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## G = C5⋊Q32order 160 = 25·5

### The semidirect product of C5 and Q32 acting via Q32/Q16=C2

Aliases: C52Q32, Q16.D5, C20.6D4, C8.7D10, C10.11D8, C40.5C22, Dic20.2C2, C52C16.1C2, C2.7(D4⋊D5), C4.4(C5⋊D4), (C5×Q16).1C2, SmallGroup(160,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C5⋊Q32
 Chief series C1 — C5 — C10 — C20 — C40 — Dic20 — C5⋊Q32
 Lower central C5 — C10 — C20 — C40 — C5⋊Q32
 Upper central C1 — C2 — C4 — C8 — Q16

Generators and relations for C5⋊Q32
G = < a,b,c | a5=b16=1, c2=b8, bab-1=a-1, ac=ca, cbc-1=b-1 >

Character table of C5⋊Q32

 class 1 2 4A 4B 4C 5A 5B 8A 8B 10A 10B 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 40A 40B 40C 40D size 1 1 2 8 40 2 2 2 2 2 2 10 10 10 10 4 4 8 8 8 8 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 2 0 0 2 2 -2 -2 2 2 0 0 0 0 2 2 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ6 2 2 2 2 0 -1+√5/2 -1-√5/2 2 2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ7 2 2 2 2 0 -1-√5/2 -1+√5/2 2 2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ8 2 2 2 -2 0 -1-√5/2 -1+√5/2 2 2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ9 2 2 2 -2 0 -1+√5/2 -1-√5/2 2 2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ10 2 2 -2 0 0 2 2 0 0 2 2 -√2 √2 -√2 √2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ11 2 2 -2 0 0 2 2 0 0 2 2 √2 -√2 √2 -√2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ12 2 -2 0 0 0 2 2 -√2 √2 -2 -2 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 0 0 0 0 0 0 -√2 √2 -√2 √2 symplectic lifted from Q32, Schur index 2 ρ13 2 -2 0 0 0 2 2 √2 -√2 -2 -2 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 -ζ165+ζ163 0 0 0 0 0 0 √2 -√2 √2 -√2 symplectic lifted from Q32, Schur index 2 ρ14 2 -2 0 0 0 2 2 -√2 √2 -2 -2 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 -ζ1615+ζ169 0 0 0 0 0 0 -√2 √2 -√2 √2 symplectic lifted from Q32, Schur index 2 ρ15 2 -2 0 0 0 2 2 √2 -√2 -2 -2 -ζ1615+ζ169 -ζ165+ζ163 ζ1615-ζ169 ζ165-ζ163 0 0 0 0 0 0 √2 -√2 √2 -√2 symplectic lifted from Q32, Schur index 2 ρ16 2 2 2 0 0 -1-√5/2 -1+√5/2 -2 -2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 -ζ53+ζ52 1-√5/2 1+√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ17 2 2 2 0 0 -1+√5/2 -1-√5/2 -2 -2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 -ζ54+ζ5 1+√5/2 1-√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ18 2 2 2 0 0 -1-√5/2 -1+√5/2 -2 -2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 ζ53-ζ52 1-√5/2 1+√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ19 2 2 2 0 0 -1+√5/2 -1-√5/2 -2 -2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 ζ54-ζ5 1+√5/2 1-√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ20 4 4 -4 0 0 -1+√5 -1-√5 0 0 -1-√5 -1+√5 0 0 0 0 1-√5 1+√5 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊D5, Schur index 2 ρ21 4 4 -4 0 0 -1-√5 -1+√5 0 0 -1+√5 -1-√5 0 0 0 0 1+√5 1-√5 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊D5, Schur index 2 ρ22 4 -4 0 0 0 -1+√5 -1-√5 2√2 -2√2 1+√5 1-√5 0 0 0 0 0 0 0 0 0 0 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 symplectic faithful, Schur index 2 ρ23 4 -4 0 0 0 -1+√5 -1-√5 -2√2 2√2 1+√5 1-√5 0 0 0 0 0 0 0 0 0 0 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 symplectic faithful, Schur index 2 ρ24 4 -4 0 0 0 -1-√5 -1+√5 -2√2 2√2 1-√5 1+√5 0 0 0 0 0 0 0 0 0 0 ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 symplectic faithful, Schur index 2 ρ25 4 -4 0 0 0 -1-√5 -1+√5 2√2 -2√2 1-√5 1+√5 0 0 0 0 0 0 0 0 0 0 ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5 ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52 -ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52 ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5 symplectic faithful, Schur index 2

