direct product, non-abelian, soluble
Aliases: C17×SL2(𝔽3), Q8⋊C51, C34.A4, (Q8×C17)⋊C3, C2.(A4×C17), SmallGroup(408,14)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C17×SL2(𝔽3) |
Generators and relations for C17×SL2(𝔽3)
G = < a,b,c,d | a17=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 C17×SL2(𝔽3)
►On 136 points
Generators in S136
(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)
(1 39 53 101)(2 40 54 102)(3 41 55 86)(4 42 56 87)(5 43 57 88)(6 44 58 89)(7 45 59 90)(8 46 60 91)(9 47 61 92)(10 48 62 93)(11 49 63 94)(12 50 64 95)(13 51 65 96)(14 35 66 97)(15 36 67 98)(16 37 68 99)(17 38 52 100)(18 127 76 115)(19 128 77 116)(20 129 78 117)(21 130 79 118)(22 131 80 119)(23 132 81 103)(24 133 82 104)(25 134 83 105)(26 135 84 106)(27 136 85 107)(28 120 69 108)(29 121 70 109)(30 122 71 110)(31 123 72 111)(32 124 73 112)(33 125 74 113)(34 126 75 114)
(1 116 53 128)(2 117 54 129)(3 118 55 130)(4 119 56 131)(5 103 57 132)(6 104 58 133)(7 105 59 134)(8 106 60 135)(9 107 61 136)(10 108 62 120)(11 109 63 121)(12 110 64 122)(13 111 65 123)(14 112 66 124)(15 113 67 125)(16 114 68 126)(17 115 52 127)(18 38 76 100)(19 39 77 101)(20 40 78 102)(21 41 79 86)(22 42 80 87)(23 43 81 88)(24 44 82 89)(25 45 83 90)(26 46 84 91)(27 47 85 92)(28 48 69 93)(29 49 70 94)(30 50 71 95)(31 51 72 96)(32 35 73 97)(33 36 74 98)(34 37 75 99)
(18 127 100)(19 128 101)(20 129 102)(21 130 86)(22 131 87)(23 132 88)(24 133 89)(25 134 90)(26 135 91)(27 136 92)(28 120 93)(29 121 94)(30 122 95)(31 123 96)(32 124 97)(33 125 98)(34 126 99)(35 73 112)(36 74 113)(37 75 114)(38 76 115)(39 77 116)(40 78 117)(41 79 118)(42 80 119)(43 81 103)(44 82 104)(45 83 105)(46 84 106)(47 85 107)(48 69 108)(49 70 109)(50 71 110)(51 72 111)
G:=sub<Sym(136)| (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), (1,39,53,101)(2,40,54,102)(3,41,55,86)(4,42,56,87)(5,43,57,88)(6,44,58,89)(7,45,59,90)(8,46,60,91)(9,47,61,92)(10,48,62,93)(11,49,63,94)(12,50,64,95)(13,51,65,96)(14,35,66,97)(15,36,67,98)(16,37,68,99)(17,38,52,100)(18,127,76,115)(19,128,77,116)(20,129,78,117)(21,130,79,118)(22,131,80,119)(23,132,81,103)(24,133,82,104)(25,134,83,105)(26,135,84,106)(27,136,85,107)(28,120,69,108)(29,121,70,109)(30,122,71,110)(31,123,72,111)(32,124,73,112)(33,125,74,113)(34,126,75,114), (1,116,53,128)(2,117,54,129)(3,118,55,130)(4,119,56,131)(5,103,57,132)(6,104,58,133)(7,105,59,134)(8,106,60,135)(9,107,61,136)(10,108,62,120)(11,109,63,121)(12,110,64,122)(13,111,65,123)(14,112,66,124)(15,113,67,125)(16,114,68,126)(17,115,52,127)(18,38,76,100)(19,39,77,101)(20,40,78,102)(21,41,79,86)(22,42,80,87)(23,43,81,88)(24,44,82,89)(25,45,83,90)(26,46,84,91)(27,47,85,92)(28,48,69,93)(29,49,70,94)(30,50,71,95)(31,51,72,96)(32,35,73,97)(33,36,74,98)(34,37,75,99), (18,127,100)(19,128,101)(20,129,102)(21,130,86)(22,131,87)(23,132,88)(24,133,89)(25,134,90)(26,135,91)(27,136,92)(28,120,93)(29,121,94)(30,122,95)(31,123,96)(32,124,97)(33,125,98)(34,126,99)(35,73,112)(36,74,113)(37,75,114)(38,76,115)(39,77,116)(40,78,117)(41,79,118)(42,80,119)(43,81,103)(44,82,104)(45,83,105)(46,84,106)(47,85,107)(48,69,108)(49,70,109)(50,71,110)(51,72,111)>;
