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G = C3×SL2(𝔽5)  order 360 = 23·32·5

Direct product of C3 and SL2(𝔽5)

direct product, non-abelian, not soluble

Aliases: C3×SL2(𝔽5), C6.A5, C2.(C3×A5), SmallGroup(360,51)

Series: ChiefDerived Lower central Upper central

C1C3C6 — C3×SL2(𝔽5)
SL2(𝔽5) — C3×SL2(𝔽5)
SL2(𝔽5) — C3×SL2(𝔽5)
C1C6

10C3
20C3
6C5
15C4
10C6
20C6
10C32
6C10
6C15
5Q8
10Dic3
15C12
10C3×C6
6Dic5
6C30
5SL2(𝔽3)
5SL2(𝔽3)
5C3×Q8
5SL2(𝔽3)
10C3×Dic3
6C3×Dic5
5C3×SL2(𝔽3)

Character table of C3×SL2(𝔽5)

 class 123A3B3C3D3E45A5B6A6B6C6D6E10A10B12A12B15A15B15C15D30A30B30C30D
 size 111120202030121211202020121230301212121212121212
ρ1111111111111111111111111111    trivial
ρ211ζ32ζ3ζ32ζ31111ζ3ζ321ζ32ζ311ζ3ζ32ζ3ζ32ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ311ζ3ζ32ζ3ζ321111ζ32ζ31ζ3ζ3211ζ32ζ3ζ32ζ3ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ42-222-1-1-10-1+5/2-1-5/2-2-21111+5/21-5/200-1-5/2-1+5/2-1+5/2-1-5/21-5/21+5/21-5/21+5/2    symplectic lifted from SL2(𝔽5), Schur index 2
ρ52-222-1-1-10-1-5/2-1+5/2-2-21111-5/21+5/200-1+5/2-1-5/2-1-5/2-1+5/21+5/21-5/21+5/21-5/2    symplectic lifted from SL2(𝔽5), Schur index 2
ρ62-2-1+-3-1--3ζ65ζ6-10-1+5/2-1-5/21+-31--31ζ3ζ321+5/21-5/200ζ32ζ5332ζ52ζ3ζ543ζ5ζ32ζ5432ζ5ζ3ζ533ζ523ζ543ζ53ζ533ζ5232ζ5432ζ532ζ5332ζ52    complex faithful
ρ72-2-1--3-1+-3ζ6ζ65-10-1+5/2-1-5/21--31+-31ζ32ζ31+5/21-5/200ζ3ζ533ζ52ζ32ζ5432ζ5ζ3ζ543ζ5ζ32ζ5332ζ5232ζ5432ζ532ζ5332ζ523ζ543ζ53ζ533ζ52    complex faithful
ρ82-2-1--3-1+-3ζ6ζ65-10-1-5/2-1+5/21--31+-31ζ32ζ31-5/21+5/200ζ3ζ543ζ5ζ32ζ5332ζ52ζ3ζ533ζ52ζ32ζ5432ζ532ζ5332ζ5232ζ5432ζ53ζ533ζ523ζ543ζ5    complex faithful
ρ92-2-1+-3-1--3ζ65ζ6-10-1-5/2-1+5/21+-31--31ζ3ζ321-5/21+5/200ζ32ζ5432ζ5ζ3ζ533ζ52ζ32ζ5332ζ52ζ3ζ543ζ53ζ533ζ523ζ543ζ532ζ5332ζ5232ζ5432ζ5    complex faithful
ρ103333000-11-5/21+5/2330001+5/21-5/2-1-11+5/21-5/21-5/21+5/21-5/21+5/21-5/21+5/2    orthogonal lifted from A5
ρ113333000-11+5/21-5/2330001-5/21+5/2-1-11-5/21+5/21+5/21-5/21+5/21-5/21+5/21-5/2    orthogonal lifted from A5
ρ1233-3-3-3/2-3+3-3/2000-11+5/21-5/2-3+3-3/2-3-3-3/20001-5/21+5/2ζ65ζ63ζ543ζ532ζ5332ζ523ζ533ζ5232ζ5432ζ532ζ5332ζ5232ζ5432ζ53ζ533ζ523ζ543ζ5    complex lifted from C3×A5
ρ1333-3-3-3/2-3+3-3/2000-11-5/21+5/2-3+3-3/2-3-3-3/20001+5/21-5/2ζ65ζ63ζ533ζ5232ζ5432ζ53ζ543ζ532ζ5332ζ5232ζ5432ζ532ζ5332ζ523ζ543ζ53ζ533ζ52    complex lifted from C3×A5
ρ1433-3+3-3/2-3-3-3/2000-11-5/21+5/2-3-3-3/2-3+3-3/20001+5/21-5/2ζ6ζ6532ζ5332ζ523ζ543ζ532ζ5432ζ53ζ533ζ523ζ543ζ53ζ533ζ5232ζ5432ζ532ζ5332ζ52    complex lifted from C3×A5
ρ1533-3+3-3/2-3-3-3/2000-11+5/21-5/2-3-3-3/2-3+3-3/20001-5/21+5/2ζ6ζ6532ζ5432ζ53ζ533ζ5232ζ5332ζ523ζ543ζ53ζ533ζ523ζ543ζ532ζ5332ζ5232ζ5432ζ5    complex lifted from C3×A5
ρ1644441110-1-144111-1-100-1-1-1-1-1-1-1-1    orthogonal lifted from A5
ρ174-4441110-1-1-4-4-1-1-11100-1-1-1-11111    symplectic lifted from SL2(𝔽5), Schur index 2
ρ1844-2+2-3-2-2-3ζ3ζ3210-1-1-2-2-3-2+2-31ζ3ζ32-1-100ζ6ζ65ζ6ζ65ζ65ζ65ζ6ζ6    complex lifted from C3×A5
ρ194-4-2-2-3-2+2-3ζ32ζ310-1-12-2-32+2-3-1ζ6ζ651100ζ65ζ6ζ65ζ6ζ32ζ32ζ3ζ3    complex faithful
ρ2044-2-2-3-2+2-3ζ32ζ310-1-1-2+2-3-2-2-31ζ32ζ3-1-100ζ65ζ6ζ65ζ6ζ6ζ6ζ65ζ65    complex lifted from C3×A5
ρ214-4-2+2-3-2-2-3ζ3ζ3210-1-12+2-32-2-3-1ζ65ζ61100ζ6ζ65ζ6ζ65ζ3ζ3ζ32ζ32    complex faithful
ρ225555-1-1-110055-1-1-1001100000000    orthogonal lifted from A5
ρ2355-5-5-3/2-5+5-3/2ζ6ζ65-1100-5+5-3/2-5-5-3/2-1ζ6ζ6500ζ3ζ3200000000    complex lifted from C3×A5
ρ2455-5+5-3/2-5-5-3/2ζ65ζ6-1100-5-5-3/2-5+5-3/2-1ζ65ζ600ζ32ζ300000000    complex lifted from C3×A5
ρ256-666000011-6-6000-1-1001111-1-1-1-1    symplectic lifted from SL2(𝔽5), Schur index 2
ρ266-6-3+3-3-3-3-30000113+3-33-3-3000-1-100ζ32ζ3ζ32ζ3ζ65ζ65ζ6ζ6    complex faithful
ρ276-6-3-3-3-3+3-30000113-3-33+3-3000-1-100ζ3ζ32ζ3ζ32ζ6ζ6ζ65ζ65    complex faithful

