direct product, abelian, monomial, 5-elementary
Aliases: C5×C35, SmallGroup(175,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5×C35 |
| C1 — C5×C35 |
| C1 — C5×C35 |
Generators and relations for C5×C35
G = < a,b | a5=b35=1, ab=ba >
Smallest permutation representation of C5×C35
►Regular action on 175 points
Generators in S175
(1 144 113 88 68)(2 145 114 89 69)(3 146 115 90 70)(4 147 116 91 36)(5 148 117 92 37)(6 149 118 93 38)(7 150 119 94 39)(8 151 120 95 40)(9 152 121 96 41)(10 153 122 97 42)(11 154 123 98 43)(12 155 124 99 44)(13 156 125 100 45)(14 157 126 101 46)(15 158 127 102 47)(16 159 128 103 48)(17 160 129 104 49)(18 161 130 105 50)(19 162 131 71 51)(20 163 132 72 52)(21 164 133 73 53)(22 165 134 74 54)(23 166 135 75 55)(24 167 136 76 56)(25 168 137 77 57)(26 169 138 78 58)(27 170 139 79 59)(28 171 140 80 60)(29 172 106 81 61)(30 173 107 82 62)(31 174 108 83 63)(32 175 109 84 64)(33 141 110 85 65)(34 142 111 86 66)(35 143 112 87 67)
(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)
G:=sub<Sym(175)| (1,144,113,88,68)(2,145,114,89,69)(3,146,115,90,70)(4,147,116,91,36)(5,148,117,92,37)(6,149,118,93,38)(7,150,119,94,39)(8,151,120,95,40)(9,152,121,96,41)(10,153,122,97,42)(11,154,123,98,43)(12,155,124,99,44)(13,156,125,100,45)(14,157,126,101,46)(15,158,127,102,47)(16,159,128,103,48)(17,160,129,104,49)(18,161,130,105,50)(19,162,131,71,51)(20,163,132,72,52)(21,164,133,73,53)(22,165,134,74,54)(23,166,135,75,55)(24,167,136,76,56)(25,168,137,77,57)(26,169,138,78,58)(27,170,139,79,59)(28,171,140,80,60)(29,172,106,81,61)(30,173,107,82,62)(31,174,108,83,63)(32,175,109,84,64)(33,141,110,85,65)(34,142,111,86,66)(35,143,112,87,67), (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)>;
G:=Group( (1,144,113,88,68)(2,145,114,89,69)(3,146,115,90,70)(4,147,116,91,36)(5,148,117,92,37)(6,149,118,93,38)(7,150,119,94,39)(8,151,120,95,40)(9,152,121,96,41)(10,153,122,97,42)(11,154,123,98,43)(12,155,124,99,44)(13,156,125,100,45)(14,157,126,101,46)(15,158,127,102,47)(16,159,128,103,48)(17,160,129,104,49)(18,161,130,105,50)(19,162,131,71,51)(20,163,132,72,52)(21,164,133,73,53)(22,165,134,74,54)(23,166,135,75,55)(24,167,136,76,56)(25,168,137,77,57)(26,169,138,78,58)(27,170,139,79,59)(28,171,140,80,60)(29,172,106,81,61)(30,173,107,82,62)(31,174,108,83,63)(32,175,109,84,64)(33,141,110,85,65)(34,142,111,86,66)(35,143,112,87,67), (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) );
G=PermutationGroup([[(1,144,113,88,68),(2,145,114,89,69),(3,146,115,90,70),(4,147,116,91,36),(5,148,117,92,37),(6,149,118,93,38),(7,150,119,94,39),(8,151,120,95,40),(9,152,121,96,41),(10,153,122,97,42),(11,154,123,98,43),(12,155,124,99,44),(13,156,125,100,45),(14,157,126,101,46),(15,158,127,102,47),(16,159,128,103,48),(17,160,129,104,49),(18,161,130,105,50),(19,162,131,71,51),(20,163,132,72,52),(21,164,133,73,53),(22,165,134,74,54),(23,166,135,75,55),(24,167,136,76,56),(25,168,137,77,57),(26,169,138,78,58),(27,170,139,79,59),(28,171,140,80,60),(29,172,106,81,61),(30,173,107,82,62),(31,174,108,83,63),(32,175,109,84,64),(33,141,110,85,65),(34,142,111,86,66),(35,143,112,87,67)], [(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)]])
C5×C35 is a maximal subgroup of
C5⋊D35
175 conjugacy classes
| class | 1 | 5A | ··· | 5X | 7A | ··· | 7F | 35A | ··· | 35EN |
| order | 1 | 5 | ··· | 5 | 7 | ··· | 7 | 35 | ··· | 35 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C5 | C7 | C35 |
| kernel | C5×C35 | C35 | C52 | C5 |
| # reps | 1 | 24 | 6 | 144 |
Matrix representation of C5×C35 ►in GL2(𝔽71) generated by
| 25 | 0 |
| 0 | 54 |
| 18 | 0 |
| 0 | 43 |
G:=sub<GL(2,GF(71))| [25,0,0,54],[18,0,0,43] >;
C5×C35 in GAP, Magma, Sage, TeX
C_5\times C_{35} % in TeX
G:=Group("C5xC35"); // GroupNames label
G:=SmallGroup(175,2);
// by ID
G=gap.SmallGroup(175,2);
# by ID
G:=PCGroup([3,-5,-5,-7]);
// Polycyclic
G:=Group<a,b|a^5=b^35=1,a*b=b*a>;
// generators/relations
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