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G = C5⋊C32order 160 = 25·5

The semidirect product of C5 and C32 acting via C32/C8=C4

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C5⋊C32, C10.C16, C8.4F5, C20.2C8, C40.3C4, C2.(C5⋊C16), C4.2(C5⋊C8), C52C16.2C2, SmallGroup(160,3)

Series: Derived Chief Lower central Upper central

C1C5 — C5⋊C32
C1C5C10C20C40C52C16 — C5⋊C32
C5 — C5⋊C32
C1C8

Generators and relations for C5⋊C32
 G = < a,b | a5=b32=1, bab-1=a3 >

5C16
5C32

Smallest permutation representation of C5⋊C32
Regular action on 160 points
Generators in S160
(1 156 60 99 67)(2 100 157 68 61)(3 69 101 62 158)(4 63 70 159 102)(5 160 64 103 71)(6 104 129 72 33)(7 73 105 34 130)(8 35 74 131 106)(9 132 36 107 75)(10 108 133 76 37)(11 77 109 38 134)(12 39 78 135 110)(13 136 40 111 79)(14 112 137 80 41)(15 81 113 42 138)(16 43 82 139 114)(17 140 44 115 83)(18 116 141 84 45)(19 85 117 46 142)(20 47 86 143 118)(21 144 48 119 87)(22 120 145 88 49)(23 89 121 50 146)(24 51 90 147 122)(25 148 52 123 91)(26 124 149 92 53)(27 93 125 54 150)(28 55 94 151 126)(29 152 56 127 95)(30 128 153 96 57)(31 65 97 58 154)(32 59 66 155 98)
(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)| (1,156,60,99,67)(2,100,157,68,61)(3,69,101,62,158)(4,63,70,159,102)(5,160,64,103,71)(6,104,129,72,33)(7,73,105,34,130)(8,35,74,131,106)(9,132,36,107,75)(10,108,133,76,37)(11,77,109,38,134)(12,39,78,135,110)(13,136,40,111,79)(14,112,137,80,41)(15,81,113,42,138)(16,43,82,139,114)(17,140,44,115,83)(18,116,141,84,45)(19,85,117,46,142)(20,47,86,143,118)(21,144,48,119,87)(22,120,145,88,49)(23,89,121,50,146)(24,51,90,147,122)(25,148,52,123,91)(26,124,149,92,53)(27,93,125,54,150)(28,55,94,151,126)(29,152,56,127,95)(30,128,153,96,57)(31,65,97,58,154)(32,59,66,155,98), (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( (1,156,60,99,67)(2,100,157,68,61)(3,69,101,62,158)(4,63,70,159,102)(5,160,64,103,71)(6,104,129,72,33)(7,73,105,34,130)(8,35,74,131,106)(9,132,36,107,75)(10,108,133,76,37)(11,77,109,38,134)(12,39,78,135,110)(13,136,40,111,79)(14,112,137,80,41)(15,81,113,42,138)(16,43,82,139,114)(17,140,44,115,83)(18,116,141,84,45)(19,85,117,46,142)(20,47,86,143,118)(21,144,48,119,87)(22,120,145,88,49)(23,89,121,50,146)(24,51,90,147,122)(25,148,52,123,91)(26,124,149,92,53)(27,93,125,54,150)(28,55,94,151,126)(29,152,56,127,95)(30,128,153,96,57)(31,65,97,58,154)(32,59,66,155,98), (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([(1,156,60,99,67),(2,100,157,68,61),(3,69,101,62,158),(4,63,70,159,102),(5,160,64,103,71),(6,104,129,72,33),(7,73,105,34,130),(8,35,74,131,106),(9,132,36,107,75),(10,108,133,76,37),(11,77,109,38,134),(12,39,78,135,110),(13,136,40,111,79),(14,112,137,80,41),(15,81,113,42,138),(16,43,82,139,114),(17,140,44,115,83),(18,116,141,84,45),(19,85,117,46,142),(20,47,86,143,118),(21,144,48,119,87),(22,120,145,88,49),(23,89,121,50,146),(24,51,90,147,122),(25,148,52,123,91),(26,124,149,92,53),(27,93,125,54,150),(28,55,94,151,126),(29,152,56,127,95),(30,128,153,96,57),(31,65,97,58,154),(32,59,66,155,98)], [(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)])

C5⋊C32 is a maximal subgroup of   D5⋊C32  C80.C4  C5⋊M6(2)  C15⋊C32
C5⋊C32 is a maximal quotient of   C5⋊C64  C15⋊C32

40 conjugacy classes

class 1  2 4A4B 5 8A8B8C8D 10 16A···16H20A20B32A···32P40A40B40C40D
order1244588881016···16202032···3240404040
size11114111145···5445···54444

40 irreducible representations

dim1111114444
type+++-
imageC1C2C4C8C16C32F5C5⋊C8C5⋊C16C5⋊C32
kernelC5⋊C32C52C16C40C20C10C5C8C4C2C1
# reps11248161124

Matrix representation of C5⋊C32 in GL4(𝔽641) generated by

0100
0010
0001
640640640640
,
4683351954
325360306133
587414281606
508192227173
G:=sub<GL(4,GF(641))| [0,0,0,640,1,0,0,640,0,1,0,640,0,0,1,640],[468,325,587,508,335,360,414,192,19,306,281,227,54,133,606,173] >;

C5⋊C32 in GAP, Magma, Sage, TeX

C_5\rtimes C_{32}
% in TeX

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

G:=SmallGroup(160,3);
// by ID

G=gap.SmallGroup(160,3);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,12,31,50,69,2309,2315]);
// Polycyclic

G:=Group<a,b|a^5=b^32=1,b*a*b^-1=a^3>;
// generators/relations

Export

Subgroup lattice of C5⋊C32 in TeX

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