direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D47, C47⋊C23, C94⋊C22, (C2×C94)⋊3C2, SmallGroup(376,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C47 — C22×D47 |
Generators and relations for C22×D47
G = < a,b,c,d | a2=b2=c47=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of C22×D47
►On 188 points
Generators in S188
(1 146)(2 147)(3 148)(4 149)(5 150)(6 151)(7 152)(8 153)(9 154)(10 155)(11 156)(12 157)(13 158)(14 159)(15 160)(16 161)(17 162)(18 163)(19 164)(20 165)(21 166)(22 167)(23 168)(24 169)(25 170)(26 171)(27 172)(28 173)(29 174)(30 175)(31 176)(32 177)(33 178)(34 179)(35 180)(36 181)(37 182)(38 183)(39 184)(40 185)(41 186)(42 187)(43 188)(44 142)(45 143)(46 144)(47 145)(48 139)(49 140)(50 141)(51 95)(52 96)(53 97)(54 98)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 121)(78 122)(79 123)(80 124)(81 125)(82 126)(83 127)(84 128)(85 129)(86 130)(87 131)(88 132)(89 133)(90 134)(91 135)(92 136)(93 137)(94 138)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(95 180)(96 181)(97 182)(98 183)(99 184)(100 185)(101 186)(102 187)(103 188)(104 142)(105 143)(106 144)(107 145)(108 146)(109 147)(110 148)(111 149)(112 150)(113 151)(114 152)(115 153)(116 154)(117 155)(118 156)(119 157)(120 158)(121 159)(122 160)(123 161)(124 162)(125 163)(126 164)(127 165)(128 166)(129 167)(130 168)(131 169)(132 170)(133 171)(134 172)(135 173)(136 174)(137 175)(138 176)(139 177)(140 178)(141 179)
(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)
(1 145)(2 144)(3 143)(4 142)(5 188)(6 187)(7 186)(8 185)(9 184)(10 183)(11 182)(12 181)(13 180)(14 179)(15 178)(16 177)(17 176)(18 175)(19 174)(20 173)(21 172)(22 171)(23 170)(24 169)(25 168)(26 167)(27 166)(28 165)(29 164)(30 163)(31 162)(32 161)(33 160)(34 159)(35 158)(36 157)(37 156)(38 155)(39 154)(40 153)(41 152)(42 151)(43 150)(44 149)(45 148)(46 147)(47 146)(48 123)(49 122)(50 121)(51 120)(52 119)(53 118)(54 117)(55 116)(56 115)(57 114)(58 113)(59 112)(60 111)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 141)(78 140)(79 139)(80 138)(81 137)(82 136)(83 135)(84 134)(85 133)(86 132)(87 131)(88 130)(89 129)(90 128)(91 127)(92 126)(93 125)(94 124)
G:=sub<Sym(188)| (1,146)(2,147)(3,148)(4,149)(5,150)(6,151)(7,152)(8,153)(9,154)(10,155)(11,156)(12,157)(13,158)(14,159)(15,160)(16,161)(17,162)(18,163)(19,164)(20,165)(21,166)(22,167)(23,168)(24,169)(25,170)(26,171)(27,172)(28,173)(29,174)(30,175)(31,176)(32,177)(33,178)(34,179)(35,180)(36,181)(37,182)(38,183)(39,184)(40,185)(41,186)(42,187)(43,188)(44,142)(45,143)(46,144)(47,145)(48,139)(49,140)(50,141)(51,95)(52,96)(53,97)(54,98)(55,99)(56,100)(57,101)(58,102)(59,103)(60,104)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,121)(78,122)(79,123)(80,124)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(95,180)(96,181)(97,182)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148)(111,149)(112,150)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158)(121,159)(122,160)(123,161)(124,162)(125,163)(126,164)(127,165)(128,166)(129,167)(130,168)(131,169)(132,170)(133,171)(134,172)(135,173)(136,174)(137,175)(138,176)(139,177)(140,178)(141,179), (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), (1,145)(2,144)(3,143)(4,142)(5,188)(6,187)(7,186)(8,185)(9,184)(10,183)(11,182)(12,181)(13,180)(14,179)(15,178)(16,177)(17,176)(18,175)(19,174)(20,173)(21,172)(22,171)(23,170)(24,169)(25,168)(26,167)(27,166)(28,165)(29,164)(30,163)(31,162)(32,161)(33,160)(34,159)(35,158)(36,157)(37,156)(38,155)(39,154)(40,153)(41,152)(42,151)(43,150)(44,149)(45,148)(46,147)(47,146)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,141)(78,140)(79,139)(80,138)(81,137)(82,136)(83,135)(84,134)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124)>;
G:=Group( (1,146)(2,147)(3,148)(4,149)(5,150)(6,151)(7,152)(8,153)(9,154)(10,155)(11,156)(12,157)(13,158)(14,159)(15,160)(16,161)(17,162)(18,163)(19,164)(20,165)(21,166)(22,167)(23,168)(24,169)(25,170)(26,171)(27,172)(28,173)(29,174)(30,175)(31,176)(32,177)(33,178)(34,179)(35,180)(36,181)(37,182)(38,183)(39,184)(40,185)(41,186)(42,187)(43,188)(44,142)(45,143)(46,144)(47,145)(48,139)(49,140)(50,141)(51,95)(52,96)(53,97)(54,98)(55,99)(56,100)(57,101)(58,102)(59,103)(60,104)(61,105)(62,106)(63,107)(64,108)(65,109)(66,110)(67,111)(68,112)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,121)(78,122)(79,123)(80,124)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(95,180)(96,181)(97,182)(98,183)(99,184)(100,