direct product, non-abelian, soluble
Aliases: C19×SL2(𝔽3), Q8⋊C57, C38.2A4, C2.(A4×C19), (Q8×C19)⋊1C3, SmallGroup(456,22)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C19×SL2(𝔽3) |
Generators and relations for C19×SL2(𝔽3)
G = < a,b,c,d | a19=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >
Smallest permutation representation of C19×SL2(𝔽3)
►On 152 points
Generators in S152
(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 87 60 43)(2 88 61 44)(3 89 62 45)(4 90 63 46)(5 91 64 47)(6 92 65 48)(7 93 66 49)(8 94 67 50)(9 95 68 51)(10 77 69 52)(11 78 70 53)(12 79 71 54)(13 80 72 55)(14 81 73 56)(15 82 74 57)(16 83 75 39)(17 84 76 40)(18 85 58 41)(19 86 59 42)(20 145 107 130)(21 146 108 131)(22 147 109 132)(23 148 110 133)(24 149 111 115)(25 150 112 116)(26 151 113 117)(27 152 114 118)(28 134 96 119)(29 135 97 120)(30 136 98 121)(31 137 99 122)(32 138 100 123)(33 139 101 124)(34 140 102 125)(35 141 103 126)(36 142 104 127)(37 143 105 128)(38 144 106 129)
(1 106 60 38)(2 107 61 20)(3 108 62 21)(4 109 63 22)(5 110 64 23)(6 111 65 24)(7 112 66 25)(8 113 67 26)(9 114 68 27)(10 96 69 28)(11 97 70 29)(12 98 71 30)(13 99 72 31)(14 100 73 32)(15 101 74 33)(16 102 75 34)(17 103 76 35)(18 104 58 36)(19 105 59 37)(39 125 83 140)(40 126 84 141)(41 127 85 142)(42 128 86 143)(43 129 87 144)(44 130 88 145)(45 131 89 146)(46 132 90 147)(47 133 91 148)(48 115 92 149)(49 116 93 150)(50 117 94 151)(51 118 95 152)(52 119 77 134)(53 120 78 135)(54 121 79 136)(55 122 80 137)(56 123 81 138)(57 124 82 139)
(20 44 130)(21 45 131)(22 46 132)(23 47 133)(24 48 115)(25 49 116)(26 50 117)(27 51 118)(28 52 119)(29 53 120)(30 54 121)(31 55 122)(32 56 123)(33 57 124)(34 39 125)(35 40 126)(36 41 127)(37 42 128)(38 43 129)(77 134 96)(78 135 97)(79 136 98)(80 137 99)(81 138 100)(82 139 101)(83 140 102)(84 141 103)(85 142 104)(86 143 105)(87 144 106)(88 145 107)(89 146 108)(90 147 109)(91 148 110)(92 149 111)(93 150 112)(94 151 113)(95 152 114)
G:=sub<Sym(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), (1,87,60,43)(2,88,61,44)(3,89,62,45)(4,90,63,46)(5,91,64,47)(6,92,65,48)(7,93,66,49)(8,94,67,50)(9,95,68,51)(10,77,69,52)(11,78,70,53)(12,79,71,54)(13,80,72,55)(14,81,73,56)(15,82,74,57)(16,83,75,39)(17,84,76,40)(18,85,58,41)(19,86,59,42)(20,145,107,130)(21,146,108,131)(22,147,109,132)(23,148,110,133)(24,149,111,115)(25,150,112,116)(26,151,113,117)(27,152,114,118)(28,134,96,119)(29,135,97,120)(30,136,98,121)(31,137,99,122)(32,138,100,123)(33,139,101,124)(34,140,102,125)(35,141,103,126)(36,142,104,127)(37,143,105,128)(38,144,106,129), (1,106,60,38)(2,107,61,20)(3,108,62,21)(4,109,63,22)(5,110,64,23)(6,111,65,24)(7,112,66,25)(8,113,67,26)(9,114,68,27)(10,96,69