direct product, non-abelian, soluble
Aliases: C13×SL2(𝔽3), Q8⋊C39, C26.2A4, C2.(A4×C13), (Q8×C13)⋊1C3, SmallGroup(312,25)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C13×SL2(𝔽3) |
Generators and relations for C13×SL2(𝔽3)
G = < a,b,c,d | a13=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 C13×SL2(𝔽3)
►On 104 points
Generators in S104
(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)
(1 21 61 33)(2 22 62 34)(3 23 63 35)(4 24 64 36)(5 25 65 37)(6 26 53 38)(7 14 54 39)(8 15 55 27)(9 16 56 28)(10 17 57 29)(11 18 58 30)(12 19 59 31)(13 20 60 32)(40 70 93 91)(41 71 94 79)(42 72 95 80)(43 73 96 81)(44 74 97 82)(45 75 98 83)(46 76 99 84)(47 77 100 85)(48 78 101 86)(49 66 102 87)(50 67 103 88)(51 68 104 89)(52 69 92 90)
(1 87 61 66)(2 88 62 67)(3 89 63 68)(4 90 64 69)(5 91 65 70)(6 79 53 71)(7 80 54 72)(8 81 55 73)(9 82 56 74)(10 83 57 75)(11 84 58 76)(12 85 59 77)(13 86 60 78)(14 95 39 42)(15 96 27 43)(16 97 28 44)(17 98 29 45)(18 99 30 46)(19 100 31 47)(20 101 32 48)(21 102 33 49)(22 103 34 50)(23 104 35 51)(24 92 36 52)(25 93 37 40)(26 94 38 41)
(14 95 80)(15 96 81)(16 97 82)(17 98 83)(18 99 84)(19 100 85)(20 101 86)(21 102 87)(22 103 88)(23 104 89)(24 92 90)(25 93 91)(26 94 79)(27 43 73)(28 44 74)(29 45 75)(30 46 76)(31 47 77)(32 48 78)(33 49 66)(34 50 67)(35 51 68)(36 52 69)(37 40 70)(38 41 71)(39 42 72)
G:=sub<Sym(104)| (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), (1,21,61,33)(2,22,62,34)(3,23,63,35)(4,24,64,36)(5,25,65,37)(6,26,53,38)(7,14,54,39)(8,15,55,27)(9,16,56,28)(10,17,57,29)(11,18,58,30)(12,19,59,31)(13,20,60,32)(40,70,93,91)(41,71,94,79)(42,72,95,80)(43,73,96,81)(44,74,97,82)(45,75,98,83)(46,76,99,84)(47,77,100,85)(48,78,101,86)(49,66,102,87)(50,67,103,88)(51,68,104,89)(52,69,92,90), (1,87,61,66)(2,88,62,67)(3,89,63,68)(4,90,64,69)(5,91,65,70)(6,79,53,71)(7,80,54,72)(8,81,55,73)(9,82,56,74)(10,83,57,75)(11,84,58,76)(12,85,59,77)(13,86,60,78)(14,95,39,42)(15,96,27,43)(16,97,28,44)(17,98,29,45)(18,99,30,46)(19,100,31,47)(20,101,32,48)(21,102,33,49)(22,103,34,50)(23,104,35,51)(24,92,36,52)(25,93,37,40)(26,94,38,41), (14,95,80)(15,96,81)(16,97,82)(17,98,83)(18,99,84)(19,100,85)(20,101,86)(21,102,87)(22,103,88)(23,104,89)(24,92,90)(25,93,91)(26,94,79)(27,43,73)(28,44,74)(29,45,75)(30,46,76)(31,47,77)(32,48,78)(33,49,66)(34,50,67)(35,51,68)(36,52,69)(37,40,70)(38,41,71)(39,42,72)>;
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), (1,21,61,33)(2,22,62,34)(3,23,63,35)(4,24,64,36)(5,25,65,37)(6,26,53,38)(7,14,54,39)(8,15,55,27)(9,16,56,28)(10,17,57,29)(11,18,58,30)(12,19,59,31)(13,20,60,32)(40,70,93,91)(41,71,94,79)(42,72,95,80)(43,73,96,81)(44,74,97,82)(45,75,98,83)(46,76,99,84)(47,77,100,85)(48,78,101,86)(49,66,102,87)(50,67,103,88)(51,68,104,89)(52,69,92,90), (1,87,61,66)(2,88,62,67)(3,89,63,68)(4,90,64,69)(5,91,65,70)(6,79,53,71)(7,80,54,72)(8,81,55,73)(9,82,56,74)(10,83,57,