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G = C4×D59order 472 = 23·59

Direct product of C4 and D59

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

Aliases: C4×D59, C2362C2, D118.C2, C2.1D118, Dic592C2, C118.2C22, C591(C2×C4), SmallGroup(472,4)

Series: Derived Chief Lower central Upper central

C1C59 — C4×D59
C1C59C118D118 — C4×D59
C59 — C4×D59
C1C4

Generators and relations for C4×D59
 G = < a,b,c | a4=b59=c2=1, ab=ba, ac=ca, cbc=b-1 >

59C2
59C2
59C22
59C4
59C2×C4

Smallest permutation representation of C4×D59
On 236 points
Generators in S236
(1 198 78 175)(2 199 79 176)(3 200 80 177)(4 201 81 119)(5 202 82 120)(6 203 83 121)(7 204 84 122)(8 205 85 123)(9 206 86 124)(10 207 87 125)(11 208 88 126)(12 209 89 127)(13 210 90 128)(14 211 91 129)(15 212 92 130)(16 213 93 131)(17 214 94 132)(18 215 95 133)(19 216 96 134)(20 217 97 135)(21 218 98 136)(22 219 99 137)(23 220 100 138)(24 221 101 139)(25 222 102 140)(26 223 103 141)(27 224 104 142)(28 225 105 143)(29 226 106 144)(30 227 107 145)(31 228 108 146)(32 229 109 147)(33 230 110 148)(34 231 111 149)(35 232 112 150)(36 233 113 151)(37 234 114 152)(38 235 115 153)(39 236 116 154)(40 178 117 155)(41 179 118 156)(42 180 60 157)(43 181 61 158)(44 182 62 159)(45 183 63 160)(46 184 64 161)(47 185 65 162)(48 186 66 163)(49 187 67 164)(50 188 68 165)(51 189 69 166)(52 190 70 167)(53 191 71 168)(54 192 72 169)(55 193 73 170)(56 194 74 171)(57 195 75 172)(58 196 76 173)(59 197 77 174)
(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)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(60 95)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(96 118)(97 117)(98 116)(99 115)(100 114)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(119 171)(120 170)(121 169)(122 168)(123 167)(124 166)(125 165)(126 164)(127 163)(128 162)(129 161)(130 160)(131 159)(132 158)(133 157)(134 156)(135 155)(136 154)(137 153)(138 152)(139 151)(140 150)(141 149)(142 148)(143 147)(144 146)(172 177)(173 176)(174 175)(178 217)(179 216)(180 215)(181 214)(182 213)(183 212)(184 211)(185 210)(186 209)(187 208)(188 207)(189 206)(190 205)(191 204)(192 203)(193 202)(194 201)(195 200)(196 199)(197 198)(218 236)(219 235)(220 234)(221 233)(222 232)(223 231)(224 230)(225 229)(226 228)

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

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

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

124 conjugacy classes

class 1 2A2B2C4A4B4C4D59A···59AC118A···118AC236A···236BF
order1222444459···59118···118236···236
size1159591159592···22···22···2

124 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D59D118C4×D59
kernelC4×D59Dic59C236D118D59C4C2C1
# reps11114292958

Matrix representation of C4×D59 in GL3(𝔽709) generated by

61300
07080
00708
,
100
0271
0554572
,
70800
0572708
0334137
G:=sub<GL(3,GF(709))| [613,0,0,0,708,0,0,0,708],[1,0,0,0,27,554,0,1,572],[708,0,0,0,572,334,0,708,137] >;

C4×D59 in GAP, Magma, Sage, TeX

C_4\times D_{59}
% in TeX

G:=Group("C4xD59");
// GroupNames label

G:=SmallGroup(472,4);
// by ID

G=gap.SmallGroup(472,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-59,21,7427]);
// Polycyclic

G:=Group<a,b,c|a^4=b^59=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D59 in TeX

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