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G = C3xA5order 180 = 22·32·5

Direct product of C3 and A5

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

Aliases: C3xA5, GL2(F4), CO3(F4), CSO3(F4), SmallGroup(180,19)

Series: ChiefDerived Lower central Upper central

C1C3 — C3xA5
A5 — C3xA5
A5 — C3xA5
C1C3

Subgroups: 148 in 21 conjugacy classes, 4 normal (all characteristic)
Quotients: C1, C3, A5, C3xA5
15C2
10C3
20C3
6C5
5C22
10S3
15C6
10C32
6D5
6C15
5A4
5C2xC6
5A4
5A4
10C3xS3
6C3xD5
5C3xA4

Character table of C3xA5

 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 C3xA5
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 14 10 12 8 2 9 5 7 18 3 4 15 17 13)(6 11 16)

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

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

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

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

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

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

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

G:=TransitiveGroup(30,45);

C3xA5 is a maximal subgroup of   ΓL2(F4)

Polynomial with Galois group C3xA5 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 C3xA5 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] >;

C3xA5 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 C3xA5 in TeX
Character table of C3xA5 in TeX

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