direct product, non-abelian, not soluble, A-group
Aliases: C3×A5, GL2(𝔽4), CO3(𝔽4), CSO3(𝔽4), SmallGroup(180,19)
Series: Chief►Derived ►Lower central ►Upper central
| A5 — C3×A5 |
| A5 — C3×A5 |
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 |
►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);
C3×A5 is a maximal subgroup of
ΓL2(𝔽4)
| action | f(x) | Disc(f) |
|---|---|---|
| 15T15 | x15-3x14-54x13+277x12+426x11-13272x10+11178x9+318432x8+50352x7-2539227x6-1001076x5-239988x4-53305596x3-145749015x2-165940785x-101738531 | 316·710·112·2310·294·2712·806272·12762132·999575212 |
| 15T16 | x15-x14+3x13+8x12+36x11-17x10-40x9-3x8+140x7-90x6-32x5+46x4-8x3-13x2-2x+1 | 312·710·236·2112·236332·9301192 |
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
Export
Subgroup lattice of C3×A5 in TeX
Character table of C3×A5 in TeX