direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C23×D19, C19⋊C24, C38⋊C23, (C22×C38)⋊3C2, (C2×C38)⋊4C22, SmallGroup(304,41)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C23×D19 |
Generators and relations for C23×D19
G = < a,b,c,d,e | a2=b2=c2=d19=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1052 in 134 conjugacy classes, 83 normal (5 characteristic)
C1, C2, C2, C22, C22, C23, C23, C24, C19, D19, C38, D38, C2×C38, C22×D19, C22×C38, C23×D19
Quotients: C1, C2, C22, C23, C24, D19, D38, C22×D19, C23×D19
►On 152 points
Generators in S152
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 77)(19 78)(20 109)(21 110)(22 111)(23 112)(24 113)(25 114)(26 96)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 115)(40 116)(41 117)(42 118)(43 119)(44 120)(45 121)(46 122)(47 123)(48 124)(49 125)(50 126)(51 127)(52 128)(53 129)(54 130)(55 131)(56 132)(57 133)(58 140)(59 141)(60 142)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 134)(72 135)(73 136)(74 137)(75 138)(76 139)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 58)(17 59)(18 60)(19 61)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(77 142)(78 143)(79 144)(80 145)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 131)(97 132)(98 133)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(105 121)(106 122)(107 123)(108 124)(109 125)(110 126)(111 127)(112 128)(113 129)(114 130)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 20)(19 21)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(77 109)(78 110)(79 111)(80 112)(81 113)(82 114)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)(115 151)(116 152)(117 134)(118 135)(119 136)(120 137)(121 138)(122 139)(123 140)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(131 148)(132 149)(133 150)
(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 78)(2 77)(3 95)(4 94)(5 93)(6 92)(7 91)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 81)(18 80)(19 79)(20 112)(21 111)(22 110)(23 109)(24 108)(25 107)(26 106)(27 105)(28 104)(29 103)(30 102)(31 101)(32 100)(33 99)(34 98)(35 97)(36 96)(37 114)(38 113)(39 119)(40 118)(41 117)(42 116)(43 115)(44 133)(45 132)(46 131)(47 130)(48 129)(49 128)(50 127)(51 126)(52 125)(53 124)(54 123)(55 122)(56 121)(57 120)(58 147)(59 146)(60 145)(61 144)(62 143)(63 142)(64 141)(65 140)(66 139)(67 138)(68 137)(69 136)(70 135)(71 134)(72 152)(73 151)(74 150)(75 149)(76 148)
G:=sub<Sym(152)| (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,77)(19,78)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,96)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(51,127)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,58)(17,59)(18,60)(19,61)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,131)(97,132)(98,133)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)(112,128)(113,129)(114,130), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,20)(19,21)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(77,109)(78,110)(79,111)(80,112)(81,113)(82,114)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(115,151)(116,152)(117,134)(118,135)(119,136)(120,137)(121,138)(122,139)(123,140)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150), (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), (1,78)(2,77)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,103)(30,102)(31,101)(32,100)(33,99)(34,98)(35,97)(36,96)(37,114)(38,113)(39,119)(40,118)(41,117)(42,116)(43,115)(44,133)(45,132)(46,131)(47,130)(48,129)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,120)(58,147)(59,146)(60,145)(61,144)(62,143)(63,142)(64,141)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(71,134)(72,152)(73,151)(74,150)(75,149)(76,148)>;
G:=Group( (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,77)(19,78)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,96)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(51,127)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,58)(17,59)(18,60)(19,61)(20,49)(21,50)(22,51)(23,52)(24,53)(25,54)(26,55)(27,56)(28,57)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,131)(97,132)(98,133)(99,115)(100,116)(101,117)(102,118)(103,119)(104,120)(105,121)(106,122)(107,123)(108,124)(109,125)(110,126)(111,127)(112,128)(113,129)(114,130), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,20)(19,21)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(77,109)(78,110)(79,111)(80,112)(81,113)(82,114)(83,96)(84,97)(85,98)(86,99)(87,100)(88,101)(89,102)(90,103)(91,104)(92,105)(93,106)(94,107)(95,108)(115,151)(116,152)(117,134)(118,135)(119,136)(120,137)(121,138)(122,139)(123,140)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150), (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), (1,78)(2,77)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,103)(30,102)(31,101)(32,100)(33,99)(34,98)(35,97)(36,96)(37,114)(38,113)(39,119)(40,118)(41,117)(42,116)(43,115)(44,133)(45,132)(46,131)(47,130)(48,129)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,120)(58,147)(59,146)(60,145)(61,144)(62,143)(63,142)(64,141)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(71,134)(72,152)(73,151)(74,150)(75,149)(76,148) );
G=PermutationGroup([[(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,77),(19,78),(20,109),(21,110),(22,111),(23,112),(24,113),(25,114),(26,96),(27,97),(28,98),(29,99),(30,100),(31,101),(32,102),(33,103),(34,104),(35,105),(36,106),(37,107),(38,108),(39,115),(40,116),(41,117),(42,118),(43,119),(44,120),(45,121),(46,122),(47,123),(48,124),(49,125),(50,126),(51,127),(52,128),(53,129),(54,130),(55,131),(56,132),(57,133),(58,140),(59,141),(60,142),(61,143),(62,144),(63,145),(64,146),(65,147),(66,148),(67,149),(68,150),(69,151),(70,152),(71,134),(72,135),(73,136),(74,137),(75,138),(76,139)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,76),(16,58),(17,59),(18,60),(19,61),(20,49),(21,50),(22,51),(23,52),(24,53),(25,54),(26,55),(27,56),(28,57),(29,39),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(77,142),(78,143),(79,144),(80,145),(81,146),(82,147),(83,148),(84,149),(85,150),(86,151),(87,152),(88,134),(89,135),(90,136),(91,137),(92,138),(93,139),(94,140),(95,141),(96,131),(97,132),(98,133),(99,115),(100,116),(101,117),(102,118),(103,119),(104,120),(105,121),(106,122),(107,123),(108,124),(109,125),(110,126),(111,127),(112,128),(113,129),(114,130)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,20),(19,21),(39,69),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(77,109),(78,110),(79,111),(80,112),(81,113),(82,114),(83,96),(84,97),(85,98),(86,99),(87,100),(88,101),(89,102),(90,103),(91,104),(92,105),(93,106),(94,107),(95,108),(115,151),(116,152),(117,134),(118,135),(119,136),(120,137),(121,138),(122,139),(123,140),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146),(130,147),(131,148),(132,149),(133,150)], [(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)], [(1,78),(2,77),(3,95),(4,94),(5,93),(6,92),(7,91),(8,90),(9,89),(10,88),(11,87),(12,86),(13,85),(14,84),(15,83),(16,82),(17,81),(18,80),(19,79),(20,112),(21,111),(22,110),(23,109),(24,108),(25,107),(26,106),(27,105),(28,104),(29,103),(30,102),(31,101),(32,100),(33,99),(34,98),(35,97),(36,96),(37,114),(38,113),(39,119),(40,118),(41,117),(42,116),(43,115),(44,133),(45,132),(46,131),(47,130),(48,129),(49,128),(50,127),(51,126),(52,125),(53,124),(54,123),(55,122),(56,121),(57,120),(58,147),(59,146),(60,145),(61,144),(62,143),(63,142),(64,141),(65,140),(66,139),(67,138),(68,137),(69,136),(70,135),(71,134),(72,152),(73,151),(74,150),(75,149),(76,148)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 19A | ··· | 19I | 38A | ··· | 38BK |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 19 | ··· | 19 | 38 | ··· | 38 |
| size | 1 | 1 | ··· | 1 | 19 | ··· | 19 | 2 | ··· | 2 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D19 | D38 |
| kernel | C23×D19 | C22×D19 | C22×C38 | C23 | C22 |
| # reps | 1 | 14 | 1 | 9 | 63 |
Matrix representation of C23×D19 ►in GL4(𝔽191) generated by
| 1 | 0 | 0 | 0 |
| 0 | 190 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 190 | 0 | 0 | 0 |
| 0 | 190 | 0 | 0 |
| 0 | 0 | 190 | 0 |
| 0 | 0 | 0 | 190 |
| 190 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 1 |
| 0 | 0 | 17 | 25 |
| 190 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 87 | 79 |
| 0 | 0 | 13 | 104 |
G:=sub<GL(4,GF(191))| [1,0,0,0,0,190,0,0,0,0,1,0,0,0,0,1],[190,0,0,0,0,190,0,0,0,0,190,0,0,0,0,190],[190,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,17,0,0,1,25],[190,0,0,0,0,1,0,0,0,0,87,13,0,0,79,104] >;
C23×D19 in GAP, Magma, Sage, TeX
C_2^3\times D_{19} % in TeX
G:=Group("C2^3xD19"); // GroupNames label
G:=SmallGroup(304,41);
// by ID
G=gap.SmallGroup(304,41);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,7204]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^19=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations