metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2D19, C4.5D38, Dic38⋊3C2, C76.5C22, C38.6C23, C22.1D38, D38.2C22, Dic19.4C22, (C4×D19)⋊2C2, (D4×C19)⋊3C2, C19⋊2(C4○D4), C19⋊D4⋊2C2, (C2×C38).C22, (C2×Dic19)⋊3C2, C2.7(C22×D19), SmallGroup(304,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2D19
G = < a,b,c,d | a4=b2=c19=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Smallest permutation representation of D4⋊2D19
►On 152 points
Generators in S152
(1 112 29 93)(2 113 30 94)(3 114 31 95)(4 96 32 77)(5 97 33 78)(6 98 34 79)(7 99 35 80)(8 100 36 81)(9 101 37 82)(10 102 38 83)(11 103 20 84)(12 104 21 85)(13 105 22 86)(14 106 23 87)(15 107 24 88)(16 108 25 89)(17 109 26 90)(18 110 27 91)(19 111 28 92)(39 144 69 120)(40 145 70 121)(41 146 71 122)(42 147 72 123)(43 148 73 124)(44 149 74 125)(45 150 75 126)(46 151 76 127)(47 152 58 128)(48 134 59 129)(49 135 60 130)(50 136 61 131)(51 137 62 132)(52 138 63 133)(53 139 64 115)(54 140 65 116)(55 141 66 117)(56 142 67 118)(57 143 68 119)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 149)(21 150)(22 151)(23 152)(24 134)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 145)(36 146)(37 147)(38 148)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 77)(57 78)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 113)(66 114)(67 96)(68 97)(69 98)(70 99)(71 100)(72 101)(73 102)(74 103)(75 104)(76 105)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(39 58)(40 76)(41 75)(42 74)(43 73)(44 72)(45 71)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)(53 63)(54 62)(55 61)(56 60)(57 59)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(90 95)(91 94)(92 93)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)(109 114)(110 113)(111 112)(115 138)(116 137)(117 136)(118 135)(119 134)(120 152)(121 151)(122 150)(123 149)(124 148)(125 147)(126 146)(127 145)(128 144)(129 143)(130 142)(131 141)(132 140)(133 139)
G:=sub<Sym(152)| (1,112,29,93)(2,113,30,94)(3,114,31,95)(4,96,32,77)(5,97,33,78)(6,98,34,79)(7,99,35,80)(8,100,36,81)(9,101,37,82)(10,102,38,83)(11,103,20,84)(12,104,21,85)(13,105,22,86)(14,106,23,87)(15,107,24,88)(16,108,25,89)(17,109,26,90)(18,110,27,91)(19,111,28,92)(39,144,69,120)(40,145,70,121)(41,146,71,122)(42,147,72,123)(43,148,73,124)(44,149,74,125)(45,150,75,126)(46,151,76,127)(47,152,58,128)(48,134,59,129)(49,135,60,130)(50,136,61,131)(51,137,62,132)(52,138,63,133)(53,139,64,115)(54,140,65,116)(55,141,66,117)(56,142,67,118)(57,143,68,119), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,149)(21,150)(22,151)(23,152)(24,134)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,77)(57,78)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,96)(68,97)(69,98)(70,99)(71,100)(72,101)(73,102)(74,103)(75,104)(76,105), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(39,58)(40,76)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(90,95)(91,94)(92,93)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103)(109,114)(110,113)(111,112)(115,138)(116,137)(117,136)(118,135)(119,134)(120,152)(121,151)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139)>;
G:=Group( (1,112,29,93)(2,113,30,94)(3,114,31,95)(4,96,32,77)(5,97,33,78)(6,98,34,79)(7,99,35,80)(8,100,36,81)(9,101,37,82)(10,102,38,83)(11,103,20,84)(12,104,21,85)(13,105,22,86)(14,106,23,87)(15,107,24,88)(16,108,25,89)(17,109,26,90)(18,110,27,91)(19,111,28,92)(39,144,69,120)(40,145,70,121)(41,146,71,122)(42,147,72,123)(43,148,73,124)(44,149,74,125)(45,150,75,126)(46,151,76,127)(47,152,58,128)(48,134,59,129)(49,135,60,130)(50,136,61,131)(51,137,62,132)(52,138,63,133)(53,139,64,115)(54,140,65,116)(55,141,66,117)(56,142,67,118)(57,143,68,119), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,149)(21,150)(22,151)(23,152)(24,134)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,145)(36,146)(37,147)(38,148)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(49,89)(50,90)(51,91)(52,92)(53,93)(54,94)(55,95)(56,77)(57,78)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,96)(68,97)(69,98)(70,99)(71,100)(72,101)(73,102)(74,103)(75,104)(76,105), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(39,58)(40,76)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(90,95)(91,94)(92,93)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103)(109,114)(110,113)(111,112)(115,138)(116,137)(117,136)(118,135)(119,134)(120,152)(121,151)(122,150)(123,149)(124,148)(125,147)(126,146)(127,145)(128,144)(129,143)(130,142)(131,141)(132,140)(133,139) );
