direct product, abelian, monomial
Aliases: C10×C20, SmallGroup(200,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C10×C20 |
| C1 — C10×C20 |
| 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, C4, C22, C5, C2×C4, C10, C20, C2×C10, C52, C2×C20, C5×C10, C5×C10, C5×C20, C102, C10×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, C10, C20, C2×C10, C52, C2×C20, C5×C10, C5×C20, C102, C10×C20
►Regular action on 200 points
Generators in S200
(1 172 91 133 195 141 42 109 63 37)(2 173 92 134 196 142 43 110 64 38)(3 174 93 135 197 143 44 111 65 39)(4 175 94 136 198 144 45 112 66 40)(5 176 95 137 199 145 46 113 67 21)(6 177 96 138 200 146 47 114 68 22)(7 178 97 139 181 147 48 115 69 23)(8 179 98 140 182 148 49 116 70 24)(9 180 99 121 183 149 50 117 71 25)(10 161 100 122 184 150 51 118 72 26)(11 162 81 123 185 151 52 119 73 27)(12 163 82 124 186 152 53 120 74 28)(13 164 83 125 187 153 54 101 75 29)(14 165 84 126 188 154 55 102 76 30)(15 166 85 127 189 155 56 103 77 31)(16 167 86 128 190 156 57 104 78 32)(17 168 87 129 191 157 58 105 79 33)(18 169 88 130 192 158 59 106 80 34)(19 170 89 131 193 159 60 107 61 35)(20 171 90 132 194 160 41 108 62 36)
(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)| (1,172,91,133,195,141,42,109,63,37)(2,173,92,134,196,142,43,110,64,38)(3,174,93,135,197,143,44,111,65,39)(4,175,94,136,198,144,45,112,66,40)(5,176,95,137,199,145,46,113,67,21)(6,177,96,138,200,146,47,114,68,22)(7,178,97,139,181,147,48,115,69,23)(8,179,98,140,182,148,49,116,70,24)(9,180,99,121,183,149,50,117,71,25)(10,161,100,122,184,150,51,118,72,26)(11,162,81,123,185,151,52,119,73,27)(12,163,82,124,186,152,53,120,74,28)(13,164,83,125,187,153,54,101,75,29)(14,165,84,126,188,154,55,102,76,30)(15,166,85,127,189,155,56,103,77,31)(16,167,86,128,190,156,57,104,78,32)(17,168,87,129,191,157,58,105,79,33)(18,169,88,130,192,158,59,106,80,34)(19,170,89,131,193,159,60,107,61,35)(20,171,90,132,194,160,41,108,62,36), (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( (1,172,91,133,195,141,42,109,63,37)(2,173,92,134,196,142,43,110,64,38)(3,174,93,135,197,143,44,111,65,39)(4,175,94,136,198,144,45,112,66,40)(5,176,95,137,199,145,46,113,67,21)(6,177,96,138,200,146,47,114,68,22)(7,178,97,139,181,147,48,115,69,23)(8,179,98,140,182,148,49,116,70,24)(9,180,99,121,183,149,50,117,71,25)(10,161,100,122,184,150,51,118,72,26)(11,162,81,123,185,151,52,119,73,27)(12,163,82,124,186,152,53,120,74,28)(13,164,83,125,187,153,54,101,75,29)(14,165,84,126,188,154,55,102,76,30)(15,166,85,127,189,155,56,103,77,31)(16,167,86,128,190,156,57,104,78,32)(17,168,87,129,191,157,58,105,79,33)(18,169,88,130,192,158,59,106,80,34)(19,170,89,131,193,159,60,107,61,35)(20,171,90,132,194,160,41,108,62,36), (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([[(1,172,91,133,195,141,42,109,63,37),(2,173,92,134,196,142,43,110,64,38),(3,174,93,135,197,143,44,111,65,39),(4,175,94,136,198,144,45,112,66,40),(5,176,95,137,199,145,46,113,67,21),(6,177,96,138,200,146,47,114,68,22),(7,178,97,139,181,147,48,115,69,23),(8,179,98,140,182,148,49,116,70,24),(9,180,99,121,183,149,50,117,71,25),(10,161,100,122,184,150,51,118,72,26),(11,162,81,123,185,151,52,119,73,27),(12,163,82,124,186,152,53,120,74,28),(13,164,83,125,187,153,54,101,75,29),(14,165,84,126,188,154,55,102,76,30),(15,166,85,127,189,155,56,103,77,31),(16,167,86,128,190,156,57,104,78,32),(17,168,87,129,191,157,58,105,79,33),(18,169,88,130,192,158,59,106,80,34),(19,170,89,131,193,159,60,107,61,35),(20,171,90,132,194,160,41,108,62,36)], [(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 C20⋊3Dic5 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