Copied to
clipboard

G = C2×A5order 120 = 23·3·5

Direct product of C2 and A5

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

Aliases: C2×A5, group of symmetries of a regular icosahedron (and its dual dodecahedron), SmallGroup(120,35)

Series: ChiefDerived Lower central Upper central

C1C2 — C2×A5
A5 — C2×A5
A5 — C2×A5
C1C2

15C2
15C2
10C3
6C5
5C22
15C22
15C22
10C6
10S3
10S3
6D5
6D5
6C10
5C23
5A4
10D6
6D10
5C2×A4

Character table of C2×A5

 class 12A2B2C35A5B610A10B
 size 111515201212201212
ρ11111111111    trivial
ρ21-1-11111-1-1-1    linear of order 2
ρ333-1-101+5/21-5/201-5/21+5/2    orthogonal lifted from A5
ρ433-1-101-5/21+5/201+5/21-5/2    orthogonal lifted from A5
ρ53-31-101+5/21-5/20-1+5/2-1-5/2    orthogonal faithful
ρ63-31-101-5/21+5/20-1-5/2-1+5/2    orthogonal faithful
ρ744001-1-11-1-1    orthogonal lifted from A5
ρ84-4001-1-1-111    orthogonal faithful
ρ95511-100-100    orthogonal lifted from A5
ρ105-5-11-100100    orthogonal faithful

Permutation representations of C2×A5
On 10 points - transitive group 10T11
Generators in S10
(1 2 3 4 5)(6 7 8 9 10)
(1 7)(2 8)(3 10)(4 6)(5 9)

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

G:=Group( (1,2,3,4,5)(6,7,8,9,10), (1,7)(2,8)(3,10)(4,6)(5,9) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10)], [(1,7),(2,8),(3,10),(4,6),(5,9)])

G:=TransitiveGroup(10,11);

On 12 points - transitive group 12T75
Generators in S12
(3 4 5 6 7)(8 9 10 11 12)
(1 7)(2 12)(3 8)(4 10)(5 9)(6 11)

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

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

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

G:=TransitiveGroup(12,75);

On 12 points - transitive group 12T76
Generators in S12
(3 4 5 6 7)(8 9 10 11 12)
(1 5)(2 10)(3 12)(7 8)

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

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

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

G:=TransitiveGroup(12,76);

On 20 points - transitive group 20T31
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 12)(2 14)(3 8)(4 9)(5 15)(6 16)(7 18)(10 19)(11 17)(13 20)

G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,12)(2,14)(3,8)(4,9)(5,15)(6,16)(7,18)(10,19)(11,17)(13,20)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,12)(2,14)(3,8)(4,9)(5,15)(6,16)(7,18)(10,19)(11,17)(13,20) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,12),(2,14),(3,8),(4,9),(5,15),(6,16),(7,18),(10,19),(11,17),(13,20)])

G:=TransitiveGroup(20,31);

On 20 points - transitive group 20T36
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 12)(2 18)(5 19)(8 15)(9 16)(10 14)(11 17)(13 20)

G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,12)(2,18)(5,19)(8,15)(9,16)(10,14)(11,17)(13,20)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,12)(2,18)(5,19)(8,15)(9,16)(10,14)(11,17)(13,20) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,12),(2,18),(5,19),(8,15),(9,16),(10,14),(11,17),(13,20)])

G:=TransitiveGroup(20,36);

On 24 points - transitive group 24T203
Generators in S24
(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19)(20 21 22 23 24)
(1 12)(2 8)(3 16)(4 22)(5 10)(6 14)(7 15)(9 17)(11 21)(13 23)(18 20)(19 24)

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

G:=Group( (5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19)(20,21,22,23,24), (1,12)(2,8)(3,16)(4,22)(5,10)(6,14)(7,15)(9,17)(11,21)(13,23)(18,20)(19,24) );

G=PermutationGroup([(5,6,7,8,9),(10,11,12,13,14),(15,16,17,18,19),(20,21,22,23,24)], [(1,12),(2,8),(3,16),(4,22),(5,10),(6,14),(7,15),(9,17),(11,21),(13,23),(18,20),(19,24)])

G:=TransitiveGroup(24,203);

On 30 points - transitive group 30T29
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 19)(2 22)(3 14)(4 29)(6 18)(8 30)(9 12)(10 24)(11 28)(13 23)(15 26)(17 21)(20 25)

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

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

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

G:=TransitiveGroup(30,29);

On 30 points - transitive group 30T30
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 16)(2 26)(3 28)(4 20)(5 8)(6 12)(7 23)(9 24)(10 15)(11 27)(13 17)(14 19)(18 21)(22 30)(25 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), (1,16)(2,26)(3,28)(4,20)(5,8)(6,12)(7,23)(9,24)(10,15)(11,27)(13,17)(14,19)(18,21)(22,30)(25,29)>;

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

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

G:=TransitiveGroup(30,30);

C2×A5 is a maximal subgroup of   A5⋊C4
C2×A5 is a maximal quotient of   C4.A5

Polynomial with Galois group C2×A5 over ℚ
actionf(x)Disc(f)
10T11x10-18x8+93x6-188x4+156x2-45210·314·55·734
12T75x12+7x8+7x4-8x2+1248·674
12T76x12-6x11+3x10+40x9-58x8-74x7+160x6+13x5-128x4+35x3+18x2-4x-132·52·136134

Matrix representation of C2×A5 in GL3(𝔽5) generated by

430
230
211
,
100
304
340
G:=sub<GL(3,GF(5))| [4,2,2,3,3,1,0,0,1],[1,3,3,0,0,4,0,4,0] >;

C2×A5 in GAP, Magma, Sage, TeX

C_2\times A_5
% in TeX

G:=Group("C2xA5");
// GroupNames label

G:=SmallGroup(120,35);
// by ID

G=gap.SmallGroup(120,35);
# by ID

Export

Subgroup lattice of C2×A5 in TeX
Character table of C2×A5 in TeX

׿
×
𝔽