p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C8:C22, D8:2C2, C4.14D4, SD16:1C2, D4:2C22, C4.5C23, Q8:2C22, C22.5D4, M4(2):1C2, C4oD4:2C2, (C2xD4):5C2, C2.15(C2xD4), (C2xC4).6C22, 2-Sylow(PGammaL(2,9)), Aut(D8), Hol(C8), SmallGroup(32,43)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8:C22
G = < a,b,c | a8=b2=c2=1, bab=a3, cac=a5, bc=cb >
Quotients: C1, C2, C22, D4, C23, C2xD4, C8:C22
Character table of C8:C22
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 8 points - transitive group 8T15
(1 2 3 4 5 6 7 8)
(2 4)(3 7)(6 8)
(1 5)(3 7)
G:=sub<Sym(8)| (1,2,3,4,5,6,7,8), (2,4)(3,7)(6,8), (1,5)(3,7)>;
G:=Group( (1,2,3,4,5,6,7,8), (2,4)(3,7)(6,8), (1,5)(3,7) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8)], [(2,4),(3,7),(6,8)], [(1,5),(3,7)]])
G:=TransitiveGroup(8,15);
►On 16 points - transitive group 16T35
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 9)(2 12)(3 15)(4 10)(5 13)(6 16)(7 11)(8 14)
(2 6)(4 8)(10 14)(12 16)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,12)(3,15)(4,10)(5,13)(6,16)(7,11)(8,14), (2,6)(4,8)(10,14)(12,16)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,12)(3,15)(4,10)(5,13)(6,16)(7,11)(8,14), (2,6)(4,8)(10,14)(12,16) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,9),(2,12),(3,15),(4,10),(5,13),(6,16),(7,11),(8,14)], [(2,6),(4,8),(10,14),(12,16)]])
G:=TransitiveGroup(16,35);
►On 16 points - transitive group 16T38
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 14)(2 9)(3 12)(4 15)(5 10)(6 13)(7 16)(8 11)
(1 14)(2 11)(3 16)(4 13)(5 10)(6 15)(7 12)(8 9)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,9)(3,12)(4,15)(5,10)(6,13)(7,16)(8,11), (1,14)(2,11)(3,16)(4,13)(5,10)(6,15)(7,12)(8,9)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,9)(3,12)(4,15)(5,10)(6,13)(7,16)(8,11), (1,14)(2,11)(3,16)(4,13)(5,10)(6,15)(7,12)(8,9) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,14),(2,9),(3,12),(4,15),(5,10),(6,13),(7,16),(8,11)], [(1,14),(2,11),(3,16),(4,13),(5,10),(6,15),(7,12),(8,9)]])
G:=TransitiveGroup(16,38);
►On 16 points - transitive group 16T45
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 9)(2 12)(3 15)(4 10)(5 13)(6 16)(7 11)(8 14)
(1 12)(2 9)(3 14)(4 11)(5 16)(6 13)(7 10)(8 15)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,12)(3,15)(4,10)(5,13)(6,16)(7,11)(8,14), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,12)(3,15)(4,10)(5,13)(6,16)(7,11)(8,14), (1,12)(2,9)(3,14)(4,11)(5,16)(6,13)(7,10)(8,15) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,9),(2,12),(3,15),(4,10),(5,13),(6,16),(7,11),(8,14)], [(1,12),(2,9),(3,14),(4,11),(5,16),(6,13),(7,10),(8,15)]])
G:=TransitiveGroup(16,45);
C8:C22 is a maximal subgroup of
C4.3S4 C32:D8:C2 C62.12D4
D4:D2p: D4:4D4 D8:S3 D4:D6 D8:D5 D4:D10 D8:D7 D4:D14 D4:D22 ...
D4.D2p: D4.8D4 D4.3D4 D4.4D4 D4oD8 D4oSD16 Q8:3D6 D12:6C22 D40:C2 ...
D4p:C22: D8:C22 C8:D6 C8:D10 C8:D14 C8:D22 C8:D26 ...
C8:C22 is a maximal quotient of
C23.36D4 C23.37D4 M4(2):C4 SD16:C4 D8:C4 C4:SD16 D4:2Q8 C4.Q16 D4.Q8 C23.46D4 C23.19D4 C23.48D4 C42.28C22 C42.29C22 C8:Q8 C32:D8:C2 C62.12D4
D4:D2p: C22:D8 D4:D4 C4:D8 D8:S3 D4:D6 D8:D5 D4:D10 D8:D7 ...
C8:D2p: C8:D4 C8:2D4 C8:3D4 C8:D6 Q8:3D6 C8:D10 D40:C2 C8:D14 ...
D4.D2p: C22:SD16 D4.2D4 D12:6C22 D4.D10 D4.D14 D44:6C22 D52:6C22 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 8T15 | x8-x4-1 | -216·54 |
Matrix representation of C8:C22 ►in GL4(Z) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | -1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
G:=sub<GL(4,Integers())| [0,0,0,1,0,0,1,0,1,0,0,0,0,-1,0,0],[1,0,0,0,0,-1,0,0,0,0,0,-1,0,0,-1,0],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1] >;
C8:C22 in GAP, Magma, Sage, TeX
C_8\rtimes C_2^2
% in TeX
G:=Group("C8:C2^2"); // GroupNames label
G:=SmallGroup(32,43);
// by ID
G=gap.SmallGroup(32,43);
# by ID
G:=PCGroup([5,-2,2,2,-2,-2,101,302,483,248,58]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=c^2=1,b*a*b=a^3,c*a*c=a^5,b*c=c*b>;
// generators/relations
Export
Subgroup lattice of C8:C22 in TeX
Character table of C8:C22 in TeX