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G = C4×C40order 160 = 25·5

Abelian group of type [4,40]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C40, SmallGroup(160,46)

Series: Derived Chief Lower central Upper central

C1 — C4×C40
C1C2C22C2×C4C2×C20C2×C40 — C4×C40
C1 — C4×C40
C1 — C4×C40

Generators and relations for C4×C40
 G = < a,b | a4=b40=1, ab=ba >


Smallest permutation representation of C4×C40
Regular action on 160 points
Generators in S160
(1 70 100 141)(2 71 101 142)(3 72 102 143)(4 73 103 144)(5 74 104 145)(6 75 105 146)(7 76 106 147)(8 77 107 148)(9 78 108 149)(10 79 109 150)(11 80 110 151)(12 41 111 152)(13 42 112 153)(14 43 113 154)(15 44 114 155)(16 45 115 156)(17 46 116 157)(18 47 117 158)(19 48 118 159)(20 49 119 160)(21 50 120 121)(22 51 81 122)(23 52 82 123)(24 53 83 124)(25 54 84 125)(26 55 85 126)(27 56 86 127)(28 57 87 128)(29 58 88 129)(30 59 89 130)(31 60 90 131)(32 61 91 132)(33 62 92 133)(34 63 93 134)(35 64 94 135)(36 65 95 136)(37 66 96 137)(38 67 97 138)(39 68 98 139)(40 69 99 140)
(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)| (1,70,100,141)(2,71,101,142)(3,72,102,143)(4,73,103,144)(5,74,104,145)(6,75,105,146)(7,76,106,147)(8,77,107,148)(9,78,108,149)(10,79,109,150)(11,80,110,151)(12,41,111,152)(13,42,112,153)(14,43,113,154)(15,44,114,155)(16,45,115,156)(17,46,116,157)(18,47,117,158)(19,48,118,159)(20,49,119,160)(21,50,120,121)(22,51,81,122)(23,52,82,123)(24,53,83,124)(25,54,84,125)(26,55,85,126)(27,56,86,127)(28,57,87,128)(29,58,88,129)(30,59,89,130)(31,60,90,131)(32,61,91,132)(33,62,92,133)(34,63,93,134)(35,64,94,135)(36,65,95,136)(37,66,96,137)(38,67,97,138)(39,68,98,139)(40,69,99,140), (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( (1,70,100,141)(2,71,101,142)(3,72,102,143)(4,73,103,144)(5,74,104,145)(6,75,105,146)(7,76,106,147)(8,77,107,148)(9,78,108,149)(10,79,109,150)(11,80,110,151)(12,41,111,152)(13,42,112,153)(14,43,113,154)(15,44,114,155)(16,45,115,156)(17,46,116,157)(18,47,117,158)(19,48,118,159)(20,49,119,160)(21,50,120,121)(22,51,81,122)(23,52,82,123)(24,53,83,124)(25,54,84,125)(26,55,85,126)(27,56,86,127)(28,57,87,128)(29,58,88,129)(30,59,89,130)(31,60,90,131)(32,61,91,132)(33,62,92,133)(34,63,93,134)(35,64,94,135)(36,65,95,136)(37,66,96,137)(38,67,97,138)(39,68,98,139)(40,69,99,140), (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([(1,70,100,141),(2,71,101,142),(3,72,102,143),(4,73,103,144),(5,74,104,145),(6,75,105,146),(7,76,106,147),(8,77,107,148),(9,78,108,149),(10,79,109,150),(11,80,110,151),(12,41,111,152),(13,42,112,153),(14,43,113,154),(15,44,114,155),(16,45,115,156),(17,46,116,157),(18,47,117,158),(19,48,118,159),(20,49,119,160),(21,50,120,121),(22,51,81,122),(23,52,82,123),(24,53,83,124),(25,54,84,125),(26,55,85,126),(27,56,86,127),(28,57,87,128),(29,58,88,129),(30,59,89,130),(31,60,90,131),(32,61,91,132),(33,62,92,133),(34,63,93,134),(35,64,94,135),(36,65,95,136),(37,66,96,137),(38,67,97,138),(39,68,98,139),(40,69,99,140)], [(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)])

C4×C40 is a maximal subgroup of
C42.279D10  C408C8  Dic103C8  C406C8  C405C8  D203C8  C40.10C8  C203C16  C40.7C8  C4011Q8  C409Q8  C20.14Q16  C408Q8  C40.13Q8  C42.282D10  C86D20  D10.5C42  C42.243D10  C85D20  C4.5D40  C204D8  C8.8D20  C42.264D10  C204Q16  D4017C4

160 conjugacy classes

class 1 2A2B2C4A···4L5A5B5C5D8A···8P10A···10L20A···20AV40A···40BL
order12224···455558···810···1020···2040···40
size11111···111111···11···11···11···1

160 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C5C8C10C10C20C20C40
kernelC4×C40C4×C20C2×C40C40C2×C20C4×C8C20C42C2×C8C8C2×C4C4
# reps1128441648321664

Matrix representation of C4×C40 in GL2(𝔽41) generated by

320
040
,
320
024
G:=sub<GL(2,GF(41))| [32,0,0,40],[32,0,0,24] >;

C4×C40 in GAP, Magma, Sage, TeX

C_4\times C_{40}
% in TeX

G:=Group("C4xC40");
// GroupNames label

G:=SmallGroup(160,46);
// by ID

G=gap.SmallGroup(160,46);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-2,120,247,117]);
// Polycyclic

G:=Group<a,b|a^4=b^40=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C40 in TeX

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