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## G = C5⋊D32order 320 = 26·5

### The semidirect product of C5 and D32 acting via D32/D16=C2

Aliases: C52D32, D161D5, D803C2, C40.9D4, C20.5D8, C10.8D16, C16.4D10, C80.2C22, C52C321C2, (C5×D16)⋊1C2, C4.1(D4⋊D5), C8.9(C5⋊D4), C2.4(C5⋊D16), SmallGroup(320,77)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C80 — C5⋊D32
 Chief series C1 — C5 — C10 — C20 — C40 — C80 — D80 — C5⋊D32
 Lower central C5 — C10 — C20 — C40 — C80 — C5⋊D32
 Upper central C1 — C2 — C4 — C8 — C16 — D16

Generators and relations for C5⋊D32
G = < a,b,c | a5=b32=c2=1, bab-1=cac=a-1, cbc=b-1 >

Smallest permutation representation of C5⋊D32
On 160 points
Generators in S160
```(1 73 129 110 62)(2 63 111 130 74)(3 75 131 112 64)(4 33 113 132 76)(5 77 133 114 34)(6 35 115 134 78)(7 79 135 116 36)(8 37 117 136 80)(9 81 137 118 38)(10 39 119 138 82)(11 83 139 120 40)(12 41 121 140 84)(13 85 141 122 42)(14 43 123 142 86)(15 87 143 124 44)(16 45 125 144 88)(17 89 145 126 46)(18 47 127 146 90)(19 91 147 128 48)(20 49 97 148 92)(21 93 149 98 50)(22 51 99 150 94)(23 95 151 100 52)(24 53 101 152 96)(25 65 153 102 54)(26 55 103 154 66)(27 67 155 104 56)(28 57 105 156 68)(29 69 157 106 58)(30 59 107 158 70)(31 71 159 108 60)(32 61 109 160 72)
(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)
(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 96)(40 95)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(97 142)(98 141)(99 140)(100 139)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)(109 130)(110 129)(111 160)(112 159)(113 158)(114 157)(115 156)(116 155)(117 154)(118 153)(119 152)(120 151)(121 150)(122 149)(123 148)(124 147)(125 146)(126 145)(127 144)(128 143)```

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

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

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

41 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 8A 8B 10A 10B 10C 10D 10E 10F 16A 16B 16C 16D 20A 20B 32A ··· 32H 40A 40B 40C 40D 80A ··· 80H order 1 2 2 2 4 5 5 8 8 10 10 10 10 10 10 16 16 16 16 20 20 32 ··· 32 40 40 40 40 80 ··· 80 size 1 1 16 80 2 2 2 2 2 2 2 16 16 16 16 2 2 2 2 4 4 10 ··· 10 4 4 4 4 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 D4 D5 D8 D10 D16 C5⋊D4 D32 D4⋊D5 C5⋊D16 C5⋊D32 kernel C5⋊D32 C5⋊2C32 D80 C5×D16 C40 D16 C20 C16 C10 C8 C5 C4 C2 C1 # reps 1 1 1 1 1 2 2 2 4 4 8 2 4 8

Matrix representation of C5⋊D32 in GL4(𝔽641) generated by

 0 1 0 0 640 278 0 0 0 0 1 0 0 0 0 1
,
 640 0 0 0 363 1 0 0 0 0 597 499 0 0 71 98
,
 1 0 0 0 278 640 0 0 0 0 640 639 0 0 0 1
`G:=sub<GL(4,GF(641))| [0,640,0,0,1,278,0,0,0,0,1,0,0,0,0,1],[640,363,0,0,0,1,0,0,0,0,597,71,0,0,499,98],[1,278,0,0,0,640,0,0,0,0,640,0,0,0,639,1] >;`

C5⋊D32 in GAP, Magma, Sage, TeX

`C_5\rtimes D_{32}`
`% in TeX`

`G:=Group("C5:D32");`
`// GroupNames label`

`G:=SmallGroup(320,77);`
`// by ID`

`G=gap.SmallGroup(320,77);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,254,135,142,675,346,192,1684,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^5=b^32=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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