185)(101,186)(102,187)(103,188)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148)(111,149)(112,150)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158)(121,159)(122,160)(123,161)(124,162)(125,163)(126,164)(127,165)(128,166)(129,167)(130,168)(131,169)(132,170)(133,171)(134,172)(135,173)(136,174)(137,175)(138,176)(139,177)(140,178)(141,179), (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), (1,145)(2,144)(3,143)(4,142)(5,188)(6,187)(7,186)(8,185)(9,184)(10,183)(11,182)(12,181)(13,180)(14,179)(15,178)(16,177)(17,176)(18,175)(19,174)(20,173)(21,172)(22,171)(23,170)(24,169)(25,168)(26,167)(27,166)(28,165)(29,164)(30,163)(31,162)(32,161)(33,160)(34,159)(35,158)(36,157)(37,156)(38,155)(39,154)(40,153)(41,152)(42,151)(43,150)(44,149)(45,148)(46,147)(47,146)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,141)(78,140)(79,139)(80,138)(81,137)(82,136)(83,135)(84,134)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124) );
G=PermutationGroup([[(1,146),(2,147),(3,148),(4,149),(5,150),(6,151),(7,152),(8,153),(9,154),(10,155),(11,156),(12,157),(13,158),(14,159),(15,160),(16,161),(17,162),(18,163),(19,164),(20,165),(21,166),(22,167),(23,168),(24,169),(25,170),(26,171),(27,172),(28,173),(29,174),(30,175),(31,176),(32,177),(33,178),(34,179),(35,180),(36,181),(37,182),(38,183),(39,184),(40,185),(41,186),(42,187),(43,188),(44,142),(45,143),(46,144),(47,145),(48,139),(49,140),(50,141),(51,95),(52,96),(53,97),(54,98),(55,99),(56,100),(57,101),(58,102),(59,103),(60,104),(61,105),(62,106),(63,107),(64,108),(65,109),(66,110),(67,111),(68,112),(69,113),(70,114),(71,115),(72,116),(73,117),(74,118),(75,119),(76,120),(77,121),(78,122),(79,123),(80,124),(81,125),(82,126),(83,127),(84,128),(85,129),(86,130),(87,131),(88,132),(89,133),(90,134),(91,135),(92,136),(93,137),(94,138)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,48),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(95,180),(96,181),(97,182),(98,183),(99,184),(100,185),(101,186),(102,187),(103,188),(104,142),(105,143),(106,144),(107,145),(108,146),(109,147),(110,148),(111,149),(112,150),(113,151),(114,152),(115,153),(116,154),(117,155),(118,156),(119,157),(120,158),(121,159),(122,160),(123,161),(124,162),(125,163),(126,164),(127,165),(128,166),(129,167),(130,168),(131,169),(132,170),(133,171),(134,172),(135,173),(136,174),(137,175),(138,176),(139,177),(140,178),(141,179)], [(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)], [(1,145),(2,144),(3,143),(4,142),(5,188),(6,187),(7,186),(8,185),(9,184),(10,183),(11,182),(12,181),(13,180),(14,179),(15,178),(16,177),(17,176),(18,175),(19,174),(20,173),(21,172),(22,171),(23,170),(24,169),(25,168),(26,167),(27,166),(28,165),(29,164),(30,163),(31,162),(32,161),(33,160),(34,159),(35,158),(36,157),(37,156),(38,155),(39,154),(40,153),(41,152),(42,151),(43,150),(44,149),(45,148),(46,147),(47,146),(48,123),(49,122),(50,121),(51,120),(52,119),(53,118),(54,117),(55,116),(56,115),(57,114),(58,113),(59,112),(60,111),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,100),(72,99),(73,98),(74,97),(75,96),(76,95),(77,141),(78,140),(79,139),(80,138),(81,137),(82,136),(83,135),(84,134),(85,133),(86,132),(87,131),(88,130),(89,129),(90,128),(91,127),(92,126),(93,125),(94,124)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 47A | ··· | 47W | 94A | ··· | 94BQ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 47 | ··· | 47 | 94 | ··· | 94 |
| size | 1 | 1 | 1 | 1 | 47 | 47 | 47 | 47 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D47 | D94 |
| kernel | C22×D47 | D94 | C2×C94 | C22 | C2 |
| # reps | 1 | 6 | 1 | 23 | 69 |
Matrix representation of C22×D47 ►in GL4(𝔽283) generated by
| 282 | 0 | 0 | 0 |
| 0 | 282 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 282 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 282 | 99 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(283))| [282,0,0,0,0,282,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,282,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,282,0,0,1,99],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
C22×D47 in GAP, Magma, Sage, TeX
C_2^2\times D_{47} % in TeX
G:=Group("C2^2xD47"); // GroupNames label
G:=SmallGroup(376,11);
// by ID
G=gap.SmallGroup(376,11);
# by ID
G:=PCGroup([4,-2,-2,-2,-47,5891]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^47=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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