Smallest permutation representation of C3×SL2(𝔽5)
On 72 points
Generators in S72
(1 36 37)(2 31 32)(3 26 27)(4 21 22)(5 16 17)(6 41 42)(7 57 58)(8 52 53)(9 47 48)(10 72 43)(11 67 68)(12 62 63)(13 35 45)(14 69 61)(15 55 23)(18 40 50)(19 44 66)(20 60 28)(24 49 71)(25 65 33)(29 54 46)(30 70 38)(34 59 51)(39 64 56)
(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)

G:=sub<Sym(72)| (1,36,37)(2,31,32)(3,26,27)(4,21,22)(5,16,17)(6,41,42)(7,57,58)(8,52,53)(9,47,48)(10,72,43)(11,67,68)(12,62,63)(13,35,45)(14,69,61)(15,55,23)(18,40,50)(19,44,66)(20,60,28)(24,49,71)(25,65,33)(29,54,46)(30,70,38)(34,59,51)(39,64,56), (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)>;

G:=Group( (1,36,37)(2,31,32)(3,26,27)(4,21,22)(5,16,17)(6,41,42)(7,57,58)(8,52,53)(9,47,48)(10,72,43)(11,67,68)(12,62,63)(13,35,45)(14,69,61)(15,55,23)(18,40,50)(19,44,66)(20,60,28)(24,49,71)(25,65,33)(29,54,46)(30,70,38)(34,59,51)(39,64,56), (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) );

G=PermutationGroup([(1,36,37),(2,31,32),(3,26,27),(4,21,22),(5,16,17),(6,41,42),(7,57,58),(8,52,53),(9,47,48),(10,72,43),(11,67,68),(12,62,63),(13,35,45),(14,69,61),(15,55,23),(18,40,50),(19,44,66),(20,60,28),(24,49,71),(25,65,33),(29,54,46),(30,70,38),(34,59,51),(39,64,56)], [(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)])

Matrix representation of C3×SL2(𝔽5) in GL2(𝔽19) generated by

413
68
,
113
139
G:=sub<GL(2,GF(19))| [4,6,13,8],[1,13,13,9] >;

C3×SL2(𝔽5) in GAP, Magma, Sage, TeX

C_3\times {\rm SL}_2({\mathbb F}_5)
% in TeX

G:=Group("C3xSL(2,5)");
// GroupNames label

G:=SmallGroup(360,51);
// by ID

G=gap.SmallGroup(360,51);
# by ID

Export

Subgroup lattice of C3×SL2(𝔽5) in TeX
Character table of C3×SL2(𝔽5) in TeX

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