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G = C3×A5order 180 = 22·32·5

Direct product of C3 and A5

direct product, non-abelian, not soluble, A-group

Aliases: C3×A5, GL2(𝔽4), CO3(𝔽4), CSO3(𝔽4), SmallGroup(180,19)

Series: ChiefDerived Lower central Upper central

C1C3 — C3×A5
A5 — C3×A5
A5 — C3×A5
C1C3

15C2
10C3
20C3
6C5
5C22
10S3
15C6
10C32
6D5
6C15
5A4
5C2×C6
5A4
5A4
10C3×S3
6C3×D5
5C3×A4

Character table of C3×A5

 class 123A3B3C3D3E5A5B6A6B15A15B15C15D
 size 115112020201212151512121212
ρ1111111111111111    trivial
ρ211ζ32ζ3ζ3ζ32111ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ311ζ3ζ32ζ32ζ3111ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ43-1330001-5/21+5/2-1-11+5/21-5/21-5/21+5/2    orthogonal lifted from A5
ρ53-1330001+5/21-5/2-1-11-5/21+5/21+5/21-5/2    orthogonal lifted from A5
ρ63-1-3-3-3/2-3+3-3/20001+5/21-5/2ζ6ζ653ζ543ζ532ζ5332ζ523ζ533ζ5232ζ5432ζ5    complex faithful
ρ73-1-3-3-3/2-3+3-3/20001-5/21+5/2ζ6ζ653ζ533ζ5232ζ5432ζ53ζ543ζ532ζ5332ζ52    complex faithful
ρ83-1-3+3-3/2-3-3-3/20001-5/21+5/2ζ65ζ632ζ5332ζ523ζ543ζ532ζ5432ζ53ζ533ζ52    complex faithful
ρ93-1-3+3-3/2-3-3-3/20001+5/21-5/2ζ65ζ632ζ5432ζ53ζ533ζ5232ζ5332ζ523ζ543ζ5    complex faithful
ρ104044111-1-100-1-1-1-1    orthogonal lifted from A5
ρ1140-2+2-3-2-2-3ζ32ζ31-1-100ζ6ζ65ζ6ζ65    complex faithful
ρ1240-2-2-3-2+2-3ζ3ζ321-1-100ζ65ζ6ζ65ζ6    complex faithful
ρ135155-1-1-100110000    orthogonal lifted from A5
ρ1451-5-5-3/2-5+5-3/2ζ65ζ6-100ζ32ζ30000    complex faithful
ρ1551-5+5-3/2-5-5-3/2ζ6ζ65-100ζ3ζ320000    complex faithful

Permutation representations of C3×A5
On 15 points - transitive group 15T15
Generators in S15
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)
(1 10 13 9 7 11 5 8 4 2 6 15 3 14 12)

G:=sub<Sym(15)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (1,10,13,9,7,11,5,8,4,2,6,15,3,14,12)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (1,10,13,9,7,11,5,8,4,2,6,15,3,14,12) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)], [(1,10,13,9,7,11,5,8,4,2,6,15,3,14,12)])

G:=TransitiveGroup(15,15);

On 15 points - transitive group 15T16
Generators in S15
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)
(1 15 8 4 12 11 10 3 14 7 6 5 13 9 2)

G:=sub<Sym(15)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (1,15,8,4,12,11,10,3,14,7,6,5,13,9,2)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15), (1,15,8,4,12,11,10,3,14,7,6,5,13,9,2) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)], [(1,15,8,4,12,11,10,3,14,7,6,5,13,9,2)])

G:=TransitiveGroup(15,16);

On 18 points - transitive group 18T90
Generators in S18
(1 2 3)(4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)
(1 10 6 8 4 2 5 16 18 14 3 15 11 13 9)(7 12 17)

G:=sub<Sym(18)| (1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18), (1,10,6,8,4,2,5,16,18,14,3,15,11,13,9)(7,12,17)>;

G:=Group( (1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18), (1,10,6,8,4,2,5,16,18,14,3,15,11,13,9)(7,12,17) );

G=PermutationGroup([(1,2,3),(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)], [(1,10,6,8,4,2,5,16,18,14,3,15,11,13,9),(7,12,17)])

G:=TransitiveGroup(18,90);

On 30 points - transitive group 30T45
Generators in S30
(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)
(1 12 8 25 18 11 7 3 20 28 6 2 13 30 23)(4 27 26 10 24 14 22 21 5 19 9 17 16 15 29)

G:=sub<Sym(30)| (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), (1,12,8,25,18,11,7,3,20,28,6,2,13,30,23)(4,27,26,10,24,14,22,21,5,19,9,17,16,15,29)>;

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), (1,12,8,25,18,11,7,3,20,28,6,2,13,30,23)(4,27,26,10,24,14,22,21,5,19,9,17,16,15,29) );

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)], [(1,12,8,25,18,11,7,3,20,28,6,2,13,30,23),(4,27,26,10,24,14,22,21,5,19,9,17,16,15,29)])

G:=TransitiveGroup(30,45);

C3×A5 is a maximal subgroup of   ΓL2(𝔽4)

Polynomial with Galois group C3×A5 over ℚ
actionf(x)Disc(f)
15T15x15-3x14-54x13+277x12+426x11-13272x10+11178x9+318432x8+50352x7-2539227x6-1001076x5-239988x4-53305596x3-145749015x2-165940785x-101738531316·710·112·2310·294·2712·806272·12762132·999575212
15T16x15-x14+3x13+8x12+36x11-17x10-40x9-3x8+140x7-90x6-32x5+46x4-8x3-13x2-2x+1312·710·236·2112·236332·9301192

Matrix representation of C3×A5 in GL4(𝔽2) generated by

1011
1100
1111
1110
,
0100
1110
1001
0010
G:=sub<GL(4,GF(2))| [1,1,1,1,0,1,1,1,1,0,1,1,1,0,1,0],[0,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0] >;

C3×A5 in GAP, Magma, Sage, TeX

C_3\times A_5
% in TeX

G:=Group("C3xA5");
// GroupNames label

G:=SmallGroup(180,19);
// by ID

G=gap.SmallGroup(180,19);
# by ID

Export

Subgroup lattice of C3×A5 in TeX
Character table of C3×A5 in TeX

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