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## G = C3×A5order 180 = 22·32·5

### Direct product of C3 and A5

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

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C3 — C3×A5
 Derived series A5 — C3×A5
 Lower central A5 — C3×A5
 Upper central C1 — C3

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

Character table of C3×A5

 class 1 2 3A 3B 3C 3D 3E 5A 5B 6A 6B 15A 15B 15C 15D size 1 15 1 1 20 20 20 12 12 15 15 12 12 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ3 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 3 -1 3 3 0 0 0 1-√5/2 1+√5/2 -1 -1 1+√5/2 1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from A5 ρ5 3 -1 3 3 0 0 0 1+√5/2 1-√5/2 -1 -1 1-√5/2 1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from A5 ρ6 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 1+√5/2 1-√5/2 ζ6 ζ65 -ζ3ζ54-ζ3ζ5 -ζ32ζ53-ζ32ζ52 -ζ3ζ53-ζ3ζ52 -ζ32ζ54-ζ32ζ5 complex faithful ρ7 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 1-√5/2 1+√5/2 ζ6 ζ65 -ζ3ζ53-ζ3ζ52 -ζ32ζ54-ζ32ζ5 -ζ3ζ54-ζ3ζ5 -ζ32ζ53-ζ32ζ52 complex faithful ρ8 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 1-√5/2 1+√5/2 ζ65 ζ6 -ζ32ζ53-ζ32ζ52 -ζ3ζ54-ζ3ζ5 -ζ32ζ54-ζ32ζ5 -ζ3ζ53-ζ3ζ52 complex faithful ρ9 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 1+√5/2 1-√5/2 ζ65 ζ6 -ζ32ζ54-ζ32ζ5 -ζ3ζ53-ζ3ζ52 -ζ32ζ53-ζ32ζ52 -ζ3ζ54-ζ3ζ5 complex faithful ρ10 4 0 4 4 1 1 1 -1 -1 0 0 -1 -1 -1 -1 orthogonal lifted from A5 ρ11 4 0 -2+2√-3 -2-2√-3 ζ32 ζ3 1 -1 -1 0 0 ζ6 ζ65 ζ6 ζ65 complex faithful ρ12 4 0 -2-2√-3 -2+2√-3 ζ3 ζ32 1 -1 -1 0 0 ζ65 ζ6 ζ65 ζ6 complex faithful ρ13 5 1 5 5 -1 -1 -1 0 0 1 1 0 0 0 0 orthogonal lifted from A5 ρ14 5 1 -5-5√-3/2 -5+5√-3/2 ζ65 ζ6 -1 0 0 ζ32 ζ3 0 0 0 0 complex faithful ρ15 5 1 -5+5√-3/2 -5-5√-3/2 ζ6 ζ65 -1 0 0 ζ3 ζ32 0 0 0 0 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

 1 0 1 1 1 1 0 0 1 1 1 1 1 1 1 0
,
 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 0
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

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