direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D47, D94.C2, C188⋊2C2, C2.1D94, Dic47⋊2C2, C94.2C22, C47⋊1(C2×C4), SmallGroup(376,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C47 — C4×D47 |
Generators and relations for C4×D47
G = < a,b,c | a4=b47=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D47
►On 188 points
Generators in S188
(1 160 81 124)(2 161 82 125)(3 162 83 126)(4 163 84 127)(5 164 85 128)(6 165 86 129)(7 166 87 130)(8 167 88 131)(9 168 89 132)(10 169 90 133)(11 170 91 134)(12 171 92 135)(13 172 93 136)(14 173 94 137)(15 174 48 138)(16 175 49 139)(17 176 50 140)(18 177 51 141)(19 178 52 95)(20 179 53 96)(21 180 54 97)(22 181 55 98)(23 182 56 99)(24 183 57 100)(25 184 58 101)(26 185 59 102)(27 186 60 103)(28 187 61 104)(29 188 62 105)(30 142 63 106)(31 143 64 107)(32 144 65 108)(33 145 66 109)(34 146 67 110)(35 147 68 111)(36 148 69 112)(37 149 70 113)(38 150 71 114)(39 151 72 115)(40 152 73 116)(41 153 74 117)(42 154 75 118)(43 155 76 119)(44 156 77 120)(45 157 78 121)(46 158 79 122)(47 159 80 123)
(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 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 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(95 105)(96 104)(97 103)(98 102)(99 101)(106 141)(107 140)(108 139)(109 138)(110 137)(111 136)(112 135)(113 134)(114 133)(115 132)(116 131)(117 130)(118 129)(119 128)(120 127)(121 126)(122 125)(123 124)(142 177)(143 176)(144 175)(145 174)(146 173)(147 172)(148 171)(149 170)(150 169)(151 168)(152 167)(153 166)(154 165)(155 164)(156 163)(157 162)(158 161)(159 160)(178 188)(179 187)(180 186)(181 185)(182 184)
G:=sub<Sym(188)| (1,160,81,124)(2,161,82,125)(3,162,83,126)(4,163,84,127)(5,164,85,128)(6,165,86,129)(7,166,87,130)(8,167,88,131)(9,168,89,132)(10,169,90,133)(11,170,91,134)(12,171,92,135)(13,172,93,136)(14,173,94,137)(15,174,48,138)(16,175,49,139)(17,176,50,140)(18,177,51,141)(19,178,52,95)(20,179,53,96)(21,180,54,97)(22,181,55,98)(23,182,56,99)(24,183,57,100)(25,184,58,101)(26,185,59,102)(27,186,60,103)(28,187,61,104)(29,188,62,105)(30,142,63,106)(31,143,64,107)(32,144,65,108)(33,145,66,109)(34,146,67,110)(35,147,68,111)(36,148,69,112)(37,149,70,113)(38,150,71,114)(39,151,72,115)(40,152,73,116)(41,153,74,117)(42,154,75,118)(43,155,76,119)(44,156,77,120)(45,157,78,121)(46,158,79,122)(47,159,80,123), (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), (1,47)(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)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(95,105)(96,104)(97,103)(98,102)(99,101)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)(142,177)(143,176)(144,175)(145,174)(146,173)(147,172)(148,171)(149,170)(150,169)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(178,188)(179,187)(180,186)(181,185)(182,184)>;
G:=Group( (1,160,81,124)(2,161,82,125)(3,162,83,126)(4,163,84,127)(5,164,85,128)(6,165,86,129)(7,166,87,130)(8,167,88,131)(9,168,89,132)(10,169,90,133)(11,170,91,134)(12,171,92,135)(13,172,93,136)(14,173,94,137)(15,174,48,138)(16,175,49,139)(17,176,50,140)(18,177,51,141)(19,178,52,95)(20,179,53,96)(21,180,54,97)(22,181,55,98)(23,182,56,99)(24,183,57,100)(25,184,58,101)(26,185,59,102)(27,186,60,103)(28,187,61,104)(29,188,62,105)(30,142,63,106)(31,143,64,107)(32,144,65,108)(33,145,66,109)(34,146,67,110)(35,147,68,111)(36,148,69,112)(37,149,70,113)(38,150,71,114)(39,151,72,115)(40,152,73,116)(41,153,74,117)(42,154,75,118)(43,155,76,119)(44,156,77,120)(45,157,78,121)(46,158,79,122)(47,159,80,123), (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), (1,47)(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)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(95,105)(96,104)(97,103)(98,102)(99,101)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)(142,177)(143,176)(144,175)(145,174)(146,173)(147,172)(148,171)(149,170)(150,169)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(178,188)(179,187)(180,186)(181,185)(182,184) );
G=PermutationGroup([[(1,160,81,124),(2,161,82,125),(3,162,83,126),(4,163,84,127),(5,164,85,128),(6,165,86,129),(7,166,87,130),(8,167,88,131),(9,168,89,132),(10,169,90,133),(11,170,91,134),(12,171,92,135),(13,172,93,136),(14,173,94,137),(15,174,48,138),(16,175,49,139),(17,176,50,140),(18,177,51,141),(19,178,52,95),(20,179,53,96),(21,180,54,97),(22,181,55,98),(23,182,56,99),(24,183,57,100),(25,184,58,101),(26,185,59,102),(27,186,60,103),(28,187,61,104),(29,188,62,105),(30,142,63,106),(31,143,64,107),(32,144,65,108),(33,145,66,109),(34,146,67,110),(35,147,68,111),(36,148,69,112),(37,149,70,113),(38,150,71,114),(39,151,72,115),(40,152,73,116),(41,153,74,117),(42,154,75,118),(43,155,76,119),(44,156,77,120),(45,157,78,121),(46,158,79,122),(47,159,80,123)], [(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)], [(1,47),(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),(48,66),(49,65),(50,64),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58),(67,94),(68,93),(69,92),(70,91),(71,90),(72,89),(73,88),(74,87),(75,86),(76,85),(77,84),(78,83),(79,82),(80,81),(95,105),(96,104),(97,103),(98,102),(99,101),(106,141),(107,140),(108,139),(109,138),(110,137),(111,136),(112,135),(113,134),(114,133),(115,132),(116,131),(117,130),(118,129),(119,128),(120,127),(121,126),(122,125),(123,124),(142,177),(143,176),(144,175),(145,174),(146,173),(147,172),(148,171),(149,170),(150,169),(151,168),(152,167),(153,166),(154,165),(155,164),(156,163),(157,162),(158,161),(159,160),(178,188),(179,187),(180,186),(181,185),(182,184)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 47A | ··· | 47W | 94A | ··· | 94W | 188A | ··· | 188AT |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 47 | ··· | 47 | 94 | ··· | 94 | 188 | ··· | 188 |
| size | 1 | 1 | 47 | 47 | 1 | 1 | 47 | 47 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D47 | D94 | C4×D47 |
| kernel | C4×D47 | Dic47 | C188 | D94 | D47 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 23 | 23 | 46 |
Matrix representation of C4×D47 ►in GL3(𝔽941) generated by
| 844 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 940 | 334 |
| 940 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(941))| [844,0,0,0,1,0,0,0,1],[1,0,0,0,0,940,0,1,334],[940,0,0,0,0,1,0,1,0] >;
C4×D47 in GAP, Magma, Sage, TeX
C_4\times D_{47} % in TeX
G:=Group("C4xD47"); // GroupNames label
G:=SmallGroup(376,4);
// by ID
G=gap.SmallGroup(376,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-47,21,5891]);
// Polycyclic
G:=Group<a,b,c|a^4=b^47=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export