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## G = C32⋊D5order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C32⋊D5
 Chief series C1 — C5 — C10 — C20 — C40 — C80 — D5×C16 — C32⋊D5
 Lower central C5 — C10 — C32⋊D5
 Upper central C1 — C16 — C32

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

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 2A 2B 4A 4B 4C 5A 5B 8A 8B 8C 8D 8E 8F 10A 10B 16A ··· 16H 16I 16J 16K 16L 20A 20B 20C 20D 32A ··· 32H 32I ··· 32P 40A ··· 40H 80A ··· 80P 160A ··· 160AF order 1 2 2 4 4 4 5 5 8 8 8 8 8 8 10 10 16 ··· 16 16 16 16 16 20 20 20 20 32 ··· 32 32 ··· 32 40 ··· 40 80 ··· 80 160 ··· 160 size 1 1 10 1 1 10 2 2 1 1 1 1 10 10 2 2 1 ··· 1 10 10 10 10 2 2 2 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 C16 C16 D5 D10 C4×D5 M6(2) C8×D5 D5×C16 C32⋊D5 kernel C32⋊D5 C5⋊2C32 C160 D5×C16 C5⋊2C16 C8×D5 C5⋊2C8 C4×D5 Dic5 D10 C32 C16 C8 C5 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 8 8 2 2 4 8 8 16 32

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

 540 252 389 101
,
 0 1 640 362
,
 0 1 1 0
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

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