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G = C32⋊D5order 320 = 26·5

3rd semidirect product of C32 and D5 acting via D5/C5=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C323D5, C1606C2, C53M6(2), D10.1C16, C16.20D10, C80.25C22, Dic5.1C16, C52C324C2, C52C8.2C8, (C4×D5).2C8, (C8×D5).2C4, C8.37(C4×D5), C4.17(C8×D5), C2.3(D5×C16), C52C16.2C4, C20.56(C2×C8), C40.95(C2×C4), (D5×C16).4C2, C10.12(C2×C16), SmallGroup(320,5)

Series: Derived Chief Lower central Upper central

C1C10 — C32⋊D5
C1C5C10C20C40C80D5×C16 — C32⋊D5
C5C10 — C32⋊D5
C1C16C32

Generators and relations for C32⋊D5
 G = < a,b,c | a32=b5=c2=1, ab=ba, cac=a17, cbc=b-1 >

10C2
5C22
5C4
2D5
5C2×C4
5C8
5C2×C8
5C16
5C2×C16
5C32
5M6(2)

Smallest permutation representation of C32⋊D5
On 160 points
Generators in S160
(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 82 125 55 159)(2 83 126 56 160)(3 84 127 57 129)(4 85 128 58 130)(5 86 97 59 131)(6 87 98 60 132)(7 88 99 61 133)(8 89 100 62 134)(9 90 101 63 135)(10 91 102 64 136)(11 92 103 33 137)(12 93 104 34 138)(13 94 105 35 139)(14 95 106 36 140)(15 96 107 37 141)(16 65 108 38 142)(17 66 109 39 143)(18 67 110 40 144)(19 68 111 41 145)(20 69 112 42 146)(21 70 113 43 147)(22 71 114 44 148)(23 72 115 45 149)(24 73 116 46 150)(25 74 117 47 151)(26 75 118 48 152)(27 76 119 49 153)(28 77 120 50 154)(29 78 121 51 155)(30 79 122 52 156)(31 80 123 53 157)(32 81 124 54 158)
(1 159)(2 144)(3 129)(4 146)(5 131)(6 148)(7 133)(8 150)(9 135)(10 152)(11 137)(12 154)(13 139)(14 156)(15 141)(16 158)(17 143)(18 160)(19 145)(20 130)(21 147)(22 132)(23 149)(24 134)(25 151)(26 136)(27 153)(28 138)(29 155)(30 140)(31 157)(32 142)(33 92)(34 77)(35 94)(36 79)(37 96)(38 81)(39 66)(40 83)(41 68)(42 85)(43 70)(44 87)(45 72)(46 89)(47 74)(48 91)(49 76)(50 93)(51 78)(52 95)(53 80)(54 65)(55 82)(56 67)(57 84)(58 69)(59 86)(60 71)(61 88)(62 73)(63 90)(64 75)(98 114)(100 116)(102 118)(104 120)(106 122)(108 124)(110 126)(112 128)

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

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

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

104 conjugacy classes

class 1 2A2B4A4B4C5A5B8A8B8C8D8E8F10A10B16A···16H16I16J16K16L20A20B20C20D32A···32H32I···32P40A···40H80A···80P160A···160AF
order12244455888888101016···16161616162020202032···3232···3240···4080···80160···160
size111011102211111010221···11010101022222···210···102···22···22···2

104 irreducible representations

dim11111111112222222
type++++++
imageC1C2C2C2C4C4C8C8C16C16D5D10C4×D5M6(2)C8×D5D5×C16C32⋊D5
kernelC32⋊D5C52C32C160D5×C16C52C16C8×D5C52C8C4×D5Dic5D10C32C16C8C5C4C2C1
# reps1111224488224881632

Matrix representation of C32⋊D5 in GL2(𝔽641) generated by

540252
389101
,
01
640362
,
01
10
G:=sub<GL(2,GF(641))| [540,389,252,101],[0,640,1,362],[0,1,1,0] >;

C32⋊D5 in GAP, Magma, Sage, TeX

C_{32}\rtimes D_5
% in TeX

G:=Group("C32:D5");
// GroupNames label

G:=SmallGroup(320,5);
// by ID

G=gap.SmallGroup(320,5);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,36,58,80,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^32=b^5=c^2=1,a*b=b*a,c*a*c=a^17,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C32⋊D5 in TeX

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