋊D10 kernel C4⋊D40 D20⋊5C4 C5×C4⋊C8 C4×D20 C20⋊4D4 C2×D40 D20 C2×C20 C4⋊C8 C20 C20 C42 C2×C8 C2×C4 C4 C10 C4 C4 C2 # reps 1 2 1 1 1 2 2 2 2 4 2 2 4 8 16 1 2 2 4

Matrix representation of C4⋊D40 in GL4(𝔽41) generated by

 0 32 0 0 32 0 0 0 0 0 40 0 0 0 0 40
,
 0 1 0 0 40 0 0 0 0 0 8 35 0 0 11 38
,
 40 0 0 0 0 1 0 0 0 0 35 7 0 0 36 6
`G:=sub<GL(4,GF(41))| [0,32,0,0,32,0,0,0,0,0,40,0,0,0,0,40],[0,40,0,0,1,0,0,0,0,0,8,11,0,0,35,38],[40,0,0,0,0,1,0,0,0,0,35,36,0,0,7,6] >;`

C4⋊D40 in GAP, Magma, Sage, TeX

`C_4\rtimes D_{40}`
`% in TeX`

`G:=Group("C4:D40");`
`// GroupNames label`

`G:=SmallGroup(320,470);`
`// by ID`

`G=gap.SmallGroup(320,470);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,226,1123,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^40=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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