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G = D5×C49order 490 = 2·5·72

Direct product of C49 and D5

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

Aliases: D5×C49, C5⋊C98, C2453C2, C35.C14, C7.(C7×D5), (C7×D5).C7, SmallGroup(490,1)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C49
C1C5C35C245 — D5×C49
C5 — D5×C49
C1C49

Generators and relations for D5×C49
 G = < a,b,c | a49=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C14
5C98

Smallest permutation representation of D5×C49
On 245 points
Generators in S245
(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)
(1 165 200 139 68)(2 166 201 140 69)(3 167 202 141 70)(4 168 203 142 71)(5 169 204 143 72)(6 170 205 144 73)(7 171 206 145 74)(8 172 207 146 75)(9 173 208 147 76)(10 174 209 99 77)(11 175 210 100 78)(12 176 211 101 79)(13 177 212 102 80)(14 178 213 103 81)(15 179 214 104 82)(16 180 215 105 83)(17 181 216 106 84)(18 182 217 107 85)(19 183 218 108 86)(20 184 219 109 87)(21 185 220 110 88)(22 186 221 111 89)(23 187 222 112 90)(24 188 223 113 91)(25 189 224 114 92)(26 190 225 115 93)(27 191 226 116 94)(28 192 227 117 95)(29 193 228 118 96)(30 194 229 119 97)(31 195 230 120 98)(32 196 231 121 50)(33 148 232 122 51)(34 149 233 123 52)(35 150 234 124 53)(36 151 235 125 54)(37 152 236 126 55)(38 153 237 127 56)(39 154 238 128 57)(40 155 239 129 58)(41 156 240 130 59)(42 157 241 131 60)(43 158 242 132 61)(44 159 243 133 62)(45 160 244 134 63)(46 161 245 135 64)(47 162 197 136 65)(48 163 198 137 66)(49 164 199 138 67)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(99 174)(100 175)(101 176)(102 177)(103 178)(104 179)(105 180)(106 181)(107 182)(108 183)(109 184)(110 185)(111 186)(112 187)(113 188)(114 189)(115 190)(116 191)(117 192)(118 193)(119 194)(120 195)(121 196)(122 148)(123 149)(124 150)(125 151)(126 152)(127 153)(128 154)(129 155)(130 156)(131 157)(132 158)(133 159)(134 160)(135 161)(136 162)(137 163)(138 164)(139 165)(140 166)(141 167)(142 168)(143 169)(144 170)(145 171)(146 172)(147 173)

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

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

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

196 conjugacy classes

class 1  2 5A5B7A···7F14A···14F35A···35L49A···49AP98A···98AP245A···245CF
order12557···714···1435···3549···4998···98245···245
size15221···15···52···21···15···52···2

196 irreducible representations

dim111111222
type+++
imageC1C2C7C14C49C98D5C7×D5D5×C49
kernelD5×C49C245C7×D5C35D5C5C49C7C1
# reps1166424221284

Matrix representation of D5×C49 in GL2(𝔽491) generated by

510
051
,
4171
4900
,
01
10
G:=sub<GL(2,GF(491))| [51,0,0,51],[417,490,1,0],[0,1,1,0] >;

D5×C49 in GAP, Magma, Sage, TeX

D_5\times C_{49}
% in TeX

G:=Group("D5xC49");
// GroupNames label

G:=SmallGroup(490,1);
// by ID

G=gap.SmallGroup(490,1);
# by ID

G:=PCGroup([4,-2,-7,-7,-5,61,6275]);
// Polycyclic

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

Export

Subgroup lattice of D5×C49 in TeX

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