direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C4×C19⋊C3, C76⋊4C6, C38⋊2C12, (C2×C76)⋊C3, C19⋊3(C2×C12), (C2×C38).2C6, C38.6(C2×C6), C22.(C2×C19⋊C3), C2.1(C22×C19⋊C3), (C22×C19⋊C3).2C2, (C2×C19⋊C3).6C22, SmallGroup(456,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C19 — C38 — C2×C19⋊C3 — C22×C19⋊C3 — C2×C4×C19⋊C3 |
| C19 — C2×C4×C19⋊C3 |
Generators and relations for C2×C4×C19⋊C3
G = < a,b,c,d | a2=b4=c19=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c11 >
Smallest permutation representation of C2×C4×C19⋊C3
►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 39 20 58)(2 40 21 59)(3 41 22 60)(4 42 23 61)(5 43 24 62)(6 44 25 63)(7 45 26 64)(8 46 27 65)(9 47 28 66)(10 48 29 67)(11 49 30 68)(12 50 31 69)(13 51 32 70)(14 52 33 71)(15 53 34 72)(16 54 35 73)(17 55 36 74)(18 56 37 75)(19 57 38 76)(77 115 96 134)(78 116 97 135)(79 117 98 136)(80 118 99 137)(81 119 100 138)(82 120 101 139)(83 121 102 140)(84 122 103 141)(85 123 104 142)(86 124 105 143)(87 125 106 144)(88 126 107 145)(89 127 108 146)(90 128 109 147)(91 129 110 148)(92 130 111 149)(93 131 112 150)(94 132 113 151)(95 133 114 152)
(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)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)(40 46 50)(41 53 42)(43 48 45)(44 55 56)(47 57 51)(49 52 54)(59 65 69)(60 72 61)(62 67 64)(63 74 75)(66 76 70)(68 71 73)(78 84 88)(79 91 80)(81 86 83)(82 93 94)(85 95 89)(87 90 92)(97 103 107)(98 110 99)(100 105 102)(101 112 113)(104 114 108)(106 109 111)(116 122 126)(117 129 118)(119 124 121)(120 131 132)(123 133 127)(125 128 130)(135 141 145)(136 148 137)(138 143 140)(139 150 151)(142 152 146)(144 147 149)
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), (1,39,20,58)(2,40,21,59)(3,41,22,60)(4,42,23,61)(5,43,24,62)(6,44,25,63)(7,45,26,64)(8,46,27,65)(9,47,28,66)(10,48,29,67)(11,49,30,68)(12,50,31,69)(13,51,32,70)(14,52,33,71)(15,53,34,72)(16,54,35,73)(17,55,36,74)(18,56,37,75)(19,57,38,76)(77,115,96,134)(78,116,97,135)(79,117,98,136)(80,118,99,137)(81,119,100,138)(82,120,101,139)(83,121,102,140)(84,122,103,141)(85,123,104,142)(86,124,105,143)(87,125,106,144)(88,126,107,145)(89,127,108,146)(90,128,109,147)(91,129,110,148)(92,130,111,149)(93,131,112,150)(94,132,113,151)(95,133,114,152), (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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73)(78,84,88)(79,91,80)(81,86,83)(82,93,94)(85,95,89)(87,90,92)(97,103,107)(98,110,99)(100,105,102)(101,112,113)(104,114,108)(106,109,111)(116,122,126)(117,129,118)(119,124,121)(120,131,132)(123,133,127)(125,128,130)(135,141,145)(136,148,137)(138,143,140)(139,150,151)(142,152,146)(144,147,149)>;
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), (1,39,20,58)(2,40,21,59)(3,41,22,60)(4,42,23,61)(5,43,24,62)(6,44,25,63)(7,45,26,64)(8,46,27,65)(9,47,28,66)(10,48,29,67)(11,49,30,68)(12,50,31,69)(13,51,32,70)(14,52,33,71)(15,53,34,72)(16,54,35,73)(17,55,36,74)(18,56,37,75)(19,57,38,76)(77,115,96,134)(78,116,97,135)(79,117,98,136)(80,118,99,137)(81,119,100,138)(82,120,101,139)(83,121,102,140)(84,122,103,141)(85,123,104,142)(86,124,105,143)(87,125,106,144)(88,126,107,145)(89,127,108,146)(90,128,109,147)(91,129,110,148)(92,130,111,149)(93,131,112,150)(94,132,113,151)(95,133,114,152), (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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)(40,46,50)(41,53,42)(43,48,45)(44,55,56)(47,57,51)(49,52,54)(59,65,69)(60,72,61)(62,67,64)(63,74,75)(66,76,70)(68,71,73)(78,84,88)(79,91,80)(81,86,83)(82,93,94)(85,95,89)(87,90,92)(97,103,107)(98,110,99)(100,105,102)(101,112,113)(104,114,108)(106,109,111)(116,122,126)(117,129,118)(119,124,121)(120,131,132)(123,133,127)(125,128,130)(135,141,145)(136,148,137)(138,143,140)(139,150,151)(142,152,146)(144,147,149) );
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)], [(1,39,20,58),(2,40,21,59),(3,41,22,60),(4,42,23,61),(5,43,24,62),(6,44,25,63),(7,45,26,64),(8,46,27,65),(9,47,28,66),(10,48,29,67),(11,49,30,68),(12,50,31,69),(13,51,32,70),(14,52,33,71),(15,53,34,72),(16,54,35,73),(17,55,36,74),(18,56,37,75),(19,57,38,76),(77,115,96,134),(78,116,97,135),(79,117,98,136),(80,118,99,137),(81,119,100,138),(82,120,101,139),(83,121,102,140),(84,122,103,141),(85,123,104,142),(86,124,105,143),(87,125,106,144),(88,126,107,145),(89,127,108,146),(90,128,109,147),(91,129,110,148),(92,130,111,149),(93,131,112,150),(94,132,113,151),(95,133,114,152)], [(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)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(21,27,31),(22,34,23),(24,29,26),(25,36,37),(28,38,32),(30,33,35),(40,46,50),(41,53,42),(43,48,45),(44,55,56),(47,57,51),(49,52,54),(59,65,69),(60,72,61),(62,67,64),(63,74,75),(66,76,70),(68,71,73),(78,84,88),(79,91,80),(81,86,83),(82,93,94),(85,95,89),(87,90,92),(97,103,107),(98,110,99),(100,105,102),(101,112,113),(104,114,108),(106,109,111),(116,122,126),(117,129,118),(119,124,121),(120,131,132),(123,133,127),(125,128,130),(135,141,145),(136,148,137),(138,143,140),(139,150,151),(142,152,146),(144,147,149)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 12A | ··· | 12H | 19A | ··· | 19F | 38A | ··· | 38R | 76A | ··· | 76X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 19 | 19 | 1 | 1 | 1 | 1 | 19 | ··· | 19 | 19 | ··· | 19 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | C19⋊C3 | C2×C19⋊C3 | C2×C19⋊C3 | C4×C19⋊C3 |
| kernel | C2×C4×C19⋊C3 | C4×C19⋊C3 | C22×C19⋊C3 | C2×C76 | C2×C19⋊C3 | C76 | C2×C38 | C38 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 6 | 12 | 6 | 24 |
Matrix representation of C2×C4×C19⋊C3 ►in GL4(𝔽229) generated by
| 228 | 0 | 0 | 0 |
| 0 | 228 | 0 | 0 |
| 0 | 0 | 228 | 0 |
| 0 | 0 | 0 | 228 |
| 1 | 0 | 0 | 0 |
| 0 | 122 | 0 | 0 |
| 0 | 0 | 122 | 0 |
| 0 | 0 | 0 | 122 |
| 1 | 0 | 0 | 0 |
| 0 | 101 | 223 | 1 |
| 0 | 102 | 223 | 1 |
| 0 | 101 | 224 | 1 |
| 94 | 0 | 0 | 0 |
| 0 | 208 | 149 | 7 |
| 0 | 0 | 0 | 1 |
| 0 | 51 | 118 | 21 |
G:=sub<GL(4,GF(229))| [228,0,0,0,0,228,0,0,0,0,228,0,0,0,0,228],[1,0,0,0,0,122,0,0,0,0,122,0,0,0,0,122],[1,0,0,0,0,101,102,101,0,223,223,224,0,1,1,1],[94,0,0,0,0,208,0,51,0,149,0,118,0,7,1,21] >;
C2×C4×C19⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_{19}\rtimes C_3 % in TeX
G:=Group("C2xC4xC19:C3"); // GroupNames label
G:=SmallGroup(456,19);
// by ID
G=gap.SmallGroup(456,19);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-19,66,1064]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^19=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^11>;
// generators/relations
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