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G = C5×D49order 490 = 2·5·72

Direct product of C5 and D49

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C5×D49, C49⋊C10, C2452C2, C35.2D7, C7.(C5×D7), SmallGroup(490,2)

Series: Derived Chief Lower central Upper central

C1C49 — C5×D49
C1C7C49C245 — C5×D49
C49 — C5×D49
C1C5

Generators and relations for C5×D49
 G = < a,b,c | a5=b49=c2=1, ab=ba, ac=ca, cbc=b-1 >

49C2
49C10
7D7
7C5×D7

Smallest permutation representation of C5×D49
On 245 points
Generators in S245
(1 233 189 147 63)(2 234 190 99 64)(3 235 191 100 65)(4 236 192 101 66)(5 237 193 102 67)(6 238 194 103 68)(7 239 195 104 69)(8 240 196 105 70)(9 241 148 106 71)(10 242 149 107 72)(11 243 150 108 73)(12 244 151 109 74)(13 245 152 110 75)(14 197 153 111 76)(15 198 154 112 77)(16 199 155 113 78)(17 200 156 114 79)(18 201 157 115 80)(19 202 158 116 81)(20 203 159 117 82)(21 204 160 118 83)(22 205 161 119 84)(23 206 162 120 85)(24 207 163 121 86)(25 208 164 122 87)(26 209 165 123 88)(27 210 166 124 89)(28 211 167 125 90)(29 212 168 126 91)(30 213 169 127 92)(31 214 170 128 93)(32 215 171 129 94)(33 216 172 130 95)(34 217 173 131 96)(35 218 174 132 97)(36 219 175 133 98)(37 220 176 134 50)(38 221 177 135 51)(39 222 178 136 52)(40 223 179 137 53)(41 224 180 138 54)(42 225 181 139 55)(43 226 182 140 56)(44 227 183 141 57)(45 228 184 142 58)(46 229 185 143 59)(47 230 186 144 60)(48 231 187 145 61)(49 232 188 146 62)
(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)
(1 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(50 75)(51 74)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(76 98)(77 97)(78 96)(79 95)(80 94)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(99 145)(100 144)(101 143)(102 142)(103 141)(104 140)(105 139)(106 138)(107 137)(108 136)(109 135)(110 134)(111 133)(112 132)(113 131)(114 130)(115 129)(116 128)(117 127)(118 126)(119 125)(120 124)(121 123)(146 147)(148 180)(149 179)(150 178)(151 177)(152 176)(153 175)(154 174)(155 173)(156 172)(157 171)(158 170)(159 169)(160 168)(161 167)(162 166)(163 165)(181 196)(182 195)(183 194)(184 193)(185 192)(186 191)(187 190)(188 189)(197 219)(198 218)(199 217)(200 216)(201 215)(202 214)(203 213)(204 212)(205 211)(206 210)(207 209)(220 245)(221 244)(222 243)(223 242)(224 241)(225 240)(226 239)(227 238)(228 237)(229 236)(230 235)(231 234)(232 233)

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

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

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

130 conjugacy classes

class 1  2 5A5B5C5D7A7B7C10A10B10C10D35A···35L49A···49U245A···245CF
order1255557771010101035···3549···49245···245
size1491111222494949492···22···22···2

130 irreducible representations

dim11112222
type++++
imageC1C2C5C10D7C5×D7D49C5×D49
kernelC5×D49C245D49C49C35C7C5C1
# reps11443122184

Matrix representation of C5×D49 in GL2(𝔽491) generated by

1010
0101
,
373477
14114
,
127332
472364
G:=sub<GL(2,GF(491))| [101,0,0,101],[373,14,477,114],[127,472,332,364] >;

C5×D49 in GAP, Magma, Sage, TeX

C_5\times D_{49}
% in TeX

G:=Group("C5xD49");
// GroupNames label

G:=SmallGroup(490,2);
// by ID

G=gap.SmallGroup(490,2);
# by ID

G:=PCGroup([4,-2,-5,-7,-7,1562,514,6723]);
// Polycyclic

G:=Group<a,b,c|a^5=b^49=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D49 in TeX

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