,28)(11,97,70,29)(12,98,71,30)(13,99,72,31)(14,100,73,32)(15,101,74,33)(16,102,75,34)(17,103,76,35)(18,104,58,36)(19,105,59,37)(39,125,83,140)(40,126,84,141)(41,127,85,142)(42,128,86,143)(43,129,87,144)(44,130,88,145)(45,131,89,146)(46,132,90,147)(47,133,91,148)(48,115,92,149)(49,116,93,150)(50,117,94,151)(51,118,95,152)(52,119,77,134)(53,120,78,135)(54,121,79,136)(55,122,80,137)(56,123,81,138)(57,124,82,139), (20,44,130)(21,45,131)(22,46,132)(23,47,133)(24,48,115)(25,49,116)(26,50,117)(27,51,118)(28,52,119)(29,53,120)(30,54,121)(31,55,122)(32,56,123)(33,57,124)(34,39,125)(35,40,126)(36,41,127)(37,42,128)(38,43,129)(77,134,96)(78,135,97)(79,136,98)(80,137,99)(81,138,100)(82,139,101)(83,140,102)(84,141,103)(85,142,104)(86,143,105)(87,144,106)(88,145,107)(89,146,108)(90,147,109)(91,148,110)(92,149,111)(93,150,112)(94,151,113)(95,152,114)>;
G:=Group( (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), (1,87,60,43)(2,88,61,44)(3,89,62,45)(4,90,63,46)(5,91,64,47)(6,92,65,48)(7,93,66,49)(8,94,67,50)(9,95,68,51)(10,77,69,52)(11,78,70,53)(12,79,71,54)(13,80,72,55)(14,81,73,56)(15,82,74,57)(16,83,75,39)(17,84,76,40)(18,85,58,41)(19,86,59,42)(20,145,107,130)(21,146,108,131)(22,147,109,132)(23,148,110,133)(24,149,111,115)(25,150,112,116)(26,151,113,117)(27,152,114,118)(28,134,96,119)(29,135,97,120)(30,136,98,121)(31,137,99,122)(32,138,100,123)(33,139,101,124)(34,140,102,125)(35,141,103,126)(36,142,104,127)(37,143,105,128)(38,144,106,129), (1,106,60,38)(2,107,61,20)(3,108,62,21)(4,109,63,22)(5,110,64,23)(6,111,65,24)(7,112,66,25)(8,113,67,26)(9,114,68,27)(10,96,69,28)(11,97,70,29)(12,98,71,30)(13,99,72,31)(14,100,73,32)(15,101,74,33)(16,102,75,34)(17,103,76,35)(18,104,58,36)(19,105,59,37)(39,125,83,140)(40,126,84,141)(41,127,85,142)(42,128,86,143)(43,129,87,144)(44,130,88,145)(45,131,89,146)(46,132,90,147)(47,133,91,148)(48,115,92,149)(49,116,93,150)(50,117,94,151)(51,118,95,152)(52,119,77,134)(53,120,78,135)(54,121,79,136)(55,122,80,137)(56,123,81,138)(57,124,82,139), (20,44,130)(21,45,131)(22,46,132)(23,47,133)(24,48,115)(25,49,116)(26,50,117)(27,51,118)(28,52,119)(29,53,120)(30,54,121)(31,55,122)(32,56,123)(33,57,124)(34,39,125)(35,40,126)(36,41,127)(37,42,128)(38,43,129)(77,134,96)(78,135,97)(79,136,98)(80,137,99)(81,138,100)(82,139,101)(83,140,102)(84,141,103)(85,142,104)(86,143,105)(87,144,106)(88,145,107)(89,146,108)(90,147,109)(91,148,110)(92,149,111)(93,150,112)(94,151,113)(95,152,114) );