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), (1,39,53,101)(2,40,54,102)(3,41,55,86)(4,42,56,87)(5,43,57,88)(6,44,58,89)(7,45,59,90)(8,46,60,91)(9,47,61,92)(10,48,62,93)(11,49,63,94)(12,50,64,95)(13,51,65,96)(14,35,66,97)(15,36,67,98)(16,37,68,99)(17,38,52,100)(18,127,76,115)(19,128,77,116)(20,129,78,117)(21,130,79,118)(22,131,80,119)(23,132,81,103)(24,133,82,104)(25,134,83,105)(26,135,84,106)(27,136,85,107)(28,120,69,108)(29,121,70,109)(30,122,71,110)(31,123,72,111)(32,124,73,112)(33,125,74,113)(34,126,75,114), (1,116,53,128)(2,117,54,129)(3,118,55,130)(4,119,56,131)(5,103,57,132)(6,104,58,133)(7,105,59,134)(8,106,60,135)(9,107,61,136)(10,108,62,120)(11,109,63,121)(12,110,64,122)(13,111,65,123)(14,112,66,124)(15,113,67,125)(16,114,68,126)(17,115,52,127)(18,38,76,100)(19,39,77,101)(20,40,78,102)(21,41,79,86)(22,42,80,87)(23,43,81,88)(24,44,82,89)(25,45,83,90)(26,46,84,91)(27,47,85,92)(28,48,69,93)(29,49,70,94)(30,50,71,95)(31,51,72,96)(32,35,73,97)(33,36,74,98)(34,37,75,99), (18,127,100)(19,128,101)(20,129,102)(21,130,86)(22,131,87)(23,132,88)(24,133,89)(25,134,90)(26,135,91)(27,136,92)(28,120,93)(29,121,94)(30,122,95)(31,123,96)(32,124,97)(33,125,98)(34,126,99)(35,73,112)(36,74,113)(37,75,114)(38,76,115)(39,77,116)(40,78,117)(41,79,118)(42,80,119)(43,81,103)(44,82,104)(45,83,105)(46,84,106)(47,85,107)(48,69,108)(49,70,109)(50,71,110)(51,72,111) );
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)], [(1,39,53,101),(2,40,54,102),(3,41,55,86),(4,42,56,87),(5,43,57,88),(6,44,58,89),(7,45,59,90),(8,46,60,91),(9,47,61,92),(10,48,62,93),(11,49,63,94),(12,50,64,95),(13,51,65,96),(14,35,66,97),(15,36,67,98),(16,37,68,99),(17,38,52,100),(18,127,76,115),(19,128,77,116),(20,129,78,117),(21,130,79,118),(22,131,80,119),(23,132,81,103),(24,133,82,104),(25,134,83,105),(26,135,84,106),(27,136,85,107),(28,120,69,108),(29,121,70,109),(30,122,71,110),(31,123,72,111),(32,124,73,112),(33,125,74,113),(34,126,75,114)], [(1,116,53,128),(2,117,54,129),(3,118,55,130),(4,119,56,131),(5,103,57,132),(6,104,58,133),(7,105,59,134),(8,106,60,135),(9,107,61,136),(10,108,62,120),(11,109,63,121),(12,110,64,122),(13,111,65,123),(14,112,66,124),(15,113,67,125),(16,114,68,126),(17,115,52,127),(18,38,76,100),(19,39,77,101),(20,40,78,102),(21,41,79,86),(22,42,80,87),(23,43,81,88),(24,44,82,89),(25,45,83,90),(26,46,84,91),(27,47,85,92),(28,48,69,93),(29,49,70,94),(30,50,71,95),(31,51,72,96),(32,35,73,97),(33,36,74,98),(34,37,75,99)], [(18,127,100),(19,128,101),(20,129,102),(21,130,86),(22,131,87),(23,132,88),(24,133,89),(25,134,90),(26,135,91),(27,136,92),(28,120,93),(29,121,94),(30,122,95),(31,123,96),(32,124,97),(33,125,98),(34,126,99),(35,73,112),(36,74,113),(37,75,114),(38,76,115),(39,77,116),(40,78,117),(41,79,118),(42,80,119),(43,81,103),(44,82,104),(45,83,105),(46,84,106),(47,85,107),(48,69,108),(49,70,109),(50,71,110),(51,72,111)]])
119 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4 | 6A | 6B | 17A | ··· | 17P | 34A | ··· | 34P | 51A | ··· | 51AF | 68A | ··· | 68P | 102A | ··· | 102AF |
| order | 1 | 2 | 3 | 3 | 4 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 |
| size | 1 | 1 | 4 | 4 | 6 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
119 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 |
| type | + | - | + | ||||||
| image | C1 | C3 | C17 | C51 | SL2(𝔽3) | SL2(𝔽3) | C17×SL2(𝔽3) | A4 | A4×C17 |
| kernel | C17×SL2(𝔽3) | Q8×C17 | SL2(𝔽3) | Q8 | C17 | C17 | C1 | C34 | C2 |
| # reps | 1 | 2 | 16 | 32 | 1 | 2 | 48 | 1 | 16 |
Matrix representation of C17×SL2(𝔽3) ►in GL2(𝔽409) generated by
| 36 | 0 |
| 0 | 36 |
| 53 | 355 |
| 355 | 356 |
| 0 | 408 |
| 1 | 0 |
| 1 | 53 |
| 0 | 355 |
G:=sub<GL(2,GF(409))| [36,0,0,36],[53,355,355,356],[0,1,408,0],[1,0,53,355] >;
C17×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_{17}\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C17xSL(2,3)"); // GroupNames label
G:=SmallGroup(408,14);
// by ID
G=gap.SmallGroup(408,14);
# by ID
G:=PCGroup([5,-3,-17,-2,2,-2,1532,72,3063,133,58]);
// Polycyclic
G:=Group<a,b,c,d|a^17=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