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G = C5×D47order 470 = 2·5·47

Direct product of C5 and D47

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

Aliases: C5×D47, C47⋊C10, C2352C2, SmallGroup(470,2)

Series: Derived Chief Lower central Upper central

C1C47 — C5×D47
C1C47C235 — C5×D47
C47 — C5×D47
C1C5

Generators and relations for C5×D47
 G = < a,b,c | a5=b47=c2=1, ab=ba, ac=ca, cbc=b-1 >

47C2
47C10

Smallest permutation representation of C5×D47
On 235 points
Generators in S235
(1 201 181 119 88)(2 202 182 120 89)(3 203 183 121 90)(4 204 184 122 91)(5 205 185 123 92)(6 206 186 124 93)(7 207 187 125 94)(8 208 188 126 48)(9 209 142 127 49)(10 210 143 128 50)(11 211 144 129 51)(12 212 145 130 52)(13 213 146 131 53)(14 214 147 132 54)(15 215 148 133 55)(16 216 149 134 56)(17 217 150 135 57)(18 218 151 136 58)(19 219 152 137 59)(20 220 153 138 60)(21 221 154 139 61)(22 222 155 140 62)(23 223 156 141 63)(24 224 157 95 64)(25 225 158 96 65)(26 226 159 97 66)(27 227 160 98 67)(28 228 161 99 68)(29 229 162 100 69)(30 230 163 101 70)(31 231 164 102 71)(32 232 165 103 72)(33 233 166 104 73)(34 234 167 105 74)(35 235 168 106 75)(36 189 169 107 76)(37 190 170 108 77)(38 191 171 109 78)(39 192 172 110 79)(40 193 173 111 80)(41 194 174 112 81)(42 195 175 113 82)(43 196 176 114 83)(44 197 177 115 84)(45 198 178 116 85)(46 199 179 117 86)(47 200 180 118 87)
(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)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)(96 141)(97 140)(98 139)(99 138)(100 137)(101 136)(102 135)(103 134)(104 133)(105 132)(106 131)(107 130)(108 129)(109 128)(110 127)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 120)(118 119)(142 172)(143 171)(144 170)(145 169)(146 168)(147 167)(148 166)(149 165)(150 164)(151 163)(152 162)(153 161)(154 160)(155 159)(156 158)(173 188)(174 187)(175 186)(176 185)(177 184)(178 183)(179 182)(180 181)(189 212)(190 211)(191 210)(192 209)(193 208)(194 207)(195 206)(196 205)(197 204)(198 203)(199 202)(200 201)(213 235)(214 234)(215 233)(216 232)(217 231)(218 230)(219 229)(220 228)(221 227)(222 226)(223 225)

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

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

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

125 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D47A···47W235A···235CN
order1255551010101047···47235···235
size1471111474747472···22···2

125 irreducible representations

dim111122
type+++
imageC1C2C5C10D47C5×D47
kernelC5×D47C235D47C47C5C1
# reps11442392

Matrix representation of C5×D47 in GL2(𝔽941) generated by

4120
0412
,
1411
9400
,
01
10
G:=sub<GL(2,GF(941))| [412,0,0,412],[141,940,1,0],[0,1,1,0] >;

C5×D47 in GAP, Magma, Sage, TeX

C_5\times D_{47}
% in TeX

G:=Group("C5xD47");
// GroupNames label

G:=SmallGroup(470,2);
// by ID

G=gap.SmallGroup(470,2);
# by ID

G:=PCGroup([3,-2,-5,-47,4142]);
// Polycyclic

G:=Group<a,b,c|a^5=b^47=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D47 in TeX

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