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G = C15×SL2(𝔽3)  order 360 = 23·32·5

Direct product of C15 and SL2(𝔽3)

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

C1C2Q8 — C15×SL2(𝔽3)
C1C2Q8C5×Q8C5×SL2(𝔽3) — C15×SL2(𝔽3)
Q8 — C15×SL2(𝔽3)
C1C30

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 >

4C3
4C3
4C3
3C4
4C6
4C6
4C6
4C32
4C15
4C15
4C15
3C12
4C3×C6
3C20
4C30
4C30
4C30
4C3×C15
3C60
4C3×C30

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 3A3B3C···3H 4 5A5B5C5D6A6B6C···6H10A10B10C10D12A12B15A···15H15I···15AF20A20B20C20D30A···30H30I···30AF60A···60H
order12333···345555666···610101010121215···1515···152020202030···3030···3060···60
size11114···461111114···41111661···14···466661···14···46···6

105 irreducible representations

dim111111222223333
type+-+
imageC1C3C3C5C15C15SL2(𝔽3)SL2(𝔽3)C3×SL2(𝔽3)C5×SL2(𝔽3)C15×SL2(𝔽3)A4C3×A4C5×A4A4×C15
kernelC15×SL2(𝔽3)C5×SL2(𝔽3)Q8×C15C3×SL2(𝔽3)SL2(𝔽3)C3×Q8C15C15C5C3C1C30C10C6C2
# reps162424812612241248

Matrix representation of C15×SL2(𝔽3) in GL2(𝔽31) generated by

280
028
,
612
1525
,
529
1326
,
112
025
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

Export

Subgroup lattice of C15×SL2(𝔽3) in TeX

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