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G = C5×Q32order 160 = 25·5

Direct product of C5 and Q32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C5×Q32, C16.C10, C80.2C2, Q16.C10, C10.17D8, C20.38D4, C40.26C22, C4.3(C5×D4), C2.5(C5×D8), C8.4(C2×C10), (C5×Q16).2C2, SmallGroup(160,63)

Series: Derived Chief Lower central Upper central

C1C8 — C5×Q32
C1C2C4C8C40C5×Q16 — C5×Q32
C1C2C4C8 — C5×Q32
C1C10C20C40 — C5×Q32

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

4C4
4C4
2Q8
2Q8
4C20
4C20
2C5×Q8
2C5×Q8

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

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

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

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

C5×Q32 is a maximal subgroup of   C5⋊SD64  C5⋊Q64  Q32⋊D5  D805C2

55 conjugacy classes

class 1  2 4A4B4C5A5B5C5D8A8B10A10B10C10D16A16B16C16D20A20B20C20D20E···20L40A···40H80A···80P
order1244455558810101010161616162020202020···2040···4080···80
size112881111221111222222228···82···22···2

55 irreducible representations

dim111111222222
type+++++-
imageC1C2C2C5C10C10D4D8Q32C5×D4C5×D8C5×Q32
kernelC5×Q32C80C5×Q16Q32C16Q16C20C10C5C4C2C1
# reps1124481244816

Matrix representation of C5×Q32 in GL2(𝔽31) generated by

160
016
,
214
73
,
023
40
G:=sub<GL(2,GF(31))| [16,0,0,16],[2,7,14,3],[0,4,23,0] >;

C5×Q32 in GAP, Magma, Sage, TeX

C_5\times Q_{32}
% in TeX

G:=Group("C5xQ32");
// GroupNames label

G:=SmallGroup(160,63);
// by ID

G=gap.SmallGroup(160,63);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,480,265,487,1443,729,165,3604,1810,88]);
// Polycyclic

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

Export

Subgroup lattice of C5×Q32 in TeX

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