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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 17 39 114)(2 18 40 115)(3 19 41 116)(4 20 42 117)(5 21 43 118)(6 22 44 119)(7 23 45 120)(8 24 31 106)(9 25 32 107)(10 26 33 108)(11 27 34 109)(12 28 35 110)(13 29 36 111)(14 30 37 112)(15 16 38 113)(46 69 102 78)(47 70 103 79)(48 71 104 80)(49 72 105 81)(50 73 91 82)(51 74 92 83)(52 75 93 84)(53 61 94 85)(54 62 95 86)(55 63 96 87)(56 64 97 88)(57 65 98 89)(58 66 99 90)(59 67 100 76)(60 68 101 77)
(1 66 39 90)(2 67 40 76)(3 68 41 77)(4 69 42 78)(5 70 43 79)(6 71 44 80)(7 72 45 81)(8 73 31 82)(9 74 32 83)(10 75 33 84)(11 61 34 85)(12 62 35 86)(13 63 36 87)(14 64 37 88)(15 65 38 89)(16 57 113 98)(17 58 114 99)(18 59 115 100)(19 60 116 101)(20 46 117 102)(21 47 118 103)(22 48 119 104)(23 49 120 105)(24 50 106 91)(25 51 107 92)(26 52 108 93)(27 53 109 94)(28 54 110 95)(29 55 111 96)(30 56 112 97)
(16 57 65)(17 58 66)(18 59 67)(19 60 68)(20 46 69)(21 47 70)(22 48 71)(23 49 72)(24 50 73)(25 51 74)(26 52 75)(27 53 61)(28 54 62)(29 55 63)(30 56 64)(76 115 100)(77 116 101)(78 117 102)(79 118 103)(80 119 104)(81 120 105)(82 106 91)(83 107 92)(84 108 93)(85 109 94)(86 110 95)(87 111 96)(88 112 97)(89 113 98)(90 114 99)

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), (1,17,39,114)(2,18,40,115)(3,19,41,116)(4,20,42,117)(5,21,43,118)(6,22,44,119)(7,23,45,120)(8,24,31,106)(9,25,32,107)(10,26,33,108)(11,27,34,109)(12,28,35,110)(13,29,36,111)(14,30,37,112)(15,16,38,113)(46,69,102,78)(47,70,103,79)(48,71,104,80)(49,72,105,81)(50,73,91,82)(51,74,92,83)(52,75,93,84)(53,61,94,85)(54,62,95,86)(55,63,96,87)(56,64,97,88)(57,65,98,89)(58,66,99,90)(59,67,100,76)(60,68,101,77), (1,66,39,90)(2,67,40,76)(3,68,41,77)(4,69,42,78)(5,70,43,79)(6,71,44,80)(7,72,45,81)(8,73,31,82)(9,74,32,83)(10,75,33,84)(11,61,34,85)(12,62,35,86)(13,63,36,87)(14,64,37,88)(15,65,38,89)(16,57,113,98)(17,58,114,99)(18,59,115,100)(19,60,116,101)(20,46,117,102)(21,47,118,103)(22,48,119,104)(23,49,120,105)(24,50,106,91)(25,51,107,92)(26,52,108,93)(27,53,109,94)(28,54,110,95)(29,55,111,96)(30,56,112,97), (16,57,65)(17,58,66)(18,59,67)(19,60,68)(20,46,69)(21,47,70)(22,48,71)(23,49,72)(24,50,73)(25,51,74)(26,52,75)(27,53,61)(28,54,62)(29,55,63)(30,56,64)(76,115,100)(77,116,101)(78,117,102)(79,118,103)(80,119,104)(81,120,105)(82,106,91)(83,107,92)(84,108,93)(85,109,94)(86,110,95)(87,111,96)(88,112,97)(89,113,98)(90,114,99)>;

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), (1,17,39,114)(2,18,40,115)(3,19,41,116)(4,20,42,117)(5,21,43,118)(6,22,44,119)(7,23,45,120)(8,24,31,106)(9,25,32,107)(10,26,33,108)(11,27,34,109)(12,28,35,110)(13,29,36,111)(14,30,37,112)(15,16,38,113)(46,69,102,78)(47,70,103,79)(48,71,104,80)(49,72,105,81)(50,73,91,82)(51,74,92,83)(52,75,93,84)(53,61,94,85)(54,62,95,86)(55,63,96,87)(56,64,97,88)(57,65,98,89)(58,66,99,90)(59,67,100,76)(60,68,101,77), (1,66,39,90)(2,67,40,76)(3,68,41,77)(4,69,42,78)(5,70,43,79)(6,71,44,80)(7,72,45,81)(8,73,31,82)(9,74,32,83)(10,75,33,84)(11,61,34,85)(12,62,35,86)(13,63,36,87)(14,64,37,88)(15,65,38,89)(16,57,113,98)(17,58,114,99)(18,59,115,100)(19,60,116,101)(20,46,117,102)(21,47,118,103)(22,48,119,104)(23,49,120,105)(24,50,106,91)(25,51,107,92)(26,52,108,93)(27,53,109,94)(28,54,110,95)(29,55,111,96)(30,56,112,97), (16,57,65)(17,58,66)(18,59,67)(19,60,68)(20,46,69)(21,47,70)(22,48,71)(23,49,72)(24,50,73)(25,51,74)(26,52,75)(27,53,61)(28,54,62)(29,55,63)(30,56,64)(76,115,100)(77,116,101)(78,117,102)(79,118,103)(80,119,104)(81,120,105)(82,106,91)(83,107,92)(84,108,93)(85,109,94)(86,110,95)(87,111,96)(88,112,97)(89,113,98)(90,114,99) );

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)], [(1,17,39,114),(2,18,40,115),(3,19,41,116),(4,20,42,117),(5,21,43,118),(6,22,44,119),(7,23,45,120),(8,24,31,106),(9,25,32,107),(10,26,33,108),(11,27,34,109),(12,28,35,110),(13,29,36,111),(14,30,37,112),(15,16,38,113),(46,69,102,78),(47,70,103,79),(48,71,104,80),(49,72,105,81),(50,73,91,82),(51,74,92,83),(52,75,93,84),(53,61,94,85),(54,62,95,86),(55,63,96,87),(56,64,97,88),(57,65,98,89),(58,66,99,90),(59,67,100,76),(60,68,101,77)], [(1,66,39,90),(2,67,40,76),(3,68,41,77),(4,69,42,78),(5,70,43,79),(6,71,44,80),(7,72,45,81),(8,73,31,82),(9,74,32,83),(10,75,33,84),(11,61,34,85),(12,62,35,86),(13,63,36,87),(14,64,37,88),(15,65,38,89),(16,57,113,98),(17,58,114,99),(18,59,115,100),(19,60,116,101),(20,46,117,102),(21,47,118,103),(22,48,119,104),(23,49,120,105),(24,50,106,91),(25,51,107,92),(26,52,108,93),(27,53,109,94),(28,54,110,95),(29,55,111,96),(30,56,112,97)], [(16,57,65),(17,58,66),(18,59,67),(19,60,68),(20,46,69),(21,47,70),(22,48,71),(23,49,72),(24,50,73),(25,51,74),(26,52,75),(27,53,61),(28,54,62),(29,55,63),(30,56,64),(76,115,100),(77,116,101),(78,117,102),(79,118,103),(80,119,104),(81,120,105),(82,106,91),(83,107,92),(84,108,93),(85,109,94),(86,110,95),(87,111,96),(88,112,97),(89,113,98),(90,114,99)])

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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