Smallest permutation representation of C5⋊Q32
Regular action on 160 points
Generators in S160
```(1 63 116 159 112)(2 97 160 117 64)(3 49 118 145 98)(4 99 146 119 50)(5 51 120 147 100)(6 101 148 121 52)(7 53 122 149 102)(8 103 150 123 54)(9 55 124 151 104)(10 105 152 125 56)(11 57 126 153 106)(12 107 154 127 58)(13 59 128 155 108)(14 109 156 113 60)(15 61 114 157 110)(16 111 158 115 62)(17 69 85 33 135)(18 136 34 86 70)(19 71 87 35 137)(20 138 36 88 72)(21 73 89 37 139)(22 140 38 90 74)(23 75 91 39 141)(24 142 40 92 76)(25 77 93 41 143)(26 144 42 94 78)(27 79 95 43 129)(28 130 44 96 80)(29 65 81 45 131)(30 132 46 82 66)(31 67 83 47 133)(32 134 48 84 68)
(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 75 9 67)(2 74 10 66)(3 73 11 65)(4 72 12 80)(5 71 13 79)(6 70 14 78)(7 69 15 77)(8 68 16 76)(17 110 25 102)(18 109 26 101)(19 108 27 100)(20 107 28 99)(21 106 29 98)(22 105 30 97)(23 104 31 112)(24 103 32 111)(33 114 41 122)(34 113 42 121)(35 128 43 120)(36 127 44 119)(37 126 45 118)(38 125 46 117)(39 124 47 116)(40 123 48 115)(49 89 57 81)(50 88 58 96)(51 87 59 95)(52 86 60 94)(53 85 61 93)(54 84 62 92)(55 83 63 91)(56 82 64 90)(129 147 137 155)(130 146 138 154)(131 145 139 153)(132 160 140 152)(133 159 141 151)(134 158 142 150)(135 157 143 149)(136 156 144 148)```

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

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

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

C5⋊Q32 is a maximal subgroup of
SD32⋊D5  SD323D5  D5×Q32  Q32⋊D5  Q16.D10  C40.30C23  C40.31C23  C15⋊Q32  C5⋊Dic24  C157Q32
C5⋊Q32 is a maximal quotient of
C40.2Q8  C10.Q32  C40.15D4  C15⋊Q32  C5⋊Dic24  C157Q32

Matrix representation of C5⋊Q32 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 240 1 0 0 50 190
,
 58 54 0 0 214 112 0 0 0 0 49 96 0 0 221 192
,
 136 183 0 0 107 105 0 0 0 0 165 103 0 0 89 76
`G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,50,0,0,1,190],[58,214,0,0,54,112,0,0,0,0,49,221,0,0,96,192],[136,107,0,0,183,105,0,0,0,0,165,89,0,0,103,76] >;`

C5⋊Q32 in GAP, Magma, Sage, TeX

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

`G:=Group("C5:Q32");`
`// GroupNames label`

`G:=SmallGroup(160,36);`
`// by ID`

`G=gap.SmallGroup(160,36);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,73,103,218,116,122,579,297,69,4613]);`
`// Polycyclic`

`G:=Group<a,b,c|a^5=b^16=1,c^2=b^8,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;`
`// generators/relations`

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