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G = C5⋊Q32  order 160 = 25·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5⋊2Q32, Q16.D5, C20.6D4, C8.7D10, C10.11D8, C40.5C22, Dic20.2C2, C5⋊2C16.1C2, 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 >

C1
C2
C5
C4
4C4
20C4
C10
C8
2Q8
10Q8
C20
4Dic5
4C20
Q16
5C16
5Q16
C40
2Dic10
2C5×Q8
5Q32
Dic20
C5⋊2C16
C5×Q16
C5⋊Q32

Character table of C5⋊Q32

 class 124A4B4C5A5B8A8B10A10B16A16B16C16D20A20B20C20D20E20F40A40B40C40D
 size 112840222222101010104488884444
ρ11111111111111111111111111    trivial
ρ2111-1-1111111111111-1-1-1-11111    linear of order 2
ρ3111-11111111-1-1-1-111-1-1-1-11111    linear of order 2
ρ41111-1111111-1-1-1-11111111111    linear of order 2
ρ52220022-2-2220000220000-2-2-2-2    orthogonal lifted from D4
ρ622220-1+√5/2-1-√5/222-1-√5/2-1+√5/20000-1+√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
ρ722220-1-√5/2-1+√5/222-1+√5/2-1-√5/20000-1-√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
ρ8222-20-1-√5/2-1+√5/222-1+√5/2-1-√5/20000-1-√5/2-1+√5/21-√5/21-√5/21+√5/21+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D10
ρ9222-20-1+√5/2-1-√5/222-1-√5/2-1+√5/20000-1+√5/2-1-√5/21+√5/21+√5/21-√5/21-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D10
ρ1022-200220022-√2√2-√2√2-2-200000000    orthogonal lifted from D8
ρ1122-200220022√2-√2√2-√2-2-200000000    orthogonal lifted from D8
ρ122-200022-√2√2-2-2ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169000000-√2√2-√2√2    symplectic lifted from Q32, Schur index 2
ρ132-200022√2-√2-2-2ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163000000√2-√2√2-√2    symplectic lifted from Q32, Schur index 2
ρ142-200022-√2√2-2-2-ζ165+ζ163ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169000000-√2√2-√2√2    symplectic lifted from Q32, Schur index 2
ρ152-200022√2-√2-2-2-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169ζ165-ζ163000000√2-√2√2-√2    symplectic lifted from Q32, Schur index 2
ρ1622200-1-√5/2-1+√5/2-2-2-1+√5/2-1-√5/20000-1-√5/2-1+√5/2ζ54-ζ5-ζ54+ζ5ζ53-ζ52-ζ53+ζ521-√5/21+√5/21+√5/21-√5/2    complex lifted from C5⋊D4
ρ1722200-1+√5/2-1-√5/2-2-2-1-√5/2-1+√5/20000-1+√5/2-1-√5/2-ζ53+ζ52ζ53-ζ52ζ54-ζ5-ζ54+ζ51+√5/21-√5/21-√5/21+√5/2    complex lifted from C5⋊D4
ρ1822200-1-√5/2-1+√5/2-2-2-1+√5/2-1-√5/20000-1-√5/2-1+√5/2-ζ54+ζ5ζ54-ζ5-ζ53+ζ52ζ53-ζ521-√5/21+√5/21+√5/21-√5/2    complex lifted from C5⋊D4
ρ1922200-1+√5/2-1-√5/2-2-2-1-√5/2-1+√5/20000-1+√5/2-1-√5/2ζ53-ζ52-ζ53+ζ52-ζ54+ζ5ζ54-ζ51+√5/21-√5/21-√5/21+√5/2    complex lifted from C5⋊D4
ρ2044-400-1+√5-1-√500-1-√5-1+√500001-√51+√500000000    orthogonal lifted from D4⋊D5, Schur index 2
ρ2144-400-1-√5-1+√500-1+√5-1-√500001+√51-√500000000    orthogonal lifted from D4⋊D5, Schur index 2
ρ224-4000-1+√5-1-√52√2-2√21+√51-√50000000000-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52    symplectic faithful, Schur index 2
ρ234-4000-1+√5-1-√5-2√22√21+√51-√50000000000ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52    symplectic faithful, Schur index 2
ρ244-4000-1-√5-1+√5-2√22√21-√51+√50000000000ζ83ζ54+ζ83ζ5-ζ8ζ54-ζ8ζ5-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5    symplectic faithful, Schur index 2
ρ254-4000-1-√5-1+√52√2-2√21-√51+√50000000000ζ87ζ54+ζ87ζ5-ζ85ζ54-ζ85ζ5ζ83ζ53+ζ83ζ52-ζ8ζ53-ζ8ζ52-ζ83ζ53-ζ83ζ52+ζ8ζ53+ζ8ζ52ζ83ζ54+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  SD32⋊3D5  D5×Q32  Q32⋊D5  Q16.D10  C40.30C23  C40.31C23  C15⋊Q32  C5⋊Dic24  C15⋊7Q32
C5⋊Q32 is a maximal quotient of
 C40.2Q8  C10.Q32  C40.15D4  C15⋊Q32  C5⋊Dic24  C15⋊7Q32

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

1000
0100
002401
0050190
,
585400
21411200
004996
00221192
,
13618300
10710500
00165103
008976
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
 

Export

Subgroup lattice of C5⋊Q32 in TeX
Character table of C5⋊Q32 in TeX

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