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## G = C5⋊2C32order 160 = 25·5

### The semidirect product of C5 and C32 acting via C32/C16=C2

Aliases: C52C32, C40.7C4, C80.3C2, C20.5C8, C16.2D5, C10.2C16, C8.3Dic5, C2.(C52C16), C4.2(C52C8), SmallGroup(160,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C5⋊2C32
 Chief series C1 — C5 — C10 — C20 — C40 — C80 — C5⋊2C32
 Lower central C5 — C5⋊2C32
 Upper central C1 — C16

Generators and relations for C52C32
G = < a,b | a5=b32=1, bab-1=a-1 >

Smallest permutation representation of C52C32
Regular action on 160 points
Generators in S160
(1 45 140 79 123)(2 124 80 141 46)(3 47 142 81 125)(4 126 82 143 48)(5 49 144 83 127)(6 128 84 145 50)(7 51 146 85 97)(8 98 86 147 52)(9 53 148 87 99)(10 100 88 149 54)(11 55 150 89 101)(12 102 90 151 56)(13 57 152 91 103)(14 104 92 153 58)(15 59 154 93 105)(16 106 94 155 60)(17 61 156 95 107)(18 108 96 157 62)(19 63 158 65 109)(20 110 66 159 64)(21 33 160 67 111)(22 112 68 129 34)(23 35 130 69 113)(24 114 70 131 36)(25 37 132 71 115)(26 116 72 133 38)(27 39 134 73 117)(28 118 74 135 40)(29 41 136 75 119)(30 120 76 137 42)(31 43 138 77 121)(32 122 78 139 44)
(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)

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

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

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

C52C32 is a maximal subgroup of
C5⋊C64  D5×C32  C32⋊D5  C80.9C4  C5⋊D32  D16.D5  C5⋊SD64  C5⋊Q64  C153C32
C52C32 is a maximal quotient of
C52C64  C153C32

64 conjugacy classes

 class 1 2 4A 4B 5A 5B 8A 8B 8C 8D 10A 10B 16A ··· 16H 20A 20B 20C 20D 32A ··· 32P 40A ··· 40H 80A ··· 80P order 1 2 4 4 5 5 8 8 8 8 10 10 16 ··· 16 20 20 20 20 32 ··· 32 40 ··· 40 80 ··· 80 size 1 1 1 1 2 2 1 1 1 1 2 2 1 ··· 1 2 2 2 2 5 ··· 5 2 ··· 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + - image C1 C2 C4 C8 C16 C32 D5 Dic5 C5⋊2C8 C5⋊2C16 C5⋊2C32 kernel C5⋊2C32 C80 C40 C20 C10 C5 C16 C8 C4 C2 C1 # reps 1 1 2 4 8 16 2 2 4 8 16

Matrix representation of C52C32 in GL2(𝔽641) generated by

 640 1 361 279
,
 500 94 301 141
G:=sub<GL(2,GF(641))| [640,361,1,279],[500,301,94,141] >;

C52C32 in GAP, Magma, Sage, TeX

C_5\rtimes_2C_{32}
% in TeX

G:=Group("C5:2C32");
// GroupNames label

G:=SmallGroup(160,1);
// by ID

G=gap.SmallGroup(160,1);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,12,31,50,69,4613]);
// Polycyclic

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

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