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G = D5×C47order 470 = 2·5·47

Direct product of C47 and D5

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

Aliases: D5×C47, C5⋊C94, C2353C2, SmallGroup(470,1)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C47
C1C5C235 — D5×C47
C5 — D5×C47
C1C47

Generators and relations for D5×C47
 G = < a,b,c | a47=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C94

Smallest permutation representation of D5×C47
On 235 points
Generators in S235
(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 89 205 174 102)(2 90 206 175 103)(3 91 207 176 104)(4 92 208 177 105)(5 93 209 178 106)(6 94 210 179 107)(7 48 211 180 108)(8 49 212 181 109)(9 50 213 182 110)(10 51 214 183 111)(11 52 215 184 112)(12 53 216 185 113)(13 54 217 186 114)(14 55 218 187 115)(15 56 219 188 116)(16 57 220 142 117)(17 58 221 143 118)(18 59 222 144 119)(19 60 223 145 120)(20 61 224 146 121)(21 62 225 147 122)(22 63 226 148 123)(23 64 227 149 124)(24 65 228 150 125)(25 66 229 151 126)(26 67 230 152 127)(27 68 231 153 128)(28 69 232 154 129)(29 70 233 155 130)(30 71 234 156 131)(31 72 235 157 132)(32 73 189 158 133)(33 74 190 159 134)(34 75 191 160 135)(35 76 192 161 136)(36 77 193 162 137)(37 78 194 163 138)(38 79 195 164 139)(39 80 196 165 140)(40 81 197 166 141)(41 82 198 167 95)(42 83 199 168 96)(43 84 200 169 97)(44 85 201 170 98)(45 86 202 171 99)(46 87 203 172 100)(47 88 204 173 101)
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 113)(13 114)(14 115)(15 116)(16 117)(17 118)(18 119)(19 120)(20 121)(21 122)(22 123)(23 124)(24 125)(25 126)(26 127)(27 128)(28 129)(29 130)(30 131)(31 132)(32 133)(33 134)(34 135)(35 136)(36 137)(37 138)(38 139)(39 140)(40 141)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 180)(49 181)(50 182)(51 183)(52 184)(53 185)(54 186)(55 187)(56 188)(57 142)(58 143)(59 144)(60 145)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(84 169)(85 170)(86 171)(87 172)(88 173)(89 174)(90 175)(91 176)(92 177)(93 178)(94 179)

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

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

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

188 conjugacy classes

class 1  2 5A5B47A···47AT94A···94AT235A···235CN
order125547···4794···94235···235
size15221···15···52···2

188 irreducible representations

dim111122
type+++
imageC1C2C47C94D5D5×C47
kernelD5×C47C235D5C5C47C1
# reps114646292

Matrix representation of D5×C47 in GL2(𝔽941) generated by

340
034
,
9401
712228
,
9400
7121
G:=sub<GL(2,GF(941))| [34,0,0,34],[940,712,1,228],[940,712,0,1] >;

D5×C47 in GAP, Magma, Sage, TeX

D_5\times C_{47}
% in TeX

G:=Group("D5xC47");
// GroupNames label

G:=SmallGroup(470,1);
// by ID

G=gap.SmallGroup(470,1);
# by ID

G:=PCGroup([3,-2,-47,-5,3386]);
// Polycyclic

G:=Group<a,b,c|a^47=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×C47 in TeX

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