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## G = C2×A5order 120 = 23·3·5

### Direct product of C2 and A5

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

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C2 — C2×A5
 Derived series A5 — C2×A5
 Lower central A5 — C2×A5
 Upper central C1 — C2

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

Character table of C2×A5

 class 1 2A 2B 2C 3 5A 5B 6 10A 10B size 1 1 15 15 20 12 12 20 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 1 -1 -1 -1 linear of order 2 ρ3 3 3 -1 -1 0 1+√5/2 1-√5/2 0 1-√5/2 1+√5/2 orthogonal lifted from A5 ρ4 3 3 -1 -1 0 1-√5/2 1+√5/2 0 1+√5/2 1-√5/2 orthogonal lifted from A5 ρ5 3 -3 1 -1 0 1+√5/2 1-√5/2 0 -1+√5/2 -1-√5/2 orthogonal faithful ρ6 3 -3 1 -1 0 1-√5/2 1+√5/2 0 -1-√5/2 -1+√5/2 orthogonal faithful ρ7 4 4 0 0 1 -1 -1 1 -1 -1 orthogonal lifted from A5 ρ8 4 -4 0 0 1 -1 -1 -1 1 1 orthogonal faithful ρ9 5 5 1 1 -1 0 0 -1 0 0 orthogonal lifted from A5 ρ10 5 -5 -1 1 -1 0 0 1 0 0 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 10)(2 6)(3 8)(4 9)(5 7)

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

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

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

G:=TransitiveGroup(10,11);

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

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

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

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

G:=TransitiveGroup(12,75);

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

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

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

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

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 17)(2 19)(3 12)(4 13)(5 20)(6 11)(7 14)(8 18)(9 15)(10 16)

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

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

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

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 14)(2 5)(3 16)(4 22)(6 10)(7 19)(8 18)(9 13)(11 20)(12 24)(15 21)(17 23)

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

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

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

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 21)(2 26)(3 19)(4 25)(7 13)(8 17)(9 28)(10 12)(11 16)(14 29)(15 23)(20 22)(24 30)

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

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

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

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

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

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

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

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

 4 3 0 2 3 0 2 1 1
,
 1 0 0 3 0 4 3 4 0
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

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