direct product, non-abelian, soluble
Aliases: C15×SL2(𝔽3), C30.3A4, Q8⋊(C3×C15), C2.(A4×C15), C10.(C3×A4), (C3×Q8)⋊C15, (Q8×C15)⋊C3, (C5×Q8)⋊C32, C6.3(C5×A4), SmallGroup(360,89)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — C15×SL2(𝔽3) |
| Q8 — C15×SL2(𝔽3) |
Generators and relations for C15×SL2(𝔽3)
G = < a,b,c,d | a15=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 C15×SL2(𝔽3)
►On 120 points
Generators in S120
(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)
(1 82 59 34)(2 83 60 35)(3 84 46 36)(4 85 47 37)(5 86 48 38)(6 87 49 39)(7 88 50 40)(8 89 51 41)(9 90 52 42)(10 76 53 43)(11 77 54 44)(12 78 55 45)(13 79 56 31)(14 80 57 32)(15 81 58 33)(16 104 108 71)(17 105 109 72)(18 91 110 73)(19 92 111 74)(20 93 112 75)(21 94 113 61)(22 95 114 62)(23 96 115 63)(24 97 116 64)(25 98 117 65)(26 99 118 66)(27 100 119 67)(28 101 120 68)(29 102 106 69)(30 103 107 70)
(1 93 59 75)(2 94 60 61)(3 95 46 62)(4 96 47 63)(5 97 48 64)(6 98 49 65)(7 99 50 66)(8 100 51 67)(9 101 52 68)(10 102 53 69)(11 103 54 70)(12 104 55 71)(13 105 56 72)(14 91 57 73)(15 92 58 74)(16 45 108 78)(17 31 109 79)(18 32 110 80)(19 33 111 81)(20 34 112 82)(21 35 113 83)(22 36 114 84)(23 37 115 85)(24 38 116 86)(25 39 117 87)(26 40 118 88)(27 41 119 89)(28 42 120 90)(29 43 106 76)(30 44 107 77)
(16 104 78)(17 105 79)(18 91 80)(19 92 81)(20 93 82)(21 94 83)(22 95 84)(23 96 85)(24 97 86)(25 98 87)(26 99 88)(27 100 89)(28 101 90)(29 102 76)(30 103 77)(31 109 72)(32 110 73)(33 111 74)(34 112 75)(35 113 61)(36 114 62)(37 115 63)(38 116 64)(39 117 65)(40 118 66)(41 119 67)(42 120 68)(43 106 69)(44 107 70)(45 108 71)
G:=sub<Sym(120)| (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), (1,82,59,34)(2,83,60,35)(3,84,46,36)(4,85,47,37)(5,86,48,38)(6,87,49,39)(7,88,50,40)(8,89,51,41)(9,90,52,42)(10,76,53,43)(11,77,54,44)(12,78,55,45)(13,79,56,31)(14,80,57,32)(15,81,58,33)(16,104,108,71)(17,105,109,72)(18,91,110,73)(19,92,111,74)(20,93,112,75)(21,94,113,61)(22,95,114,62)(23,96,115,63)(24,97,116,64)(25,98,117,65)(26,99,118,66)(27,100,119,67)(28,101,120,68)(29,102,106,69)(30,103,107,70), (1,93,59,75)(2,94,60,61)(3,95,46,62)(4,96,47,63)(5,97,48,64)(6,98,49,65)(7,99,50,66)(8,100,51,67)(9,101,52,68)(10,102,53,69)(11,103,54,70)(12,104,55,71)(13,105,56,72)(14,91,57,73)(15,92,58,74)(16,45,108,78)(17,31,109,79)(18,32,110,80)(19,33,111,81)(20,34,112,82)(21,35,113,83)(22,36,114,84)(23,37,115,85)(24,38,116,86)(25,39,117,87)(26,40,118,88)(27,41,119,89)(28,42,120,90)(29,43,106,76)(30,44,107,77), (16,104,78)(17,105,79)(18,91,80)(19,92,81)(20,93,82)(21,94,83)(22,95,84)(23,96,85)(24,97,86)(25,98,87)(26,99,88)(27,100,89)(28,101,90)(29,102,76)(30,103,77)(31,109,72)(32,110,73)(33,111,74)(34,112,75)(35,113,61)(36,114,62)(37,115,63)(38,116,64)(39,117,65)(40,118,66)(41,119,67)(42,120,68)(43,106,69)(44,107,70)(45,108,71)>;
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), (1,82,59,34)(2,83,60,35)(3,84,46,36)(4,85,47,37)(5,86,48,38)(6,87,49,39)(7,88,50,40)(8,89,51,41)(9,90,52,42)(10,76,53,43)(11,77,54,44)(12,78,55,45)(13,79,56,31)(14,80,57,32)(15,81,58,33)(16,104,108,71)(17,105,109,72)(18,91,110,73)(19,92,111,74)(20,93,112,75)(21,94,113,61)(22,95,114,62)(23,96,115,63)(24,97,116,64)(25,98,117,65)(26,99,118,66)(27,100,119,67)(28,101,120,68)(29,102,106,69)(30,103,107,70), (1,93,59,75)(2,94,60,61)(3,95,46,62)(4,96,47,63)(5,97,48,64)(6,98,49,65)(7,99,50,66)(8,100,51,67)(9,101,52,68)(10,102,53,69)(11,103,54,70)(12,104,55,71)(13,105