direct product, abelian, monomial, 5-elementary
Aliases: C5×C40, SmallGroup(200,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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
(1 137 176 112 42)(2 138 177 113 43)(3 139 178 114 44)(4 140 179 115 45)(5 141 180 116 46)(6 142 181 117 47)(7 143 182 118 48)(8 144 183 119 49)(9 145 184 120 50)(10 146 185 81 51)(11 147 186 82 52)(12 148 187 83 53)(13 149 188 84 54)(14 150 189 85 55)(15 151 190 86 56)(16 152 191 87 57)(17 153 192 88 58)(18 154 193 89 59)(19 155 194 90 60)(20 156 195 91 61)(21 157 196 92 62)(22 158 197 93 63)(23 159 198 94 64)(24 160 199 95 65)(25 121 200 96 66)(26 122 161 97 67)(27 123 162 98 68)(28 124 163 99 69)(29 125 164 100 70)(30 126 165 101 71)(31 127 166 102 72)(32 128 167 103 73)(33 129 168 104 74)(34 130 169 105 75)(35 131 170 106 76)(36 132 171 107 77)(37 133 172 108 78)(38 134 173 109 79)(39 135 174 110 80)(40 136 175 111 41)
(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)| (1,137,176,112,42)(2,138,177,113,43)(3,139,178,114,44)(4,140,179,115,45)(5,141,180,116,46)(6,142,181,117,47)(7,143,182,118,48)(8,144,183,119,49)(9,145,184,120,50)(10,146,185,81,51)(11,147,186,82,52)(12,148,187,83,53)(13,149,188,84,54)(14,150,189,85,55)(15,151,190,86,56)(16,152,191,87,57)(17,153,192,88,58)(18,154,193,89,59)(19,155,194,90,60)(20,156,195,91,61)(21,157,196,92,62)(22,158,197,93,63)(23,159,198,94,64)(24,160,199,95,65)(25,121,200,96,66)(26,122,161,97,67)(27,123,162,98,68)(28,124,163,99,69)(29,125,164,100,70)(30,126,165,101,71)(31,127,166,102,72)(32,128,167,103,73)(33,129,168,104,74)(34,130,169,105,75)(35,131,170,106,76)(36,132,171,107,77)(37,133,172,108,78)(38,134,173,109,79)(39,135,174,110,80)(40,136,175,111,41), (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( (1,137,176,112,42)(2,138,177,113,43)(3,139,178,114,44)(4,140,179,115,45)(5,141,180,116,46)(6,142,181,117,47)(7,143,182,118,48)(8,144,183,119,49)(9,145,184,120,50)(10,146,185,81,51)(11,147,186,82,52)(12,148,187,83,53)(13,149,188,84,54)(14,150,189,85,55)(15,151,190,86,56)(16,152,191,87,57)(17,153,192,88,58)(18,154,193,89,59)(19,155,194,90,60)(20,156,195,91,61)(21,157,196,92,62)(22,158,197,93,63)(23,159,198,94,64)(24,160,199,95,65)(25,121,200,96,66)(26,122,161,97,67)(27,123,162,98,68)(28,124,163,99,69)(29,125,164,100,70)(30,126,165,101,71)(31,127,166,102,72)(32,128,167,103,73)(33,129,168,104,74)(34,130,169,105,75)(35,131,170,106,76)(36,132,171,107,77)(37,133,172,108,78)(38,134,173,109,79)(39,135,174,110,80)(40,136,175,111,41), (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([[(1,137,176,112,42),(2,138,177,113,43),(3,139,178,114,44),(4,140,179,115,45),(5,141,180,116,46),(6,142,181,117,47),(7,143,182,118,48),(8,144,183,119,49),(9,145,184,120,50),(10,146,185,81,51),(11,147,186,82,52),(12,148,187,83,53),(13,149,188,84,54),(14,150,189,85,55),(15,151,190,86,56),(16,152,191,87,57),(17,153,192,88,58),(18,154,193,89,59),(19,155,194,90,60),(20,156,195,91,61),(21,157,196,92,62),(22,158,197,93,63),(23,159,198,94,64),(24,160,199,95,65),(25,121,200,96,66),(26,122,161,97,67),(27,123,162,98,68),(28,124,163,99,69),(29,125,164,100,70),(30,126,165,101,71),(31,127,166,102,72),(32,128,167,103,73),(33,129,168,104,74),(34,130,169,105,75),(35,131,170,106,76),(36,132,171,107,77),(37,133,172,108,78),(38,134,173,109,79),(39,135,174,110,80),(40,136,175,111,41)], [(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
C52⋊7C16 C40⋊D5 C40⋊2D5 C52⋊5D8 C40.D5
200 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | ··· | 5X | 8A | 8B | 8C | 8D | 10A | ··· | 10X | 20A | ··· | 20AV | 40A | ··· | 40CR |
| order | 1 | 2 | 4 | 4 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 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 | C4 | C5 | C8 | C10 | C20 | C40 |
| kernel | C5×C40 | C5×C20 | C5×C10 | C40 | C52 | C20 | C10 | C5 |
| # reps | 1 | 1 | 2 | 24 | 4 | 24 | 48 | 96 |
Matrix representation of C5×C40 ►in GL2(𝔽41) generated by
| 1 | 0 |
| 0 | 37 |
| 21 | 0 |
| 0 | 19 |
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
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