direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C19⋊D4, C38⋊2D4, C23⋊D19, C22⋊2D38, D38⋊3C22, C38.10C23, Dic19⋊2C22, C19⋊3(C2×D4), (C2×C38)⋊3C22, (C22×C38)⋊2C2, (C2×Dic19)⋊4C2, (C22×D19)⋊3C2, C2.10(C22×D19), SmallGroup(304,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C19⋊D4
G = < a,b,c,d | a2=b19=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 412 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C23, C2×D4, C19, D19, C38, C38, C38, Dic19, D38, D38, C2×C38, C2×C38, C2×C38, C2×Dic19, C19⋊D4, C22×D19, C22×C38, C2×C19⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, D19, D38, C19⋊D4, C22×D19, C2×C19⋊D4
►On 152 points
Generators in S152
(1 96)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 77)(21 78)(22 79)(23 80)(24 81)(25 82)(26 83)(27 84)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 134)(40 135)(41 136)(42 137)(43 138)(44 139)(45 140)(46 141)(47 142)(48 143)(49 144)(50 145)(51 146)(52 147)(53 148)(54 149)(55 150)(56 151)(57 152)(58 115)(59 116)(60 117)(61 118)(62 119)(63 120)(64 121)(65 122)(66 123)(67 124)(68 125)(69 126)(70 127)(71 128)(72 129)(73 130)(74 131)(75 132)(76 133)
(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 39 20 58)(2 57 21 76)(3 56 22 75)(4 55 23 74)(5 54 24 73)(6 53 25 72)(7 52 26 71)(8 51 27 70)(9 50 28 69)(10 49 29 68)(11 48 30 67)(12 47 31 66)(13 46 32 65)(14 45 33 64)(15 44 34 63)(16 43 35 62)(17 42 36 61)(18 41 37 60)(19 40 38 59)(77 115 96 134)(78 133 97 152)(79 132 98 151)(80 131 99 150)(81 130 100 149)(82 129 101 148)(83 128 102 147)(84 127 103 146)(85 126 104 145)(86 125 105 144)(87 124 106 143)(88 123 107 142)(89 122 108 141)(90 121 109 140)(91 120 110 139)(92 119 111 138)(93 118 112 137)(94 117 113 136)(95 116 114 135)
(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(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)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(115 134)(116 152)(117 151)(118 150)(119 149)(120 148)(121 147)(122 146)(123 145)(124 144)(125 143)(126 142)(127 141)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)
G:=sub<Sym(152)| (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,77)(21,78)(22,79)(23,80)(24,81)(25,82)(26,83)(27,84)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,134)(40,135)(41,136)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,144)(50,145)(51,146)(52,147)(53,148)(54,149)(55,150)(56,151)(57,152)(58,115)(59,116)(60,117)(61,118)(62,119)(63,120)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133), (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), (1,39,20,58)(2,57,21,76)(3,56,22,75)(4,55,23,74)(5,54,24,73)(6,53,25,72)(7,52,26,71)(8,51,27,70)(9,50,28,69)(10,49,29,68)(11,48,30,67)(12,47,31,66)(13,46,32,65)(14,45,33,64)(15,44,34,63)(16,43,35,62)(17,42,36,61)(18,41,37,60)(19,40,38,59)(77,115,96,134)(78,133,97,152)(79,132,98,151)(80,131,99,150)(81,130,100,149)(82,129,101,148)(83,128,102,147)(84,127,103,146)(85,126,104,145)(86,125,105,144)(87,124,106,143)(88,123,107,142)(89,122,108,141)(90,121,109,140)(91,120,110,139)(92,119,111,138)(93,118,112,137)(94,117,113,136)(95,116,114,135), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(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)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(115,134)(116,152)(117,151)(118,150)(119,149)(120,148)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)>;