G=PermutationGroup([[(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)], [(1,87,60,43),(2,88,61,44),(3,89,62,45),(4,90,63,46),(5,91,64,47),(6,92,65,48),(7,93,66,49),(8,94,67,50),(9,95,68,51),(10,77,69,52),(11,78,70,53),(12,79,71,54),(13,80,72,55),(14,81,73,56),(15,82,74,57),(16,83,75,39),(17,84,76,40),(18,85,58,41),(19,86,59,42),(20,145,107,130),(21,146,108,131),(22,147,109,132),(23,148,110,133),(24,149,111,115),(25,150,112,116),(26,151,113,117),(27,152,114,118),(28,134,96,119),(29,135,97,120),(30,136,98,121),(31,137,99,122),(32,138,100,123),(33,139,101,124),(34,140,102,125),(35,141,103,126),(36,142,104,127),(37,143,105,128),(38,144,106,129)], [(1,106,60,38),(2,107,61,20),(3,108,62,21),(4,109,63,22),(5,110,64,23),(6,111,65,24),(7,112,66,25),(8,113,67,26),(9,114,68,27),(10,96,69,28),(11,97,70,29),(12,98,71,30),(13,99,72,31),(14,100,73,32),(15,101,74,33),(16,102,75,34),(17,103,76,35),(18,104,58,36),(19,105,59,37),(39,125,83,140),(40,126,84,141),(41,127,85,142),(42,128,86,143),(43,129,87,144),(44,130,88,145),(45,131,89,146),(46,132,90,147),(47,133,91,148),(48,115,92,149),(49,116,93,150),(50,117,94,151),(51,118,95,152),(52,119,77,134),(53,120,78,135),(54,121,79,136),(55,122,80,137),(56,123,81,138),(57,124,82,139)], [(20,44,130),(21,45,131),(22,46,132),(23,47,133),(24,48,115),(25,49,116),(26,50,117),(27,51,118),(28,52,119),(29,53,120),(30,54,121),(31,55,122),(32,56,123),(33,57,124),(34,39,125),(35,40,126),(36,41,127),(37,42,128),(38,43,129),(77,134,96),(78,135,97),(79,136,98),(80,137,99),(81,138,100),(82,139,101),(83,140,102),(84,141,103),(85,142,104),(86,143,105),(87,144,106),(88,145,107),(89,146,108),(90,147,109),(91,148,110),(92,149,111),(93,150,112),(94,151,113),(95,152,114)]])
133 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4 | 6A | 6B | 19A | ··· | 19R | 38A | ··· | 38R | 57A | ··· | 57AJ | 76A | ··· | 76R | 114A | ··· | 114AJ |
| order | 1 | 2 | 3 | 3 | 4 | 6 | 6 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 76 | ··· | 76 | 114 | ··· | 114 |
| size | 1 | 1 | 4 | 4 | 6 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 |
| type | + | - | + | ||||||
| image | C1 | C3 | C19 | C57 | SL2(𝔽3) | SL2(𝔽3) | C19×SL2(𝔽3) | A4 | A4×C19 |
| kernel | C19×SL2(𝔽3) | Q8×C19 | SL2(𝔽3) | Q8 | C19 | C19 | C1 | C38 | C2 |
| # reps | 1 | 2 | 18 | 36 | 1 | 2 | 54 | 1 | 18 |
Matrix representation of C19×SL2(𝔽3) ►in GL2(𝔽229) generated by
| 165 | 0 |
| 0 | 165 |
| 134 | 94 |
| 94 | 95 |
| 0 | 1 |
| 228 | 0 |
| 0 | 1 |
| 95 | 135 |
G:=sub<GL(2,GF(229))| [165,0,0,165],[134,94,94,95],[0,228,1,0],[0,95,1,135] >;
C19×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_{19}\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C19xSL(2,3)"); // GroupNames label
G:=SmallGroup(456,22);
// by ID
G=gap.SmallGroup(456,22);
# by ID
G:=PCGroup([5,-3,-19,-2,2,-2,1712,72,3423,133,58]);
// Polycyclic
G:=Group<a,b,c,d|a^19=b^4=d^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations
Export