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G = C4xC40order 160 = 25·5

Abelian group of type [4,40]

direct product, abelian, monomial, 2-elementary

Aliases: C4xC40, SmallGroup(160,46)

Series: Derived Chief Lower central Upper central

C1 — C4xC40
C1C2C22C2xC4C2xC20C2xC40 — C4xC40
C1 — C4xC40
C1 — C4xC40

Generators and relations for C4xC40
 G = < a,b | a4=b40=1, ab=ba >

Subgroups: 44, all normal (12 characteristic)
Quotients: C1, C2, C4, C22, C5, C8, C2xC4, C10, C42, C2xC8, C20, C2xC10, C4xC8, C40, C2xC20, C4xC20, C2xC40, C4xC40

Smallest permutation representation of C4xC40
Regular action on 160 points
Generators in S160
(1 54 107 141)(2 55 108 142)(3 56 109 143)(4 57 110 144)(5 58 111 145)(6 59 112 146)(7 60 113 147)(8 61 114 148)(9 62 115 149)(10 63 116 150)(11 64 117 151)(12 65 118 152)(13 66 119 153)(14 67 120 154)(15 68 81 155)(16 69 82 156)(17 70 83 157)(18 71 84 158)(19 72 85 159)(20 73 86 160)(21 74 87 121)(22 75 88 122)(23 76 89 123)(24 77 90 124)(25 78 91 125)(26 79 92 126)(27 80 93 127)(28 41 94 128)(29 42 95 129)(30 43 96 130)(31 44 97 131)(32 45 98 132)(33 46 99 133)(34 47 100 134)(35 48 101 135)(36 49 102 136)(37 50 103 137)(38 51 104 138)(39 52 105 139)(40 53 106 140)
(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)| (1,54,107,141)(2,55,108,142)(3,56,109,143)(4,57,110,144)(5,58,111,145)(6,59,112,146)(7,60,113,147)(8,61,114,148)(9,62,115,149)(10,63,116,150)(11,64,117,151)(12,65,118,152)(13,66,119,153)(14,67,120,154)(15,68,81,155)(16,69,82,156)(17,70,83,157)(18,71,84,158)(19,72,85,159)(20,73,86,160)(21,74,87,121)(22,75,88,122)(23,76,89,123)(24,77,90,124)(25,78,91,125)(26,79,92,126)(27,80,93,127)(28,41,94,128)(29,42,95,129)(30,43,96,130)(31,44,97,131)(32,45,98,132)(33,46,99,133)(34,47,100,134)(35,48,101,135)(36,49,102,136)(37,50,103,137)(38,51,104,138)(39,52,105,139)(40,53,106,140), (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( (1,54,107,141)(2,55,108,142)(3,56,109,143)(4,57,110,144)(5,58,111,145)(6,59,112,146)(7,60,113,147)(8,61,114,148)(9,62,115,149)(10,63,116,150)(11,64,117,151)(12,65,118,152)(13,66,119,153)(14,67,120,154)(15,68,81,155)(16,69,82,156)(17,70,83,157)(18,71,84,158)(19,72,85,159)(20,73,86,160)(21,74,87,121)(22,75,88,122)(23,76,89,123)(24,77,90,124)(25,78,91,125)(26,79,92,126)(27,80,93,127)(28,41,94,128)(29,42,95,129)(30,43,96,130)(31,44,97,131)(32,45,98,132)(33,46,99,133)(34,47,100,134)(35,48,101,135)(36,49,102,136)(37,50,103,137)(38,51,104,138)(39,52,105,139)(40,53,106,140), (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([[(1,54,107,141),(2,55,108,142),(3,56,109,143),(4,57,110,144),(5,58,111,145),(6,59,112,146),(7,60,113,147),(8,61,114,148),(9,62,115,149),(10,63,116,150),(11,64,117,151),(12,65,118,152),(13,66,119,153),(14,67,120,154),(15,68,81,155),(16,69,82,156),(17,70,83,157),(18,71,84,158),(19,72,85,159),(20,73,86,160),(21,74,87,121),(22,75,88,122),(23,76,89,123),(24,77,90,124),(25,78,91,125),(26,79,92,126),(27,80,93,127),(28,41,94,128),(29,42,95,129),(30,43,96,130),(31,44,97,131),(32,45,98,132),(33,46,99,133),(34,47,100,134),(35,48,101,135),(36,49,102,136),(37,50,103,137),(38,51,104,138),(39,52,105,139),(40,53,106,140)], [(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)]])

C4xC40 is a maximal subgroup of
C42.279D10  C40:8C8  Dic10:3C8  C40:6C8  C40:5C8  D20:3C8  C40.10C8  C20:3C16  C40.7C8  C40:11Q8  C40:9Q8  C20.14Q16  C40:8Q8  C40.13Q8  C42.282D10  C8:6D20  D10.5C42  C42.243D10  C8:5D20  C4.5D40  C20:4D8  C8.8D20  C42.264D10  C20:4Q16  D40:17C4

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
kernelC4xC40C4xC20C2xC40C40C2xC20C4xC8C20C42C2xC8C8C2xC4C4
# reps1128441648321664

Matrix representation of C4xC40 in GL2(F41) generated by

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

C4xC40 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 C4xC40 in TeX

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