direct product, non-abelian, not soluble, A-group
Aliases: C22×A5, SmallGroup(240,190)
Series: Chief►Derived ►Lower central ►Upper central
| A5 — C22×A5 |
| A5 — C22×A5 |
Subgroups: 601 in 71 conjugacy classes, 10 normal (4 characteristic)
C1, C2, C2, C3, C22, C22, C5, S3, C6, C23, D5, C10, A4, D6, C2×C6, C24, D10, C2×C10, C2×A4, C22×S3, C22×D5, C22×A4, A5, C2×A5, C22×A5
Quotients: C1, C2, C22, A5, C2×A5, C22×A5
Character table of C22×A5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | |
| size | 1 | 1 | 1 | 1 | 15 | 15 | 15 | 15 | 20 | 12 | 12 | 20 | 20 | 20 | 12 | 12 | 12 | 12 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 3 | -3 | 3 | -3 | 1 | -1 | -1 | 1 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from C2×A5 |
| ρ6 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from A5 |
| ρ7 | 3 | -3 | -3 | 3 | -1 | -1 | 1 | 1 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from C2×A5 |
| ρ8 | 3 | -3 | 3 | -3 | 1 | -1 | -1 | 1 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from C2×A5 |
| ρ9 | 3 | 3 | -3 | -3 | 1 | -1 | 1 | -1 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from C2×A5 |
| ρ10 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from A5 |
| ρ11 | 3 | 3 | -3 | -3 | 1 | -1 | 1 | -1 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from C2×A5 |
| ρ12 | 3 | -3 | -3 | 3 | -1 | -1 | 1 | 1 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from C2×A5 |
| ρ13 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×A5 |
| ρ14 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×A5 |
| ρ15 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A5 |
| ρ16 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×A5 |
| ρ17 | 5 | -5 | -5 | 5 | 1 | 1 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A5 |
| ρ18 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | -1 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A5 |
| ρ19 | 5 | 5 | -5 | -5 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A5 |
| ρ20 | 5 | -5 | 5 | -5 | -1 | 1 | 1 | -1 | -1 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A5 |
►On 20 points - transitive group 20T64
Generators in S20
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 14)(2 13)(3 20)(4 17)(5 16)(6 19)(7 18)(8 15)(9 12)(10 11)
G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,14)(2,13)(3,20)(4,17)(5,16)(6,19)(7,18)(8,15)(9,12)(10,11)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,14)(2,13)(3,20)(4,17)(5,16)(6,19)(7,18)(8,15)(9,12)(10,11) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,14),(2,13),(3,20),(4,17),(5,16),(6,19),(7,18),(8,15),(9,12),(10,11)]])
G:=TransitiveGroup(20,64);
►On 24 points - transitive group 24T572
Generators in S24
(1 2)(3 4)(5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24)
(1 5)(2 10)(3 17)(4 22)(6 16)(7 24)(8 15)(9 23)(11 21)(12 19)(13 20)(14 18)
G:=sub<Sym(24)| (1,2)(3,4)(5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24), (1,5)(2,10)(3,17)(4,22)(6,16)(7,24)(8,15)(9,23)(11,21)(12,19)(13,20)(14,18)>;
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), (1,5)(2,10)(3,17)(4,22)(6,16)(7,24)(8,15)(9,23)(11,21)(12,19)(13,20)(14,18) );
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)], [(1,5),(2,10),(3,17),(4,22),(6,16),(7,24),(8,15),(9,23),(11,21),(12,19),(13,20),(14,18)]])
G:=TransitiveGroup(24,572);
►On 24 points - transitive group 24T573
Generators in S24
(1 2)(3 4)(5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24)
(1 20)(2 15)(3 12)(4 7)(5 23)(6 16)(8 24)(9 17)(10 18)(11 21)(13 19)(14 22)
G:=sub<Sym(24)| (1,2)(3,4)(5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24), (1,20)(2,15)(3,12)(4,7)(5,23)(6,16)(8,24)(9,17)(10,18)(11,21)(13,19)(14,22)>;
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), (1,20)(2,15)(3,12)(4,7)(5,23)(6,16)(8,24)(9,17)(10,18)(11,21)(13,19)(14,22) );
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)], [(1,20),(2,15),(3,12),(4,7),(5,23),(6,16),(8,24),(9,17),(10,18),(11,21),(13,19),(14,22)]])
G:=TransitiveGroup(24,573);
C22×A5 is a maximal subgroup of
C22⋊S5
C22×A5 is a maximal quotient of D4.A5 Q8.A5
Matrix representation of C22×A5 ►in GL5(𝔽31)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 | 0 |
| 0 | 0 | 29 | 3 | 10 |
| 0 | 0 | 28 | 21 | 29 |
| 0 | 0 | 1 | 0 | 0 |
| 30 | 0 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 |
G:=sub<GL(5,GF(31))| [1,0,0,0,0,0,30,0,0,0,0,0,29,28,1,0,0,3,21,0,0,0,10,29,0],[30,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,30] >;
C22×A5 in GAP, Magma, Sage, TeX
C_2^2\times A_5
% in TeX
G:=Group("C2^2xA5"); // GroupNames label
G:=SmallGroup(240,190);
// by ID
G=gap.SmallGroup(240,190);
# by ID
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