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G = C7×C35order 245 = 5·72

Abelian group of type [7,35]

direct product, abelian, monomial, 7-elementary

Aliases: C7×C35, SmallGroup(245,2)

Series: Derived Chief Lower central Upper central

C1 — C7×C35
C1C7C72 — C7×C35
C1 — C7×C35
C1 — C7×C35

Generators and relations for C7×C35
 G = < a,b | a7=b35=1, ab=ba >


Smallest permutation representation of C7×C35
Regular action on 245 points
Generators in S245
(1 86 223 158 208 46 119)(2 87 224 159 209 47 120)(3 88 225 160 210 48 121)(4 89 226 161 176 49 122)(5 90 227 162 177 50 123)(6 91 228 163 178 51 124)(7 92 229 164 179 52 125)(8 93 230 165 180 53 126)(9 94 231 166 181 54 127)(10 95 232 167 182 55 128)(11 96 233 168 183 56 129)(12 97 234 169 184 57 130)(13 98 235 170 185 58 131)(14 99 236 171 186 59 132)(15 100 237 172 187 60 133)(16 101 238 173 188 61 134)(17 102 239 174 189 62 135)(18 103 240 175 190 63 136)(19 104 241 141 191 64 137)(20 105 242 142 192 65 138)(21 71 243 143 193 66 139)(22 72 244 144 194 67 140)(23 73 245 145 195 68 106)(24 74 211 146 196 69 107)(25 75 212 147 197 70 108)(26 76 213 148 198 36 109)(27 77 214 149 199 37 110)(28 78 215 150 200 38 111)(29 79 216 151 201 39 112)(30 80 217 152 202 40 113)(31 81 218 153 203 41 114)(32 82 219 154 204 42 115)(33 83 220 155 205 43 116)(34 84 221 156 206 44 117)(35 85 222 157 207 45 118)
(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)(176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245)

G:=sub<Sym(245)| (1,86,223,158,208,46,119)(2,87,224,159,209,47,120)(3,88,225,160,210,48,121)(4,89,226,161,176,49,122)(5,90,227,162,177,50,123)(6,91,228,163,178,51,124)(7,92,229,164,179,52,125)(8,93,230,165,180,53,126)(9,94,231,166,181,54,127)(10,95,232,167,182,55,128)(11,96,233,168,183,56,129)(12,97,234,169,184,57,130)(13,98,235,170,185,58,131)(14,99,236,171,186,59,132)(15,100,237,172,187,60,133)(16,101,238,173,188,61,134)(17,102,239,174,189,62,135)(18,103,240,175,190,63,136)(19,104,241,141,191,64,137)(20,105,242,142,192,65,138)(21,71,243,143,193,66,139)(22,72,244,144,194,67,140)(23,73,245,145,195,68,106)(24,74,211,146,196,69,107)(25,75,212,147,197,70,108)(26,76,213,148,198,36,109)(27,77,214,149,199,37,110)(28,78,215,150,200,38,111)(29,79,216,151,201,39,112)(30,80,217,152,202,40,113)(31,81,218,153,203,41,114)(32,82,219,154,204,42,115)(33,83,220,155,205,43,116)(34,84,221,156,206,44,117)(35,85,222,157,207,45,118), (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)(176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245)>;

G:=Group( (1,86,223,158,208,46,119)(2,87,224,159,209,47,120)(3,88,225,160,210,48,121)(4,89,226,161,176,49,122)(5,90,227,162,177,50,123)(6,91,228,163,178,51,124)(7,92,229,164,179,52,125)(8,93,230,165,180,53,126)(9,94,231,166,181,54,127)(10,95,232,167,182,55,128)(11,96,233,168,183,56,129)(12,97,234,169,184,57,130)(13,98,235,170,185,58,131)(14,99,236,171,186,59,132)(15,100,237,172,187,60,133)(16,101,238,173,188,61,134)(17,102,239,174,189,62,135)(18,103,240,175,190,63,136)(19,104,241,141,191,64,137)(20,105,242,142,192,65,138)(21,71,243,143,193,66,139)(22,72,244,144,194,67,140)(23,73,245,145,195,68,106)(24,74,211,146,196,69,107)(25,75,212,147,197,70,108)(26,76,213,148,198,36,109)(27,77,214,149,199,37,110)(28,78,215,150,200,38,111)(29,79,216,151,201,39,112)(30,80,217,152,202,40,113)(31,81,218,153,203,41,114)(32,82,219,154,204,42,115)(33,83,220,155,205,43,116)(34,84,221,156,206,44,117)(35,85,222,157,207,45,118), (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)(176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245) );

G=PermutationGroup([(1,86,223,158,208,46,119),(2,87,224,159,209,47,120),(3,88,225,160,210,48,121),(4,89,226,161,176,49,122),(5,90,227,162,177,50,123),(6,91,228,163,178,51,124),(7,92,229,164,179,52,125),(8,93,230,165,180,53,126),(9,94,231,166,181,54,127),(10,95,232,167,182,55,128),(11,96,233,168,183,56,129),(12,97,234,169,184,57,130),(13,98,235,170,185,58,131),(14,99,236,171,186,59,132),(15,100,237,172,187,60,133),(16,101,238,173,188,61,134),(17,102,239,174,189,62,135),(18,103,240,175,190,63,136),(19,104,241,141,191,64,137),(20,105,242,142,192,65,138),(21,71,243,143,193,66,139),(22,72,244,144,194,67,140),(23,73,245,145,195,68,106),(24,74,211,146,196,69,107),(25,75,212,147,197,70,108),(26,76,213,148,198,36,109),(27,77,214,149,199,37,110),(28,78,215,150,200,38,111),(29,79,216,151,201,39,112),(30,80,217,152,202,40,113),(31,81,218,153,203,41,114),(32,82,219,154,204,42,115),(33,83,220,155,205,43,116),(34,84,221,156,206,44,117),(35,85,222,157,207,45,118)], [(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),(176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245)])

C7×C35 is a maximal subgroup of   C7⋊D35

245 conjugacy classes

class 1 5A5B5C5D7A···7AV35A···35GJ
order155557···735···35
size111111···11···1

245 irreducible representations

dim1111
type+
imageC1C5C7C35
kernelC7×C35C72C35C7
# reps1448192

Matrix representation of C7×C35 in GL2(𝔽71) generated by

300
048
,
600
08
G:=sub<GL(2,GF(71))| [30,0,0,48],[60,0,0,8] >;

C7×C35 in GAP, Magma, Sage, TeX

C_7\times C_{35}
% in TeX

G:=Group("C7xC35");
// GroupNames label

G:=SmallGroup(245,2);
// by ID

G=gap.SmallGroup(245,2);
# by ID

G:=PCGroup([3,-5,-7,-7]);
// Polycyclic

G:=Group<a,b|a^7=b^35=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C7×C35 in TeX

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