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## G = C10×C20order 200 = 23·52

### Abelian group of type [10,20]

Aliases: C10×C20, SmallGroup(200,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10×C20
 Chief series C1 — C2 — C10 — C5×C10 — C5×C20 — C10×C20
 Lower central C1 — C10×C20
 Upper central C1 — C10×C20

Generators and relations for C10×C20
G = < a,b | a10=b20=1, ab=ba >

Subgroups: 64, all normal (8 characteristic)
C1, C2, C2 [×2], C4 [×2], C22, C5 [×6], C2×C4, C10 [×18], C20 [×12], C2×C10 [×6], C52, C2×C20 [×6], C5×C10, C5×C10 [×2], C5×C20 [×2], C102, C10×C20
Quotients: C1, C2 [×3], C4 [×2], C22, C5 [×6], C2×C4, C10 [×18], C20 [×12], C2×C10 [×6], C52, C2×C20 [×6], C5×C10 [×3], C5×C20 [×2], C102, C10×C20

Smallest permutation representation of C10×C20
Regular action on 200 points
Generators in S200
(1 62 103 31 124 89 162 144 184 49)(2 63 104 32 125 90 163 145 185 50)(3 64 105 33 126 91 164 146 186 51)(4 65 106 34 127 92 165 147 187 52)(5 66 107 35 128 93 166 148 188 53)(6 67 108 36 129 94 167 149 189 54)(7 68 109 37 130 95 168 150 190 55)(8 69 110 38 131 96 169 151 191 56)(9 70 111 39 132 97 170 152 192 57)(10 71 112 40 133 98 171 153 193 58)(11 72 113 21 134 99 172 154 194 59)(12 73 114 22 135 100 173 155 195 60)(13 74 115 23 136 81 174 156 196 41)(14 75 116 24 137 82 175 157 197 42)(15 76 117 25 138 83 176 158 198 43)(16 77 118 26 139 84 177 159 199 44)(17 78 119 27 140 85 178 160 200 45)(18 79 120 28 121 86 179 141 181 46)(19 80 101 29 122 87 180 142 182 47)(20 61 102 30 123 88 161 143 183 48)
(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)| (1,62,103,31,124,89,162,144,184,49)(2,63,104,32,125,90,163,145,185,50)(3,64,105,33,126,91,164,146,186,51)(4,65,106,34,127,92,165,147,187,52)(5,66,107,35,128,93,166,148,188,53)(6,67,108,36,129,94,167,149,189,54)(7,68,109,37,130,95,168,150,190,55)(8,69,110,38,131,96,169,151,191,56)(9,70,111,39,132,97,170,152,192,57)(10,71,112,40,133,98,171,153,193,58)(11,72,113,21,134,99,172,154,194,59)(12,73,114,22,135,100,173,155,195,60)(13,74,115,23,136,81,174,156,196,41)(14,75,116,24,137,82,175,157,197,42)(15,76,117,25,138,83,176,158,198,43)(16,77,118,26,139,84,177,159,199,44)(17,78,119,27,140,85,178,160,200,45)(18,79,120,28,121,86,179,141,181,46)(19,80,101,29,122,87,180,142,182,47)(20,61,102,30,123,88,161,143,183,48), (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( (1,62,103,31,124,89,162,144,184,49)(2,63,104,32,125,90,163,145,185,50)(3,64,105,33,126,91,164,146,186,51)(4,65,106,34,127,92,165,147,187,52)(5,66,107,35,128,93,166,148,188,53)(6,67,108,36,129,94,167,149,189,54)(7,68,109,37,130,95,168,150,190,55)(8,69,110,38,131,96,169,151,191,56)(9,70,111,39,132,97,170,152,192,57)(10,71,112,40,133,98,171,153,193,58)(11,72,113,21,134,99,172,154,194,59)(12,73,114,22,135,100,173,155,195,60)(13,74,115,23,136,81,174,156,196,41)(14,75,116,24,137,82,175,157,197,42)(15,76,117,25,138,83,176,158,198,43)(16,77,118,26,139,84,177,159,199,44)(17,78,119,27,140,85,178,160,200,45)(18,79,120,28,121,86,179,141,181,46)(19,80,101,29,122,87,180,142,182,47)(20,61,102,30,123,88,161,143,183,48), (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([(1,62,103,31,124,89,162,144,184,49),(2,63,104,32,125,90,163,145,185,50),(3,64,105,33,126,91,164,146,186,51),(4,65,106,34,127,92,165,147,187,52),(5,66,107,35,128,93,166,148,188,53),(6,67,108,36,129,94,167,149,189,54),(7,68,109,37,130,95,168,150,190,55),(8,69,110,38,131,96,169,151,191,56),(9,70,111,39,132,97,170,152,192,57),(10,71,112,40,133,98,171,153,193,58),(11,72,113,21,134,99,172,154,194,59),(12,73,114,22,135,100,173,155,195,60),(13,74,115,23,136,81,174,156,196,41),(14,75,116,24,137,82,175,157,197,42),(15,76,117,25,138,83,176,158,198,43),(16,77,118,26,139,84,177,159,199,44),(17,78,119,27,140,85,178,160,200,45),(18,79,120,28,121,86,179,141,181,46),(19,80,101,29,122,87,180,142,182,47),(20,61,102,30,123,88,161,143,183,48)], [(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)])

C10×C20 is a maximal subgroup of   C20.59D10  C102.22C22  C203Dic5  C10.11D20  C20.50D10

200 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A ··· 5X 10A ··· 10BT 20A ··· 20CR order 1 2 2 2 4 4 4 4 5 ··· 5 10 ··· 10 20 ··· 20 size 1 1 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C5 C10 C10 C20 kernel C10×C20 C5×C20 C102 C5×C10 C2×C20 C20 C2×C10 C10 # reps 1 2 1 4 24 48 24 96

Matrix representation of C10×C20 in GL2(𝔽41) generated by

 4 0 0 16
,
 4 0 0 8
G:=sub<GL(2,GF(41))| [4,0,0,16],[4,0,0,8] >;

C10×C20 in GAP, Magma, Sage, TeX

C_{10}\times C_{20}
% in TeX

G:=Group("C10xC20");
// GroupNames label

G:=SmallGroup(200,37);
// by ID

G=gap.SmallGroup(200,37);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-2,500]);
// Polycyclic

G:=Group<a,b|a^10=b^20=1,a*b=b*a>;
// generators/relations

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