direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C47, C4⋊C94, C22⋊C94, C188⋊3C2, C94.6C22, (C2×C94)⋊1C2, C2.1(C2×C94), SmallGroup(376,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C47
G = < a,b,c | a47=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C47
►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 92 167 118)(2 93 168 119)(3 94 169 120)(4 48 170 121)(5 49 171 122)(6 50 172 123)(7 51 173 124)(8 52 174 125)(9 53 175 126)(10 54 176 127)(11 55 177 128)(12 56 178 129)(13 57 179 130)(14 58 180 131)(15 59 181 132)(16 60 182 133)(17 61 183 134)(18 62 184 135)(19 63 185 136)(20 64 186 137)(21 65 187 138)(22 66 188 139)(23 67 142 140)(24 68 143 141)(25 69 144 95)(26 70 145 96)(27 71 146 97)(28 72 147 98)(29 73 148 99)(30 74 149 100)(31 75 150 101)(32 76 151 102)(33 77 152 103)(34 78 153 104)(35 79 154 105)(36 80 155 106)(37 81 156 107)(38 82 157 108)(39 83 158 109)(40 84 159 110)(41 85 160 111)(42 86 161 112)(43 87 162 113)(44 88 163 114)(45 89 164 115)(46 90 165 116)(47 91 166 117)
(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 133)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 141)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)
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), (1,92,167,118)(2,93,168,119)(3,94,169,120)(4,48,170,121)(5,49,171,122)(6,50,172,123)(7,51,173,124)(8,52,174,125)(9,53,175,126)(10,54,176,127)(11,55,177,128)(12,56,178,129)(13,57,179,130)(14,58,180,131)(15,59,181,132)(16,60,182,133)(17,61,183,134)(18,62,184,135)(19,63,185,136)(20,64,186,137)(21,65,187,138)(22,66,188,139)(23,67,142,140)(24,68,143,141)(25,69,144,95)(26,70,145,96)(27,71,146,97)(28,72,147,98)(29,73,148,99)(30,74,149,100)(31,75,150,101)(32,76,151,102)(33,77,152,103)(34,78,153,104)(35,79,154,105)(36,80,155,106)(37,81,156,107)(38,82,157,108)(39,83,158,109)(40,84,159,110)(41,85,160,111)(42,86,161,112)(43,87,162,113)(44,88,163,114)(45,89,164,115)(46,90,165,116)(47,91,166,117), (48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)>;
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), (1,92,167,118)(2,93,168,119)(3,94,169,120)(4,48,170,121)(5,49,171,122)(6,50,172,123)(7,51,173,124)(8,52,174,125)(9,53,175,126)(10,54,176,127)(11,55,177,128)(12,56,178,129)(13,57,179,130)(14,58,180,131)(15,59,181,132)(16,60,182,133)(17,61,183,134)(18,62,184,135)(19,63,185,136)(20,64,186,137)(21,65,187,138)(22,66,188,139)(23,67,142,140)(24,68,143,141)(25,69,144,95)(26,70,145,96)(27,71,146,97)(28,72,147,98)(29,73,148,99)(30,74,149,100)(31,75,150,101)(32,76,151,102)(33,77,152,103)(34,78,153,104)(35,79,154,105)(36,80,155,106)(37,81,156,107)(38,82,157,108)(39,83,158,109)(40,84,159,110)(41,85,160,111)(42,86,161,112)(43,87,162,113)(44,88,163,114)(45,89,164,115)(46,90,165,116)(47,91,166,117), (48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,133)(61,134)(62,135)(63,136)(64,137)(65,138)(66,139)(67,140)(68,141)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120) );
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)], [(1,92,167,118),(2,93,168,119),(3,94,169,120),(4,48,170,121),(5,49,171,122),(6,50,172,123),(7,51,173,124),(8,52,174,125),(9,53,175,126),(10,54,176,127),(11,55,177,128),(12,56,178,129),(13,57,179,130),(14,58,180,131),(15,59,181,132),(16,60,182,133),(17,61,183,134),(18,62,184,135),(19,63,185,136),(20,64,186,137),(21,65,187,138),(22,66,188,139),(23,67,142,140),(24,68,143,141),(25,69,144,95),(26,70,145,96),(27,71,146,97),(28,72,147,98),(29,73,148,99),(30,74,149,100),(31,75,150,101),(32,76,151,102),(33,77,152,103),(34,78,153,104),(35,79,154,105),(36,80,155,106),(37,81,156,107),(38,82,157,108),(39,83,158,109),(40,84,159,110),(41,85,160,111),(42,86,161,112),(43,87,162,113),(44,88,163,114),(45,89,164,115),(46,90,165,116),(47,91,166,117)], [(48,121),(49,122),(50,123),(51,124),(52,125),(53,126),(54,127),(55,128),(56,129),(57,130),(58,131),(59,132),(60,133),(61,134),(62,135),(63,136),(64,137),(65,138),(66,139),(67,140),(68,141),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,107),(82,108),(83,109),(84,110),(85,111),(86,112),(87,113),(88,114),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120)]])
235 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 47A | ··· | 47AT | 94A | ··· | 94AT | 94AU | ··· | 94EH | 188A | ··· | 188AT |
| order | 1 | 2 | 2 | 2 | 4 | 47 | ··· | 47 | 94 | ··· | 94 | 94 | ··· | 94 | 188 | ··· | 188 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
235 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C47 | C94 | C94 | D4 | D4×C47 |
| kernel | D4×C47 | C188 | C2×C94 | D4 | C4 | C22 | C47 | C1 |
| # reps | 1 | 1 | 2 | 46 | 46 | 92 | 1 | 46 |
Matrix representation of D4×C47 ►in GL2(𝔽941) generated by
| 480 | 0 |
| 0 | 480 |
| 940 | 2 |
| 940 | 1 |
| 1 | 0 |
| 1 | 940 |
G:=sub<GL(2,GF(941))| [480,0,0,480],[940,940,2,1],[1,1,0,940] >;
D4×C47 in GAP, Magma, Sage, TeX
D_4\times C_{47} % in TeX
G:=Group("D4xC47"); // GroupNames label
G:=SmallGroup(376,9);
// by ID
G=gap.SmallGroup(376,9);
# by ID
G:=PCGroup([4,-2,-2,-47,-2,1521]);
// Polycyclic
G:=Group<a,b,c|a^47=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export