metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.D19, C38.11D4, C22⋊Dic19, C22.7D38, (C2×C38)⋊2C4, C38.9(C2×C4), C19⋊2(C22⋊C4), (C2×Dic19)⋊2C2, C2.3(C19⋊D4), (C2×C38).7C22, (C22×C38).2C2, C2.5(C2×Dic19), SmallGroup(304,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.D19
G = < a,b,c,d,e | a2=b2=c2=d19=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Smallest permutation representation of C23.D19
►On 152 points
Generators in S152
(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)(115 134)(116 135)(117 136)(118 137)(119 138)(120 139)(121 140)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)(133 152)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 39)(17 40)(18 41)(19 42)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 58)(36 59)(37 60)(38 61)(77 115)(78 116)(79 117)(80 118)(81 119)(82 120)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 145)(108 146)(109 147)(110 148)(111 149)(112 150)(113 151)(114 152)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 20)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(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 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)(115 134)(116 135)(117 136)(118 137)(119 138)(120 139)(121 140)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)(133 152)
(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)
(1 133 43 95)(2 132 44 94)(3 131 45 93)(4 130 46 92)(5 129 47 91)(6 128 48 90)(7 127 49 89)(8 126 50 88)(9 125 51 87)(10 124 52 86)(11 123 53 85)(12 122 54 84)(13 121 55 83)(14 120 56 82)(15 119 57 81)(16 118 39 80)(17 117 40 79)(18 116 41 78)(19 115 42 77)(20 144 62 106)(21 143 63 105)(22 142 64 104)(23 141 65 103)(24 140 66 102)(25 139 67 101)(26 138 68 100)(27 137 69 99)(28 136 70 98)(29 135 71 97)(30 134 72 96)(31 152 73 114)(32 151 74 113)(33 150 75 112)(34 149 76 111)(35 148 58 110)(36 147 59 109)(37 146 60 108)(38 145 61 107)
G:=sub<Sym(152)| (77,96)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,39)(17,40)(18,41)(19,42)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,58)(36,59)(37,60)(38,61)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148)(111,149)(112,150)(113,151)(114,152), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(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)(77,96)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152), (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), (1,133,43,95)(2,132,44,94)(3,131,45,93)(4,130,46,92)(5,129,47,91)(6,128,48,90)(7,127,49,89)(8,126,50,88)(9,125,51,87)(10,124,52,86)(11,123,53,85)(12,122,54,84)(13,121,55,83)(14,120,56,82)(15,119,57,81)(16,118,39,80)(17,117,40,79)(18,116,41,78)(19,115,42,77)(20,144,62,106)(21,143,63,105)(22,142,64,104)(23,141,65,103)(24,140,66,102)(25,139,67,101)(26,138,68,100)(27,137,69,99)(28,136,70,98)(29,135,71,97)(30,134,72,96)(31,152,73,114)(32,151,74,113)(33,150,75,112)(34,149,76,111)(35,148,58,110)(36,147,59,109)(37,146,60,108)(38,145,61,107)>;
G:=Group( (77,96)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,39)(17,40)(18,41)(19,42)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,58)(36,59)(37,60)(38,61)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148)(111,149)(112,150)(113,151)(114,152), (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,20)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(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)(77,96)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152), (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), (1,133,43,95)(2,132,44,94)(3,131,45,93)(4,130,46,92)(5,129,47,91)(6,128,48,90)(7,127,49,89)(8,126,50,88)(9,125,51,87)(10,124,52,86)(11,123,53,85)(12,122,54,84)(13,121,55,83)(14,120,56,82)(15,119,57,81)(16,118,39,80)(17,117,40,79)(18,116,41,78)(19,115,42,77)(20,144,62,106)(21,143,63,105)(22,142,64,104)(23,141,65,103)(24,140,66,102)(25,139,67,101)(26,138,68,100)(27,137,69,99)(28,136,70,98)(29,135,71,97)(30,134,72,96)(31,152,73,114)(32,151,74,113)(33,150,75,112)(34,149,76,111)(35,148,58,110)(36,147,59,109)(37,146,60,108)(38,145,61,107) );
G=PermutationGroup([[(77,96),(78,97),(79,98),(80,99),(81,100),(82,101),(83,102),(84,103),(85,104),(86,105),(87,106),(88,107),(89,108),(90,109),(91,110),(92,111),(93,112),(94,113),(95,114),(115,134),(116,135),(117,136),(118,137),(119,138),(120,139),(121,140),(122,141),(123,142),(124,143),(125,144),(126,145),(127,146),(128,147),(129,148),(130,149),(131,150),(132,151),(133,152)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,39),(17,40),(18,41),(19,42),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,58),(36,59),(37,60),(38,61),(77,115),(78,116),(79,117),(80,118),(81,119),(82,120),(83,121),(84,122),(85,123),(86,124),(87,125),(88,126),(89,127),(90,128),(91,129),(92,130),(93,131),(94,132),(95,133),(96,134),(97,135),(98,136),(99,137),(100,138),(101,139),(102,140),(103,141),(104,142),(105,143),(106,144),(107,145),(108,146),(109,147),(110,148),(111,149),(112,150),(113,151),(114,152)], [(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,20),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(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),(77,96),(78,97),(79,98),(80,99),(81,100),(82,101),(83,102),(84,103),(85,104),(86,105),(87,106),(88,107),(89,108),(90,109),(91,110),(92,111),(93,112),(94,113),(95,114),(115,134),(116,135),(117,136),(118,137),(119,138),(120,139),(121,140),(122,141),(123,142),(124,143),(125,144),(126,145),(127,146),(128,147),(129,148),(130,149),(131,150),(132,151),(133,152)], [(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)], [(1,133,43,95),(2,132,44,94),(3,131,45,93),(4,130,46,92),(5,129,47,91),(6,128,48,90),(7,127,49,89),(8,126,50,88),(9,125,51,87),(10,124,52,86),(11,123,53,85),(12,122,54,84),(13,121,55,83),(14,120,56,82),(15,119,57,81),(16,118,39,80),(17,117,40,79),(18,116,41,78),(19,115,42,77),(20,144,62,106),(21,143,63,105),(22,142,64,104),(23,141,65,103),(24,140,66,102),(25,139,67,101),(26,138,68,100),(27,137,69,99),(28,136,70,98),(29,135,71,97),(30,134,72,96),(31,152,73,114),(32,151,74,113),(33,150,75,112),(34,149,76,111),(35,148,58,110),(36,147,59,109),(37,146,60,108),(38,145,61,107)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 19A | ··· | 19I | 38A | ··· | 38BK |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C4 | D4 | D19 | Dic19 | D38 | C19⋊D4 |
| kernel | C23.D19 | C2×Dic19 | C22×C38 | C2×C38 | C38 | C23 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 9 | 18 | 9 | 36 |
Matrix representation of C23.D19 ►in GL3(𝔽229) generated by
| 228 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 22 | 228 |
| 228 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 228 |
| 1 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 228 |
| 1 | 0 | 0 |
| 0 | 43 | 0 |
| 0 | 68 | 16 |
| 122 | 0 | 0 |
| 0 | 120 | 114 |
| 0 | 177 | 109 |
G:=sub<GL(3,GF(229))| [228,0,0,0,1,22,0,0,228],[228,0,0,0,228,0,0,0,228],[1,0,0,0,228,0,0,0,228],[1,0,0,0,43,68,0,0,16],[122,0,0,0,120,177,0,114,109] >;
C23.D19 in GAP, Magma, Sage, TeX
C_2^3.D_{19} % in TeX
G:=Group("C2^3.D19"); // GroupNames label
G:=SmallGroup(304,18);
// by ID
G=gap.SmallGroup(304,18);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,20,101,7204]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^19=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
Export