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G = C5⋊D32order 320 = 26·5

The semidirect product of C5 and D32 acting via D32/D16=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52D32, D161D5, D803C2, C40.9D4, C20.5D8, C10.8D16, C16.4D10, C80.2C22, C52C321C2, (C5×D16)⋊1C2, C4.1(D4⋊D5), C8.9(C5⋊D4), C2.4(C5⋊D16), SmallGroup(320,77)

Series: Derived Chief Lower central Upper central

C1C80 — C5⋊D32
C1C5C10C20C40C80D80 — C5⋊D32
C5C10C20C40C80 — C5⋊D32
C1C2C4C8C16D16

Generators and relations for C5⋊D32
 G = < a,b,c | a5=b32=c2=1, bab-1=cac=a-1, cbc=b-1 >

16C2
80C2
8C22
40C22
16D5
16C10
4D4
20D4
8D10
8C2×C10
2D8
10D8
4D20
4C5×D4
5C32
5D16
2D40
2C5×D8
5D32

Smallest permutation representation of C5⋊D32
On 160 points
Generators in S160
(1 73 129 110 62)(2 63 111 130 74)(3 75 131 112 64)(4 33 113 132 76)(5 77 133 114 34)(6 35 115 134 78)(7 79 135 116 36)(8 37 117 136 80)(9 81 137 118 38)(10 39 119 138 82)(11 83 139 120 40)(12 41 121 140 84)(13 85 141 122 42)(14 43 123 142 86)(15 87 143 124 44)(16 45 125 144 88)(17 89 145 126 46)(18 47 127 146 90)(19 91 147 128 48)(20 49 97 148 92)(21 93 149 98 50)(22 51 99 150 94)(23 95 151 100 52)(24 53 101 152 96)(25 65 153 102 54)(26 55 103 154 66)(27 67 155 104 56)(28 57 105 156 68)(29 69 157 106 58)(30 59 107 158 70)(31 71 159 108 60)(32 61 109 160 72)
(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)
(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 96)(40 95)(41 94)(42 93)(43 92)(44 91)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(97 142)(98 141)(99 140)(100 139)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)(109 130)(110 129)(111 160)(112 159)(113 158)(114 157)(115 156)(116 155)(117 154)(118 153)(119 152)(120 151)(121 150)(122 149)(123 148)(124 147)(125 146)(126 145)(127 144)(128 143)

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

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

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

41 conjugacy classes

class 1 2A2B2C 4 5A5B8A8B10A10B10C10D10E10F16A16B16C16D20A20B32A···32H40A40B40C40D80A···80H
order12224558810101010101016161616202032···324040404080···80
size11168022222221616161622224410···1044444···4

41 irreducible representations

dim11112222222444
type+++++++++++++
imageC1C2C2C2D4D5D8D10D16C5⋊D4D32D4⋊D5C5⋊D16C5⋊D32
kernelC5⋊D32C52C32D80C5×D16C40D16C20C16C10C8C5C4C2C1
# reps11111222448248

Matrix representation of C5⋊D32 in GL4(𝔽641) generated by

0100
64027800
0010
0001
,
640000
363100
00597499
007198
,
1000
27864000
00640639
0001
G:=sub<GL(4,GF(641))| [0,640,0,0,1,278,0,0,0,0,1,0,0,0,0,1],[640,363,0,0,0,1,0,0,0,0,597,71,0,0,499,98],[1,278,0,0,0,640,0,0,0,0,640,0,0,0,639,1] >;

C5⋊D32 in GAP, Magma, Sage, TeX

C_5\rtimes D_{32}
% in TeX

G:=Group("C5:D32");
// GroupNames label

G:=SmallGroup(320,77);
// by ID

G=gap.SmallGroup(320,77);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,85,254,135,142,675,346,192,1684,851,102,12550]);
// Polycyclic

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

Export

Subgroup lattice of C5⋊D32 in TeX

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