direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×D19, C76⋊3C22, C38.2C23, C22.9D38, D38.4C22, Dic19⋊3C22, (C2×C76)⋊5C2, C38⋊1(C2×C4), C19⋊1(C22×C4), (C2×Dic19)⋊5C2, (C2×C38).9C22, C2.1(C22×D19), (C22×D19).2C2, SmallGroup(304,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C2×C4×D19 |
Generators and relations for C2×C4×D19
G = < a,b,c,d | a2=b4=c19=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 396 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C22×C4, C19, D19, C38, C38, Dic19, C76, D38, C2×C38, C4×D19, C2×Dic19, C2×C76, C22×D19, C2×C4×D19
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D19, D38, C4×D19, C22×D19, C2×C4×D19
►On 152 points
Generators in S152
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 77)(16 78)(17 79)(18 80)(19 81)(20 114)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 133)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 152)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 140)(66 141)(67 142)(68 143)(69 144)(70 145)(71 146)(72 147)(73 148)(74 149)(75 150)(76 151)
(1 64 20 44)(2 65 21 45)(3 66 22 46)(4 67 23 47)(5 68 24 48)(6 69 25 49)(7 70 26 50)(8 71 27 51)(9 72 28 52)(10 73 29 53)(11 74 30 54)(12 75 31 55)(13 76 32 56)(14 58 33 57)(15 59 34 39)(16 60 35 40)(17 61 36 41)(18 62 37 42)(19 63 38 43)(77 134 109 133)(78 135 110 115)(79 136 111 116)(80 137 112 117)(81 138 113 118)(82 139 114 119)(83 140 96 120)(84 141 97 121)(85 142 98 122)(86 143 99 123)(87 144 100 124)(88 145 101 125)(89 146 102 126)(90 147 103 127)(91 148 104 128)(92 149 105 129)(93 150 106 130)(94 151 107 131)(95 152 108 132)
(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 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(39 48)(40 47)(41 46)(42 45)(43 44)(49 57)(50 56)(51 55)(52 54)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(70 76)(71 75)(72 74)(77 86)(78 85)(79 84)(80 83)(81 82)(87 95)(88 94)(89 93)(90 92)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)(113 114)(115 122)(116 121)(117 120)(118 119)(123 133)(124 132)(125 131)(126 130)(127 129)(134 143)(135 142)(136 141)(137 140)(138 139)(144 152)(145 151)(146 150)(147 149)
G:=sub<Sym(152)| (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,77)(16,78)(17,79)(18,80)(19,81)(20,114)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,133)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,152)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151), (1,64,20,44)(2,65,21,45)(3,66,22,46)(4,67,23,47)(5,68,24,48)(6,69,25,49)(7,70,26,50)(8,71,27,51)(9,72,28,52)(10,73,29,53)(11,74,30,54)(12,75,31,55)(13,76,32,56)(14,58,33,57)(15,59,34,39)(16,60,35,40)(17,61,36,41)(18,62,37,42)(19,63,38,43)(77,134,109,133)(78,135,110,115)(79,136,111,116)(80,137,112,117)(81,138,113,118)(82,139,114,119)(83,140,96,120)(84,141,97,121)(85,142,98,122)(86,143,99,123)(87,144,100,124)(88,145,101,125)(89,146,102,126)(90,147,103,127)(91,148,104,128)(92,149,105,129)(93,150,106,130)(94,151,107,131)(95,152,108,132), (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)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(39,48)(40,47)(41,46)(42,45)(43,44)(49,57)(50,56)(51,55)(52,54)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(70,76)(71,75)(72,74)(77,86)(78,85)(79,84)(80,83)(81,82)(87,95)(88,94)(89,93)(90,92)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(113,114)(115,122)(116,121)(117,120)(118,119)(123,133)(124,132)(125,131)(126,130)(127,129)(134,143)(135,142)(136,141)(137,140)(138,139)(144,152)(145,151)(146,150)(147,149)>;