G=PermutationGroup([[(1,112,29,93),(2,113,30,94),(3,114,31,95),(4,96,32,77),(5,97,33,78),(6,98,34,79),(7,99,35,80),(8,100,36,81),(9,101,37,82),(10,102,38,83),(11,103,20,84),(12,104,21,85),(13,105,22,86),(14,106,23,87),(15,107,24,88),(16,108,25,89),(17,109,26,90),(18,110,27,91),(19,111,28,92),(39,144,69,120),(40,145,70,121),(41,146,71,122),(42,147,72,123),(43,148,73,124),(44,149,74,125),(45,150,75,126),(46,151,76,127),(47,152,58,128),(48,134,59,129),(49,135,60,130),(50,136,61,131),(51,137,62,132),(52,138,63,133),(53,139,64,115),(54,140,65,116),(55,141,66,117),(56,142,67,118),(57,143,68,119)], [(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,121),(8,122),(9,123),(10,124),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,131),(18,132),(19,133),(20,149),(21,150),(22,151),(23,152),(24,134),(25,135),(26,136),(27,137),(28,138),(29,139),(30,140),(31,141),(32,142),(33,143),(34,144),(35,145),(36,146),(37,147),(38,148),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(49,89),(50,90),(51,91),(52,92),(53,93),(54,94),(55,95),(56,77),(57,78),(58,106),(59,107),(60,108),(61,109),(62,110),(63,111),(64,112),(65,113),(66,114),(67,96),(68,97),(69,98),(70,99),(71,100),(72,101),(73,102),(74,103),(75,104),(76,105)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,37),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(39,58),(40,76),(41,75),(42,74),(43,73),(44,72),(45,71),(46,70),(47,69),(48,68),(49,67),(50,66),(51,65),(52,64),(53,63),(54,62),(55,61),(56,60),(57,59),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(90,95),(91,94),(92,93),(96,108),(97,107),(98,106),(99,105),(100,104),(101,103),(109,114),(110,113),(111,112),(115,138),(116,137),(117,136),(118,135),(119,134),(120,152),(121,151),(122,150),(123,149),(124,148),(125,147),(126,146),(127,145),(128,144),(129,143),(130,142),(131,141),(132,140),(133,139)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 19A | ··· | 19I | 38A | ··· | 38I | 38J | ··· | 38AA | 76A | ··· | 76I |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 2 | 2 | 38 | 2 | 19 | 19 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4○D4 | D19 | D38 | D38 | D4⋊2D19 |
| kernel | D4⋊2D19 | Dic38 | C4×D19 | C2×Dic19 | C19⋊D4 | D4×C19 | C19 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 9 | 9 | 18 | 9 |
Matrix representation of D4⋊2D19 ►in GL4(𝔽229) generated by
| 228 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 |
| 0 | 0 | 107 | 0 |
| 0 | 0 | 60 | 122 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 122 | 145 |
| 0 | 0 | 169 | 107 |
| 90 | 1 | 0 | 0 |
| 161 | 185 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 54 | 116 | 0 | 0 |
| 194 | 175 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 221 | 228 |
G:=sub<GL(4,GF(229))| [228,0,0,0,0,228,0,0,0,0,107,60,0,0,0,122],[1,0,0,0,0,1,0,0,0,0,122,169,0,0,145,107],[90,161,0,0,1,185,0,0,0,0,1,0,0,0,0,1],[54,194,0,0,116,175,0,0,0,0,1,221,0,0,0,228] >;
D4⋊2D19 in GAP, Magma, Sage, TeX
D_4\rtimes_2D_{19} % in TeX
G:=Group("D4:2D19"); // GroupNames label
G:=SmallGroup(304,32);
// by ID
G=gap.SmallGroup(304,32);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,46,182,97,7204]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^19=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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