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G = C5×C40order 200 = 23·52

Abelian group of type [5,40]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C40, SmallGroup(200,17)

Series: Derived Chief Lower central Upper central

C1 — C5×C40
C1C2C4C20C5×C20 — C5×C40
C1 — C5×C40
C1 — C5×C40

Generators and relations for C5×C40
 G = < a,b | a5=b40=1, ab=ba >


Smallest permutation representation of C5×C40
Regular action on 200 points
Generators in S200
(1 105 156 78 167)(2 106 157 79 168)(3 107 158 80 169)(4 108 159 41 170)(5 109 160 42 171)(6 110 121 43 172)(7 111 122 44 173)(8 112 123 45 174)(9 113 124 46 175)(10 114 125 47 176)(11 115 126 48 177)(12 116 127 49 178)(13 117 128 50 179)(14 118 129 51 180)(15 119 130 52 181)(16 120 131 53 182)(17 81 132 54 183)(18 82 133 55 184)(19 83 134 56 185)(20 84 135 57 186)(21 85 136 58 187)(22 86 137 59 188)(23 87 138 60 189)(24 88 139 61 190)(25 89 140 62 191)(26 90 141 63 192)(27 91 142 64 193)(28 92 143 65 194)(29 93 144 66 195)(30 94 145 67 196)(31 95 146 68 197)(32 96 147 69 198)(33 97 148 70 199)(34 98 149 71 200)(35 99 150 72 161)(36 100 151 73 162)(37 101 152 74 163)(38 102 153 75 164)(39 103 154 76 165)(40 104 155 77 166)
(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)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)

G:=sub<Sym(200)| (1,105,156,78,167)(2,106,157,79,168)(3,107,158,80,169)(4,108,159,41,170)(5,109,160,42,171)(6,110,121,43,172)(7,111,122,44,173)(8,112,123,45,174)(9,113,124,46,175)(10,114,125,47,176)(11,115,126,48,177)(12,116,127,49,178)(13,117,128,50,179)(14,118,129,51,180)(15,119,130,52,181)(16,120,131,53,182)(17,81,132,54,183)(18,82,133,55,184)(19,83,134,56,185)(20,84,135,57,186)(21,85,136,58,187)(22,86,137,59,188)(23,87,138,60,189)(24,88,139,61,190)(25,89,140,62,191)(26,90,141,63,192)(27,91,142,64,193)(28,92,143,65,194)(29,93,144,66,195)(30,94,145,67,196)(31,95,146,68,197)(32,96,147,69,198)(33,97,148,70,199)(34,98,149,71,200)(35,99,150,72,161)(36,100,151,73,162)(37,101,152,74,163)(38,102,153,75,164)(39,103,154,76,165)(40,104,155,77,166), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)>;

G:=Group( (1,105,156,78,167)(2,106,157,79,168)(3,107,158,80,169)(4,108,159,41,170)(5,109,160,42,171)(6,110,121,43,172)(7,111,122,44,173)(8,112,123,45,174)(9,113,124,46,175)(10,114,125,47,176)(11,115,126,48,177)(12,116,127,49,178)(13,117,128,50,179)(14,118,129,51,180)(15,119,130,52,181)(16,120,131,53,182)(17,81,132,54,183)(18,82,133,55,184)(19,83,134,56,185)(20,84,135,57,186)(21,85,136,58,187)(22,86,137,59,188)(23,87,138,60,189)(24,88,139,61,190)(25,89,140,62,191)(26,90,141,63,192)(27,91,142,64,193)(28,92,143,65,194)(29,93,144,66,195)(30,94,145,67,196)(31,95,146,68,197)(32,96,147,69,198)(33,97,148,70,199)(34,98,149,71,200)(35,99,150,72,161)(36,100,151,73,162)(37,101,152,74,163)(38,102,153,75,164)(39,103,154,76,165)(40,104,155,77,166), (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)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200) );

G=PermutationGroup([(1,105,156,78,167),(2,106,157,79,168),(3,107,158,80,169),(4,108,159,41,170),(5,109,160,42,171),(6,110,121,43,172),(7,111,122,44,173),(8,112,123,45,174),(9,113,124,46,175),(10,114,125,47,176),(11,115,126,48,177),(12,116,127,49,178),(13,117,128,50,179),(14,118,129,51,180),(15,119,130,52,181),(16,120,131,53,182),(17,81,132,54,183),(18,82,133,55,184),(19,83,134,56,185),(20,84,135,57,186),(21,85,136,58,187),(22,86,137,59,188),(23,87,138,60,189),(24,88,139,61,190),(25,89,140,62,191),(26,90,141,63,192),(27,91,142,64,193),(28,92,143,65,194),(29,93,144,66,195),(30,94,145,67,196),(31,95,146,68,197),(32,96,147,69,198),(33,97,148,70,199),(34,98,149,71,200),(35,99,150,72,161),(36,100,151,73,162),(37,101,152,74,163),(38,102,153,75,164),(39,103,154,76,165),(40,104,155,77,166)], [(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),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)])

C5×C40 is a maximal subgroup of   C527C16  C40⋊D5  C402D5  C525D8  C40.D5

200 conjugacy classes

class 1  2 4A4B5A···5X8A8B8C8D10A···10X20A···20AV40A···40CR
order12445···5888810···1020···2040···40
size11111···111111···11···11···1

200 irreducible representations

dim11111111
type++
imageC1C2C4C5C8C10C20C40
kernelC5×C40C5×C20C5×C10C40C52C20C10C5
# reps112244244896

Matrix representation of C5×C40 in GL2(𝔽41) generated by

10
037
,
210
019
G:=sub<GL(2,GF(41))| [1,0,0,37],[21,0,0,19] >;

C5×C40 in GAP, Magma, Sage, TeX

C_5\times C_{40}
% in TeX

G:=Group("C5xC40");
// GroupNames label

G:=SmallGroup(200,17);
// by ID

G=gap.SmallGroup(200,17);
# by ID

G:=PCGroup([5,-2,-5,-5,-2,-2,250,58]);
// Polycyclic

G:=Group<a,b|a^5=b^40=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C5×C40 in TeX

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