direct product, non-abelian, soluble
Aliases: C14×SL2(𝔽3), Q8⋊C42, (C2×Q8)⋊C21, (C7×Q8)⋊7C6, (Q8×C14)⋊1C3, C2.2(A4×C14), (C2×C14).4A4, C14.11(C2×A4), C22.2(C7×A4), SmallGroup(336,169)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C7×Q8 — C7×SL2(𝔽3) — C14×SL2(𝔽3) |
| Q8 — C14×SL2(𝔽3) |
Generators and relations for C14×SL2(𝔽3)
G = < a,b,c,d | a14=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 C14×SL2(𝔽3)
►On 112 points
Generators in S112
(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)
(1 85 66 72)(2 86 67 73)(3 87 68 74)(4 88 69 75)(5 89 70 76)(6 90 57 77)(7 91 58 78)(8 92 59 79)(9 93 60 80)(10 94 61 81)(11 95 62 82)(12 96 63 83)(13 97 64 84)(14 98 65 71)(15 37 100 48)(16 38 101 49)(17 39 102 50)(18 40 103 51)(19 41 104 52)(20 42 105 53)(21 29 106 54)(22 30 107 55)(23 31 108 56)(24 32 109 43)(25 33 110 44)(26 34 111 45)(27 35 112 46)(28 36 99 47)
(1 18 66 103)(2 19 67 104)(3 20 68 105)(4 21 69 106)(5 22 70 107)(6 23 57 108)(7 24 58 109)(8 25 59 110)(9 26 60 111)(10 27 61 112)(11 28 62 99)(12 15 63 100)(13 16 64 101)(14 17 65 102)(29 88 54 75)(30 89 55 76)(31 90 56 77)(32 91 43 78)(33 92 44 79)(34 93 45 80)(35 94 46 81)(36 95 47 82)(37 96 48 83)(38 97 49 84)(39 98 50 71)(40 85 51 72)(41 86 52 73)(42 87 53 74)
(15 96 48)(16 97 49)(17 98 50)(18 85 51)(19 86 52)(20 87 53)(21 88 54)(22 89 55)(23 90 56)(24 91 43)(25 92 44)(26 93 45)(27 94 46)(28 95 47)(29 106 75)(30 107 76)(31 108 77)(32 109 78)(33 110 79)(34 111 80)(35 112 81)(36 99 82)(37 100 83)(38 101 84)(39 102 71)(40 103 72)(41 104 73)(42 105 74)
G:=sub<Sym(112)| (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), (1,85,66,72)(2,86,67,73)(3,87,68,74)(4,88,69,75)(5,89,70,76)(6,90,57,77)(7,91,58,78)(8,92,59,79)(9,93,60,80)(10,94,61,81)(11,95,62,82)(12,96,63,83)(13,97,64,84)(14,98,65,71)(15,37,100,48)(16,38,101,49)(17,39,102,50)(18,40,103,51)(19,41,104,52)(20,42,105,53)(21,29,106,54)(22,30,107,55)(23,31,108,56)(24,32,109,43)(25,33,110,44)(26,34,111,45)(27,35,112,46)(28,36,99,47), (1,18,66,103)(2,19,67,104)(3,20,68,105)(4,21,69,106)(5,22,70,107)(6,23,57,108)(7,24,58,109)(8,25,59,110)(9,26,60,111)(10,27,61,112)(11,28,62,99)(12,15,63,100)(13,16,64,101)(14,17,65,102)(29,88,54,75)(30,89,55,76)(31,90,56,77)(32,91,43,78)(33,92,44,79)(34,93,45,80)(35,94,46,81)(36,95,47,82)(37,96,48,83)(38,97,49,84)(39,98,50,71)(40,85,51,72)(41,86,52,73)(42,87,53,74), (15,96,48)(16,97,49)(17,98,50)(18,85,51)(19,86,52)(20,87,53)(21,88,54)(22,89,55)(23,90,56)(24,91,43)(25,92,44)(26,93,45)(27,94,46)(28,95,47)(29,106,75)(30,107,76)(31,108,77)(32,109,78)(33,110,79)(34,111,80)(35,112,81)(36,99,82)(37,100,83)(38,101,84)(39,102,71)(40,103,72)(41,104,73)(42,105,74)>;
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), (1,85,66,72)(2,86,67,73)(3,87,68,74)(4,88,69,75)(5,89,70,76)(6,90,57,77)(7,91,58,78)(8,92,59,79)(9,93,60,80)(10,94,61,81)(11,95,62,82)(12,96,63,83)(13,97,64,84)(14,98,65,71)(15,37,100,48)(16,38,101,49)(17,39,102,50)(18,40,103,51)(19,41,104,52)(20,42,105,53)(21,29,106,54)(22,30,107,55)(23,31,108,56)(24,32,109,43)(25,33,110,44)(26,34,111,45)(27,35,112,46)(28,36,99,47), (1,18,66,103)(2,19,67,104)(3,20,68,105)(4,21,69,106)(5,22,70,107)(6