,56,72)(14,91,57,73)(15,92,58,74)(16,45,108,78)(17,31,109,79)(18,32,110,80)(19,33,111,81)(20,34,112,82)(21,35,113,83)(22,36,114,84)(23,37,115,85)(24,38,116,86)(25,39,117,87)(26,40,118,88)(27,41,119,89)(28,42,120,90)(29,43,106,76)(30,44,107,77), (16,104,78)(17,105,79)(18,91,80)(19,92,81)(20,93,82)(21,94,83)(22,95,84)(23,96,85)(24,97,86)(25,98,87)(26,99,88)(27,100,89)(28,101,90)(29,102,76)(30,103,77)(31,109,72)(32,110,73)(33,111,74)(34,112,75)(35,113,61)(36,114,62)(37,115,63)(38,116,64)(39,117,65)(40,118,66)(41,119,67)(42,120,68)(43,106,69)(44,107,70)(45,108,71) );
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)], [(1,82,59,34),(2,83,60,35),(3,84,46,36),(4,85,47,37),(5,86,48,38),(6,87,49,39),(7,88,50,40),(8,89,51,41),(9,90,52,42),(10,76,53,43),(11,77,54,44),(12,78,55,45),(13,79,56,31),(14,80,57,32),(15,81,58,33),(16,104,108,71),(17,105,109,72),(18,91,110,73),(19,92,111,74),(20,93,112,75),(21,94,113,61),(22,95,114,62),(23,96,115,63),(24,97,116,64),(25,98,117,65),(26,99,118,66),(27,100,119,67),(28,101,120,68),(29,102,106,69),(30,103,107,70)], [(1,93,59,75),(2,94,60,61),(3,95,46,62),(4,96,47,63),(5,97,48,64),(6,98,49,65),(7,99,50,66),(8,100,51,67),(9,101,52,68),(10,102,53,69),(11,103,54,70),(12,104,55,71),(13,105,56,72),(14,91,57,73),(15,92,58,74),(16,45,108,78),(17,31,109,79),(18,32,110,80),(19,33,111,81),(20,34,112,82),(21,35,113,83),(22,36,114,84),(23,37,115,85),(24,38,116,86),(25,39,117,87),(26,40,118,88),(27,41,119,89),(28,42,120,90),(29,43,106,76),(30,44,107,77)], [(16,104,78),(17,105,79),(18,91,80),(19,92,81),(20,93,82),(21,94,83),(22,95,84),(23,96,85),(24,97,86),(25,98,87),(26,99,88),(27,100,89),(28,101,90),(29,102,76),(30,103,77),(31,109,72),(32,110,73),(33,111,74),(34,112,75),(35,113,61),(36,114,62),(37,115,63),(38,116,64),(39,117,65),(40,118,66),(41,119,67),(42,120,68),(43,106,69),(44,107,70),(45,108,71)]])
105 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 4 | 5A | 5B | 5C | 5D | 6A | 6B | 6C | ··· | 6H | 10A | 10B | 10C | 10D | 12A | 12B | 15A | ··· | 15H | 15I | ··· | 15AF | 20A | 20B | 20C | 20D | 30A | ··· | 30H | 30I | ··· | 30AF | 60A | ··· | 60H |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | ··· | 6 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | ··· | 15 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 6 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 4 | ··· | 4 | 6 | ··· | 6 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | - | + | ||||||||||||
| image | C1 | C3 | C3 | C5 | C15 | C15 | SL2(𝔽3) | SL2(𝔽3) | C3×SL2(𝔽3) | C5×SL2(𝔽3) | C15×SL2(𝔽3) | A4 | C3×A4 | C5×A4 | A4×C15 |
| kernel | C15×SL2(𝔽3) | C5×SL2(𝔽3) | Q8×C15 | C3×SL2(𝔽3) | SL2(𝔽3) | C3×Q8 | C15 | C15 | C5 | C3 | C1 | C30 | C10 | C6 | C2 |
| # reps | 1 | 6 | 2 | 4 | 24 | 8 | 1 | 2 | 6 | 12 | 24 | 1 | 2 | 4 | 8 |
Matrix representation of C15×SL2(𝔽3) ►in GL2(𝔽31) generated by
| 28 | 0 |
| 0 | 28 |
| 6 | 12 |
| 15 | 25 |
| 5 | 29 |
| 13 | 26 |
| 1 | 12 |
| 0 | 25 |
G:=sub<GL(2,GF(31))| [28,0,0,28],[6,15,12,25],[5,13,29,26],[1,0,12,25] >;
C15×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_{15}\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C15xSL(2,3)"); // GroupNames label
G:=SmallGroup(360,89);
// by ID
G=gap.SmallGroup(360,89);
# by ID
G:=PCGroup([6,-3,-3,-5,-2,2,-2,2163,117,4054,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^15=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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