75)(11,84,58,76)(12,85,59,77)(13,86,60,78)(14,95,39,42)(15,96,27,43)(16,97,28,44)(17,98,29,45)(18,99,30,46)(19,100,31,47)(20,101,32,48)(21,102,33,49)(22,103,34,50)(23,104,35,51)(24,92,36,52)(25,93,37,40)(26,94,38,41), (14,95,80)(15,96,81)(16,97,82)(17,98,83)(18,99,84)(19,100,85)(20,101,86)(21,102,87)(22,103,88)(23,104,89)(24,92,90)(25,93,91)(26,94,79)(27,43,73)(28,44,74)(29,45,75)(30,46,76)(31,47,77)(32,48,78)(33,49,66)(34,50,67)(35,51,68)(36,52,69)(37,40,70)(38,41,71)(39,42,72) );
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)], [(1,21,61,33),(2,22,62,34),(3,23,63,35),(4,24,64,36),(5,25,65,37),(6,26,53,38),(7,14,54,39),(8,15,55,27),(9,16,56,28),(10,17,57,29),(11,18,58,30),(12,19,59,31),(13,20,60,32),(40,70,93,91),(41,71,94,79),(42,72,95,80),(43,73,96,81),(44,74,97,82),(45,75,98,83),(46,76,99,84),(47,77,100,85),(48,78,101,86),(49,66,102,87),(50,67,103,88),(51,68,104,89),(52,69,92,90)], [(1,87,61,66),(2,88,62,67),(3,89,63,68),(4,90,64,69),(5,91,65,70),(6,79,53,71),(7,80,54,72),(8,81,55,73),(9,82,56,74),(10,83,57,75),(11,84,58,76),(12,85,59,77),(13,86,60,78),(14,95,39,42),(15,96,27,43),(16,97,28,44),(17,98,29,45),(18,99,30,46),(19,100,31,47),(20,101,32,48),(21,102,33,49),(22,103,34,50),(23,104,35,51),(24,92,36,52),(25,93,37,40),(26,94,38,41)], [(14,95,80),(15,96,81),(16,97,82),(17,98,83),(18,99,84),(19,100,85),(20,101,86),(21,102,87),(22,103,88),(23,104,89),(24,92,90),(25,93,91),(26,94,79),(27,43,73),(28,44,74),(29,45,75),(30,46,76),(31,47,77),(32,48,78),(33,49,66),(34,50,67),(35,51,68),(36,52,69),(37,40,70),(38,41,71),(39,42,72)]])
91 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4 | 6A | 6B | 13A | ··· | 13L | 26A | ··· | 26L | 39A | ··· | 39X | 52A | ··· | 52L | 78A | ··· | 78X |
| order | 1 | 2 | 3 | 3 | 4 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 52 | ··· | 52 | 78 | ··· | 78 |
| size | 1 | 1 | 4 | 4 | 6 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
91 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 |
| type | + | - | + | ||||||
| image | C1 | C3 | C13 | C39 | SL2(𝔽3) | SL2(𝔽3) | C13×SL2(𝔽3) | A4 | A4×C13 |
| kernel | C13×SL2(𝔽3) | Q8×C13 | SL2(𝔽3) | Q8 | C13 | C13 | C1 | C26 | C2 |
| # reps | 1 | 2 | 12 | 24 | 1 | 2 | 36 | 1 | 12 |
Matrix representation of C13×SL2(𝔽3) ►in GL2(𝔽157) generated by
| 16 | 0 |
| 0 | 16 |
| 13 | 145 |
| 145 | 144 |
| 0 | 156 |
| 1 | 0 |
| 1 | 0 |
| 145 | 144 |
G:=sub<GL(2,GF(157))| [16,0,0,16],[13,145,145,144],[0,1,156,0],[1,145,0,144] >;
C13×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_{13}\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C13xSL(2,3)"); // GroupNames label
G:=SmallGroup(312,25);
// by ID
G=gap.SmallGroup(312,25);
# by ID
G:=PCGroup([5,-3,-13,-2,2,-2,1172,72,2343,133,58]);
// Polycyclic
G:=Group<a,b,c,d|a^13=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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