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G = C4×D61order 488 = 23·61

Direct product of C4 and D61

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

Aliases: C4×D61, C2442C2, C2.1D122, Dic612C2, D122.2C2, C122.2C22, C612(C2×C4), SmallGroup(488,5)

Series: Derived Chief Lower central Upper central

C1C61 — C4×D61
C1C61C122D122 — C4×D61
C61 — C4×D61
C1C4

Generators and relations for C4×D61
 G = < a,b,c | a4=b61=c2=1, ab=ba, ac=ca, cbc=b-1 >

61C2
61C2
61C22
61C4
61C2×C4

Smallest permutation representation of C4×D61
On 244 points
Generators in S244
(1 218 75 176)(2 219 76 177)(3 220 77 178)(4 221 78 179)(5 222 79 180)(6 223 80 181)(7 224 81 182)(8 225 82 183)(9 226 83 123)(10 227 84 124)(11 228 85 125)(12 229 86 126)(13 230 87 127)(14 231 88 128)(15 232 89 129)(16 233 90 130)(17 234 91 131)(18 235 92 132)(19 236 93 133)(20 237 94 134)(21 238 95 135)(22 239 96 136)(23 240 97 137)(24 241 98 138)(25 242 99 139)(26 243 100 140)(27 244 101 141)(28 184 102 142)(29 185 103 143)(30 186 104 144)(31 187 105 145)(32 188 106 146)(33 189 107 147)(34 190 108 148)(35 191 109 149)(36 192 110 150)(37 193 111 151)(38 194 112 152)(39 195 113 153)(40 196 114 154)(41 197 115 155)(42 198 116 156)(43 199 117 157)(44 200 118 158)(45 201 119 159)(46 202 120 160)(47 203 121 161)(48 204 122 162)(49 205 62 163)(50 206 63 164)(51 207 64 165)(52 208 65 166)(53 209 66 167)(54 210 67 168)(55 211 68 169)(56 212 69 170)(57 213 70 171)(58 214 71 172)(59 215 72 173)(60 216 73 174)(61 217 74 175)
(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)
(1 61)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(88 122)(89 121)(90 120)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(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)(168 183)(169 182)(170 181)(171 180)(172 179)(173 178)(174 177)(175 176)(184 190)(185 189)(186 188)(191 244)(192 243)(193 242)(194 241)(195 240)(196 239)(197 238)(198 237)(199 236)(200 235)(201 234)(202 233)(203 232)(204 231)(205 230)(206 229)(207 228)(208 227)(209 226)(210 225)(211 224)(212 223)(213 222)(214 221)(215 220)(216 219)(217 218)

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

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

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

128 conjugacy classes

class 1 2A2B2C4A4B4C4D61A···61AD122A···122AD244A···244BH
order1222444461···61122···122244···244
size1161611161612···22···22···2

128 irreducible representations

dim11111222
type++++++
imageC1C2C2C2C4D61D122C4×D61
kernelC4×D61Dic61C244D122D61C4C2C1
# reps11114303060

Matrix representation of C4×D61 in GL2(𝔽733) generated by

3800
0380
,
65227
732308
,
440340
515293
G:=sub<GL(2,GF(733))| [380,0,0,380],[652,732,27,308],[440,515,340,293] >;

C4×D61 in GAP, Magma, Sage, TeX

C_4\times D_{61}
% in TeX

G:=Group("C4xD61");
// GroupNames label

G:=SmallGroup(488,5);
// by ID

G=gap.SmallGroup(488,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-61,21,7683]);
// Polycyclic

G:=Group<a,b,c|a^4=b^61=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C4×D61 in TeX

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