,23,57,108)(7,24,58,109)(8,25,59,110)(9,26,60,111)(10,27,61,112)(11,28,62,99)(12,15,63,100)(13,16,64,101)(14,17,65,102)(29,88,54,75)(30,89,55,76)(31,90,56,77)(32,91,43,78)(33,92,44,79)(34,93,45,80)(35,94,46,81)(36,95,47,82)(37,96,48,83)(38,97,49,84)(39,98,50,71)(40,85,51,72)(41,86,52,73)(42,87,53,74), (15,96,48)(16,97,49)(17,98,50)(18,85,51)(19,86,52)(20,87,53)(21,88,54)(22,89,55)(23,90,56)(24,91,43)(25,92,44)(26,93,45)(27,94,46)(28,95,47)(29,106,75)(30,107,76)(31,108,77)(32,109,78)(33,110,79)(34,111,80)(35,112,81)(36,99,82)(37,100,83)(38,101,84)(39,102,71)(40,103,72)(41,104,73)(42,105,74) );
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)], [(1,85,66,72),(2,86,67,73),(3,87,68,74),(4,88,69,75),(5,89,70,76),(6,90,57,77),(7,91,58,78),(8,92,59,79),(9,93,60,80),(10,94,61,81),(11,95,62,82),(12,96,63,83),(13,97,64,84),(14,98,65,71),(15,37,100,48),(16,38,101,49),(17,39,102,50),(18,40,103,51),(19,41,104,52),(20,42,105,53),(21,29,106,54),(22,30,107,55),(23,31,108,56),(24,32,109,43),(25,33,110,44),(26,34,111,45),(27,35,112,46),(28,36,99,47)], [(1,18,66,103),(2,19,67,104),(3,20,68,105),(4,21,69,106),(5,22,70,107),(6,23,57,108),(7,24,58,109),(8,25,59,110),(9,26,60,111),(10,27,61,112),(11,28,62,99),(12,15,63,100),(13,16,64,101),(14,17,65,102),(29,88,54,75),(30,89,55,76),(31,90,56,77),(32,91,43,78),(33,92,44,79),(34,93,45,80),(35,94,46,81),(36,95,47,82),(37,96,48,83),(38,97,49,84),(39,98,50,71),(40,85,51,72),(41,86,52,73),(42,87,53,74)], [(15,96,48),(16,97,49),(17,98,50),(18,85,51),(19,86,52),(20,87,53),(21,88,54),(22,89,55),(23,90,56),(24,91,43),(25,92,44),(26,93,45),(27,94,46),(28,95,47),(29,106,75),(30,107,76),(31,108,77),(32,109,78),(33,110,79),(34,111,80),(35,112,81),(36,99,82),(37,100,83),(38,101,84),(39,102,71),(40,103,72),(41,104,73),(42,105,74)]])
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 6A | ··· | 6F | 7A | ··· | 7F | 14A | ··· | 14R | 21A | ··· | 21L | 28A | ··· | 28L | 42A | ··· | 42AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | - | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 | SL2(𝔽3) | SL2(𝔽3) | C7×SL2(𝔽3) | A4 | C2×A4 | C7×A4 | A4×C14 |
| kernel | C14×SL2(𝔽3) | C7×SL2(𝔽3) | Q8×C14 | C7×Q8 | C2×SL2(𝔽3) | SL2(𝔽3) | C2×Q8 | Q8 | C14 | C14 | C2 | C2×C14 | C14 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 2 | 4 | 36 | 1 | 1 | 6 | 6 |
Matrix representation of C14×SL2(𝔽3) ►in GL3(𝔽337) generated by
| 336 | 0 | 0 |
| 0 | 285 | 0 |
| 0 | 0 | 285 |
| 1 | 0 | 0 |
| 0 | 0 | 336 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 209 | 208 |
| 0 | 208 | 128 |
| 128 | 0 | 0 |
| 0 | 1 | 208 |
| 0 | 0 | 128 |
G:=sub<GL(3,GF(337))| [336,0,0,0,285,0,0,0,285],[1,0,0,0,0,1,0,336,0],[1,0,0,0,209,208,0,208,128],[128,0,0,0,1,0,0,208,128] >;
C14×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_{14}\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C14xSL(2,3)"); // GroupNames label
G:=SmallGroup(336,169);
// by ID
G=gap.SmallGroup(336,169);
# by ID
G:=PCGroup([6,-2,-3,-7,-2,2,-2,1017,117,1900,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^14=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
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