G:=Group( (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,77)(16,78)(17,79)(18,80)(19,81)(20,114)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,133)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,152)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151), (1,64,20,44)(2,65,21,45)(3,66,22,46)(4,67,23,47)(5,68,24,48)(6,69,25,49)(7,70,26,50)(8,71,27,51)(9,72,28,52)(10,73,29,53)(11,74,30,54)(12,75,31,55)(13,76,32,56)(14,58,33,57)(15,59,34,39)(16,60,35,40)(17,61,36,41)(18,62,37,42)(19,63,38,43)(77,134,109,133)(78,135,110,115)(79,136,111,116)(80,137,112,117)(81,138,113,118)(82,139,114,119)(83,140,96,120)(84,141,97,121)(85,142,98,122)(86,143,99,123)(87,144,100,124)(88,145,101,125)(89,146,102,126)(90,147,103,127)(91,148,104,128)(92,149,105,129)(93,150,106,130)(94,151,107,131)(95,152,108,132), (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)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(39,48)(40,47)(41,46)(42,45)(43,44)(49,57)(50,56)(51,55)(52,54)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(70,76)(71,75)(72,74)(77,86)(78,85)(79,84)(80,83)(81,82)(87,95)(88,94)(89,93)(90,92)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(113,114)(115,122)(116,121)(117,120)(118,119)(123,133)(124,132)(125,131)(126,130)(127,129)(134,143)(135,142)(136,141)(137,140)(138,139)(144,152)(145,151)(146,150)(147,149) );
G=PermutationGroup([[(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,77),(16,78),(17,79),(18,80),(19,81),(20,114),(21,96),(22,97),(23,98),(24,99),(25,100),(26,101),(27,102),(28,103),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,113),(39,133),(40,115),(41,116),(42,117),(43,118),(44,119),(45,120),(46,121),(47,122),(48,123),(49,124),(50,125),(51,126),(52,127),(53,128),(54,129),(55,130),(56,131),(57,132),(58,152),(59,134),(60,135),(61,136),(62,137),(63,138),(64,139),(65,140),(66,141),(67,142),(68,143),(69,144),(70,145),(71,146),(72,147),(73,148),(74,149),(75,150),(76,151)], [(1,64,20,44),(2,65,21,45),(3,66,22,46),(4,67,23,47),(5,68,24,48),(6,69,25,49),(7,70,26,50),(8,71,27,51),(9,72,28,52),(10,73,29,53),(11,74,30,54),(12,75,31,55),(13,76,32,56),(14,58,33,57),(15,59,34,39),(16,60,35,40),(17,61,36,41),(18,62,37,42),(19,63,38,43),(77,134,109,133),(78,135,110,115),(79,136,111,116),(80,137,112,117),(81,138,113,118),(82,139,114,119),(83,140,96,120),(84,141,97,121),(85,142,98,122),(86,143,99,123),(87,144,100,124),(88,145,101,125),(89,146,102,126),(90,147,103,127),(91,148,104,128),(92,149,105,129),(93,150,106,130),(94,151,107,131),(95,152,108,132)], [(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),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(39,48),(40,47),(41,46),(42,45),(43,44),(49,57),(50,56),(51,55),(52,54),(58,69),(59,68),(60,67),(61,66),(62,65),(63,64),(70,76),(71,75),(72,74),(77,86),(78,85),(79,84),(80,83),(81,82),(87,95),(88,94),(89,93),(90,92),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,105),(113,114),(115,122),(116,121),(117,120),(118,119),(123,133),(124,132),(125,131),(126,130),(127,129),(134,143),(135,142),(136,141),(137,140),(138,139),(144,152),(145,151),(146,150),(147,149)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 19A | ··· | 19I | 38A | ··· | 38AA | 76A | ··· | 76AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D19 | D38 | D38 | C4×D19 |
| kernel | C2×C4×D19 | C4×D19 | C2×Dic19 | C2×C76 | C22×D19 | D38 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 9 | 18 | 9 | 36 |
Matrix representation of C2×C4×D19 ►in GL3(𝔽229) generated by
| 228 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 228 |
| 1 | 0 | 0 |
| 0 | 122 | 0 |
| 0 | 0 | 122 |
| 1 | 0 | 0 |
| 0 | 176 | 1 |
| 0 | 47 | 146 |
| 228 | 0 | 0 |
| 0 | 146 | 228 |
| 0 | 18 | 83 |
G:=sub<GL(3,GF(229))| [228,0,0,0,228,0,0,0,228],[1,0,0,0,122,0,0,0,122],[1,0,0,0,176,47,0,1,146],[228,0,0,0,146,18,0,228,83] >;
C2×C4×D19 in GAP, Magma, Sage, TeX
C_2\times C_4\times D_{19} % in TeX
G:=Group("C2xC4xD19"); // GroupNames label
G:=SmallGroup(304,28);
// by ID
G=gap.SmallGroup(304,28);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,42,7204]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^19=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations