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G = C5×C35order 175 = 52·7

Abelian group of type [5,35]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C35, SmallGroup(175,2)

Series: Derived Chief Lower central Upper central

C1 — C5×C35
C1C7C35 — 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
(1 158 125 94 66)(2 159 126 95 67)(3 160 127 96 68)(4 161 128 97 69)(5 162 129 98 70)(6 163 130 99 36)(7 164 131 100 37)(8 165 132 101 38)(9 166 133 102 39)(10 167 134 103 40)(11 168 135 104 41)(12 169 136 105 42)(13 170 137 71 43)(14 171 138 72 44)(15 172 139 73 45)(16 173 140 74 46)(17 174 106 75 47)(18 175 107 76 48)(19 141 108 77 49)(20 142 109 78 50)(21 143 110 79 51)(22 144 111 80 52)(23 145 112 81 53)(24 146 113 82 54)(25 147 114 83 55)(26 148 115 84 56)(27 149 116 85 57)(28 150 117 86 58)(29 151 118 87 59)(30 152 119 88 60)(31 153 120 89 61)(32 154 121 90 62)(33 155 122 91 63)(34 156 123 92 64)(35 157 124 93 65)
(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)| (1,158,125,94,66)(2,159,126,95,67)(3,160,127,96,68)(4,161,128,97,69)(5,162,129,98,70)(6,163,130,99,36)(7,164,131,100,37)(8,165,132,101,38)(9,166,133,102,39)(10,167,134,103,40)(11,168,135,104,41)(12,169,136,105,42)(13,170,137,71,43)(14,171,138,72,44)(15,172,139,73,45)(16,173,140,74,46)(17,174,106,75,47)(18,175,107,76,48)(19,141,108,77,49)(20,142,109,78,50)(21,143,110,79,51)(22,144,111,80,52)(23,145,112,81,53)(24,146,113,82,54)(25,147,114,83,55)(26,148,115,84,56)(27,149,116,85,57)(28,150,117,86,58)(29,151,118,87,59)(30,152,119,88,60)(31,153,120,89,61)(32,154,121,90,62)(33,155,122,91,63)(34,156,123,92,64)(35,157,124,93,65), (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( (1,158,125,94,66)(2,159,126,95,67)(3,160,127,96,68)(4,161,128,97,69)(5,162,129,98,70)(6,163,130,99,36)(7,164,131,100,37)(8,165,132,101,38)(9,166,133,102,39)(10,167,134,103,40)(11,168,135,104,41)(12,169,136,105,42)(13,170,137,71,43)(14,171,138,72,44)(15,172,139,73,45)(16,173,140,74,46)(17,174,106,75,47)(18,175,107,76,48)(19,141,108,77,49)(20,142,109,78,50)(21,143,110,79,51)(22,144,111,80,52)(23,145,112,81,53)(24,146,113,82,54)(25,147,114,83,55)(26,148,115,84,56)(27,149,116,85,57)(28,150,117,86,58)(29,151,118,87,59)(30,152,119,88,60)(31,153,120,89,61)(32,154,121,90,62)(33,155,122,91,63)(34,156,123,92,64)(35,157,124,93,65), (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([(1,158,125,94,66),(2,159,126,95,67),(3,160,127,96,68),(4,161,128,97,69),(5,162,129,98,70),(6,163,130,99,36),(7,164,131,100,37),(8,165,132,101,38),(9,166,133,102,39),(10,167,134,103,40),(11,168,135,104,41),(12,169,136,105,42),(13,170,137,71,43),(14,171,138,72,44),(15,172,139,73,45),(16,173,140,74,46),(17,174,106,75,47),(18,175,107,76,48),(19,141,108,77,49),(20,142,109,78,50),(21,143,110,79,51),(22,144,111,80,52),(23,145,112,81,53),(24,146,113,82,54),(25,147,114,83,55),(26,148,115,84,56),(27,149,116,85,57),(28,150,117,86,58),(29,151,118,87,59),(30,152,119,88,60),(31,153,120,89,61),(32,154,121,90,62),(33,155,122,91,63),(34,156,123,92,64),(35,157,124,93,65)], [(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···5X7A···7F35A···35EN
order15···57···735···35
size11···11···11···1

175 irreducible representations

dim1111
type+
imageC1C5C7C35
kernelC5×C35C35C52C5
# reps1246144

Matrix representation of C5×C35 in GL2(𝔽71) generated by

250
054
,
180
043
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

Export

Subgroup lattice of C5×C35 in TeX

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