G:=Group( (1,96)(2,97)(3,98)(4,99)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,77)(21,78)(22,79)(23,80)(24,81)(25,82)(26,83)(27,84)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,134)(40,135)(41,136)(42,137)(43,138)(44,139)(45,140)(46,141)(47,142)(48,143)(49,144)(50,145)(51,146)(52,147)(53,148)(54,149)(55,150)(56,151)(57,152)(58,115)(59,116)(60,117)(61,118)(62,119)(63,120)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,127)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133), (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), (1,39,20,58)(2,57,21,76)(3,56,22,75)(4,55,23,74)(5,54,24,73)(6,53,25,72)(7,52,26,71)(8,51,27,70)(9,50,28,69)(10,49,29,68)(11,48,30,67)(12,47,31,66)(13,46,32,65)(14,45,33,64)(15,44,34,63)(16,43,35,62)(17,42,36,61)(18,41,37,60)(19,40,38,59)(77,115,96,134)(78,133,97,152)(79,132,98,151)(80,131,99,150)(81,130,100,149)(82,129,101,148)(83,128,102,147)(84,127,103,146)(85,126,104,145)(86,125,105,144)(87,124,106,143)(88,123,107,142)(89,122,108,141)(90,121,109,140)(91,120,110,139)(92,119,111,138)(93,118,112,137)(94,117,113,136)(95,116,114,135), (2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(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)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(115,134)(116,152)(117,151)(118,150)(119,149)(120,148)(121,147)(122,146)(123,145)(124,144)(125,143)(126,142)(127,141)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135) );
G=PermutationGroup([[(1,96),(2,97),(3,98),(4,99),(5,100),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,77),(21,78),(22,79),(23,80),(24,81),(25,82),(26,83),(27,84),(28,85),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,134),(40,135),(41,136),(42,137),(43,138),(44,139),(45,140),(46,141),(47,142),(48,143),(49,144),(50,145),(51,146),(52,147),(53,148),(54,149),(55,150),(56,151),(57,152),(58,115),(59,116),(60,117),(61,118),(62,119),(63,120),(64,121),(65,122),(66,123),(67,124),(68,125),(69,126),(70,127),(71,128),(72,129),(73,130),(74,131),(75,132),(76,133)], [(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)], [(1,39,20,58),(2,57,21,76),(3,56,22,75),(4,55,23,74),(5,54,24,73),(6,53,25,72),(7,52,26,71),(8,51,27,70),(9,50,28,69),(10,49,29,68),(11,48,30,67),(12,47,31,66),(13,46,32,65),(14,45,33,64),(15,44,34,63),(16,43,35,62),(17,42,36,61),(18,41,37,60),(19,40,38,59),(77,115,96,134),(78,133,97,152),(79,132,98,151),(80,131,99,150),(81,130,100,149),(82,129,101,148),(83,128,102,147),(84,127,103,146),(85,126,104,145),(86,125,105,144),(87,124,106,143),(88,123,107,142),(89,122,108,141),(90,121,109,140),(91,120,110,139),(92,119,111,138),(93,118,112,137),(94,117,113,136),(95,116,114,135)], [(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(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),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87),(97,114),(98,113),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106),(115,134),(116,152),(117,151),(118,150),(119,149),(120,148),(121,147),(122,146),(123,145),(124,144),(125,143),(126,142),(127,141),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 19A | ··· | 19I | 38A | ··· | 38BK |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 38 | 38 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D19 | D38 | C19⋊D4 |
| kernel | C2×C19⋊D4 | C2×Dic19 | C19⋊D4 | C22×D19 | C22×C38 | C38 | C23 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 9 | 27 | 36 |
Matrix representation of C2×C19⋊D4 ►in GL4(𝔽229) generated by
| 228 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 |
| 0 | 0 | 228 | 0 |
| 0 | 0 | 0 | 228 |
| 38 | 228 | 0 | 0 |
| 100 | 136 | 0 | 0 |
| 0 | 0 | 91 | 228 |
| 0 | 0 | 24 | 191 |
| 199 | 55 | 0 | 0 |
| 196 | 30 | 0 | 0 |
| 0 | 0 | 19 | 202 |
| 0 | 0 | 200 | 210 |
| 199 | 55 | 0 | 0 |
| 196 | 30 | 0 | 0 |
| 0 | 0 | 79 | 79 |
| 0 | 0 | 179 | 150 |
G:=sub<GL(4,GF(229))| [228,0,0,0,0,228,0,0,0,0,228,0,0,0,0,228],[38,100,0,0,228,136,0,0,0,0,91,24,0,0,228,191],[199,196,0,0,55,30,0,0,0,0,19,200,0,0,202,210],[199,196,0,0,55,30,0,0,0,0,79,179,0,0,79,150] >;
C2×C19⋊D4 in GAP, Magma, Sage, TeX
C_2\times C_{19}\rtimes D_4 % in TeX
G:=Group("C2xC19:D4"); // GroupNames label
G:=SmallGroup(304,36);
// by ID
G=gap.SmallGroup(304,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,182,7204]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^19=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations