metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C47⋊2D4, C22⋊D47, D94⋊2C2, Dic47⋊C2, C2.5D94, C94.5C22, (C2×C94)⋊2C2, SmallGroup(376,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C47⋊D4
G = < a,b,c | a47=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C47⋊D4
►On 188 points
Generators in S188
(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)
(1 120 73 154)(2 119 74 153)(3 118 75 152)(4 117 76 151)(5 116 77 150)(6 115 78 149)(7 114 79 148)(8 113 80 147)(9 112 81 146)(10 111 82 145)(11 110 83 144)(12 109 84 143)(13 108 85 142)(14 107 86 188)(15 106 87 187)(16 105 88 186)(17 104 89 185)(18 103 90 184)(19 102 91 183)(20 101 92 182)(21 100 93 181)(22 99 94 180)(23 98 48 179)(24 97 49 178)(25 96 50 177)(26 95 51 176)(27 141 52 175)(28 140 53 174)(29 139 54 173)(30 138 55 172)(31 137 56 171)(32 136 57 170)(33 135 58 169)(34 134 59 168)(35 133 60 167)(36 132 61 166)(37 131 62 165)(38 130 63 164)(39 129 64 163)(40 128 65 162)(41 127 66 161)(42 126 67 160)(43 125 68 159)(44 124 69 158)(45 123 70 157)(46 122 71 156)(47 121 72 155)
(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(48 51)(49 50)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)(61 85)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(95 179)(96 178)(97 177)(98 176)(99 175)(100 174)(101 173)(102 172)(103 171)(104 170)(105 169)(106 168)(107 167)(108 166)(109 165)(110 164)(111 163)(112 162)(113 161)(114 160)(115 159)(116 158)(117 157)(118 156)(119 155)(120 154)(121 153)(122 152)(123 151)(124 150)(125 149)(126 148)(127 147)(128 146)(129 145)(130 144)(131 143)(132 142)(133 188)(134 187)(135 186)(136 185)(137 184)(138 183)(139 182)(140 181)(141 180)
G:=sub<Sym(188)| (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), (1,120,73,154)(2,119,74,153)(3,118,75,152)(4,117,76,151)(5,116,77,150)(6,115,78,149)(7,114,79,148)(8,113,80,147)(9,112,81,146)(10,111,82,145)(11,110,83,144)(12,109,84,143)(13,108,85,142)(14,107,86,188)(15,106,87,187)(16,105,88,186)(17,104,89,185)(18,103,90,184)(19,102,91,183)(20,101,92,182)(21,100,93,181)(22,99,94,180)(23,98,48,179)(24,97,49,178)(25,96,50,177)(26,95,51,176)(27,141,52,175)(28,140,53,174)(29,139,54,173)(30,138,55,172)(31,137,56,171)(32,136,57,170)(33,135,58,169)(34,134,59,168)(35,133,60,167)(36,132,61,166)(37,131,62,165)(38,130,63,164)(39,129,64,163)(40,128,65,162)(41,127,66,161)(42,126,67,160)(43,125,68,159)(44,124,69,158)(45,123,70,157)(46,122,71,156)(47,121,72,155), (2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(48,51)(49,50)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(95,179)(96,178)(97,177)(98,176)(99,175)(100,174)(101,173)(102,172)(103,171)(104,170)(105,169)(106,168)(107,167)(108,166)(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,154)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,188)(134,187)(135,186)(136,185)(137,184)(138,183)(139,182)(140,181)(141,180)>;
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), (1,120,73,154)(2,119,74,153)(3,118,75,152)(4,117,76,151)(5,116,77,150)(6,115,78,149)(7,114,79,148)(8,113,80,147)(9,112,81,146)(10,111,82,145)(11,110,83,144)(12,109,84,143)(13,108,85,142)(14,107,86,188)(15,106,87,187)(16,105,88,186)(17,104,89,185)(18,103,90,184)(19,102,91,183)(20,101,92,182)(21,100,93,181)(22,99,94,180)(23,98,48,179)(24,97,49,178)(25,96,50,177)(26,95,51,176)(27,141,52,175)(28,140,53,174)(29,139,54,173)(30,138,55,172)(31,137,56,171)(32,136,57,170)(33,135,58,169)(34,134,59,168)(35,133,60,167)(36,132,61,166)(37,131,62,165)(38,130,63,164)(39,129,64,163)(40,128,65,162)(41,127,66,161)(42,126,67,160)(43,125,68,159)(44,124,69,158)(45,123,70,157)(46,122,71,156)(47,121,72,155), (2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(48,51)(49,50)(52,94)(53,93)(54,92)(55,91)(56,90)(57,89)(58,88)(59,87)(60,86)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(95,179)(96,178)(97,177)(98,176)(99,175)(100,174)(101,173)(102,172)(103,171)(104,170)(105,169)(106,168)(107,167)(108,166)(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,154)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,188)(134,187)(135,186)(136,185)(137,184)(138,183)(139,182)(140,181)(141,180) );
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)], [(1,120,73,154),(2,119,74,153),(3,118,75,152),(4,117,76,151),(5,116,77,150),(6,115,78,149),(7,114,79,148),(8,113,80,147),(9,112,81,146),(10,111,82,145),(11,110,83,144),(12,109,84,143),(13,108,85,142),(14,107,86,188),(15,106,87,187),(16,105,88,186),(17,104,89,185),(18,103,90,184),(19,102,91,183),(20,101,92,182),(21,100,93,181),(22,99,94,180),(23,98,48,179),(24,97,49,178),(25,96,50,177),(26,95,51,176),(27,141,52,175),(28,140,53,174),(29,139,54,173),(30,138,55,172),(31,137,56,171),(32,136,57,170),(33,135,58,169),(34,134,59,168),(35,133,60,167),(36,132,61,166),(37,131,62,165),(38,130,63,164),(39,129,64,163),(40,128,65,162),(41,127,66,161),(42,126,67,160),(43,125,68,159),(44,124,69,158),(45,123,70,157),(46,122,71,156),(47,121,72,155)], [(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25),(48,51),(49,50),(52,94),(53,93),(54,92),(55,91),(56,90),(57,89),(58,88),(59,87),(60,86),(61,85),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(95,179),(96,178),(97,177),(98,176),(99,175),(100,174),(101,173),(102,172),(103,171),(104,170),(105,169),(106,168),(107,167),(108,166),(109,165),(110,164),(111,163),(112,162),(113,161),(114,160),(115,159),(116,158),(117,157),(118,156),(119,155),(120,154),(121,153),(122,152),(123,151),(124,150),(125,149),(126,148),(127,147),(128,146),(129,145),(130,144),(131,143),(132,142),(133,188),(134,187),(135,186),(136,185),(137,184),(138,183),(139,182),(140,181),(141,180)]])
97 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 47A | ··· | 47W | 94A | ··· | 94BQ |
| order | 1 | 2 | 2 | 2 | 4 | 47 | ··· | 47 | 94 | ··· | 94 |
| size | 1 | 1 | 2 | 94 | 94 | 2 | ··· | 2 | 2 | ··· | 2 |
97 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D47 | D94 | C47⋊D4 |
| kernel | C47⋊D4 | Dic47 | D94 | C2×C94 | C47 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 23 | 23 | 46 |
Matrix representation of C47⋊D4 ►in GL2(𝔽941) generated by
| 0 | 1 |
| 940 | 45 |
| 841 | 109 |
| 314 | 100 |
| 1 | 0 |
| 45 | 940 |
G:=sub<GL(2,GF(941))| [0,940,1,45],[841,314,109,100],[1,45,0,940] >;
C47⋊D4 in GAP, Magma, Sage, TeX
C_{47}\rtimes D_4 % in TeX
G:=Group("C47:D4"); // GroupNames label
G:=SmallGroup(376,7);
// by ID
G=gap.SmallGroup(376,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-47,49,5891]);
// Polycyclic
G:=Group<a,b,c|a^47=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export