p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C8.C22, Q16⋊2C2, C4.15D4, SD16⋊2C2, C4.6C23, C22.6D4, M4(2)⋊2C2, D4.3C22, Q8.3C22, (C2×Q8)⋊4C2, C4○D4.2C2, C2.16(C2×D4), (C2×C4).7C22, 2-Sylow(ASigmaL(2,9)), SmallGroup(32,44)
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, cbc=a4b >
Character table of C8.C22
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 8A | 8B | |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 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 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 16 points - transitive group 16T32
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)
(1 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16)], [(1,11),(2,16),(3,13),(4,10),(5,15),(6,12),(7,9),(8,14)]])
G:=TransitiveGroup(16,32);
►On 16 points - transitive group 16T50
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 12)(2 15)(3 10)(4 13)(5 16)(6 11)(7 14)(8 9)
(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), (1,12)(2,15)(3,10)(4,13)(5,16)(6,11)(7,14)(8,9), (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), (1,12)(2,15)(3,10)(4,13)(5,16)(6,11)(7,14)(8,9), (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)], [(1,12),(2,15),(3,10),(4,13),(5,16),(6,11),(7,14),(8,9)], [(2,6),(4,8),(10,14),(12,16)]])
G:=TransitiveGroup(16,50);
C8.C22 is a maximal subgroup of
Q8.D6 C4.S4 C32⋊Q16⋊C2 C62.15D4
D4.D2p: D4.9D4 D4.10D4 D4.3D4 D4.5D4 D4○SD16 Q8○D8 D4.D6 Q8.14D6 ...
C4p.C23: D8⋊C22 C8.D6 Q16⋊S3 Q8.11D6 C8.D10 Q16⋊D5 C20.C23 C8.D14 ...
C8.C22 is a maximal quotient of
C23.36D4 C23.38D4 M4(2)⋊C4 SD16⋊C4 Q16⋊C4 Q8⋊D4 C8⋊D4 D4⋊Q8 Q8⋊Q8 Q8.Q8 C22.D8 C23.47D4 C23.20D4 C42.28C22 C42.30C22 C8⋊Q8 C32⋊Q16⋊C2 C62.15D4
D4.D2p: D4.7D4 D4.D4 D4.D6 Q8.14D6 SD16⋊D5 D4.9D10 SD16⋊D7 D4.9D14 ...
C8.D2p: C8.D4 C8.2D4 C8.D6 Q16⋊S3 C8.D10 Q16⋊D5 C8.D14 Q16⋊D7 ...
Q8.D2p: C22⋊Q16 C4⋊2Q16 Q8.D4 Q8.11D6 C20.C23 C28.C23 C44.C23 Q8.D26 ...
Matrix representation of C8.C22 ►in GL4(𝔽3) generated by
| 0 | 2 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 0 | 2 | 0 | 2 |
| 0 | 0 | 1 | 0 |
| 0 | 2 | 0 | 2 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 2 | 0 | 2 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(3))| [0,1,0,0,2,0,2,0,0,1,0,1,0,0,2,0],[0,0,0,2,2,0,1,0,0,1,0,2,2,0,0,0],[2,0,0,0,0,1,0,0,0,0,2,0,0,0,0,1] >;
C8.C22 in GAP, Magma, Sage, TeX
C_8.C_2^2
% in TeX
G:=Group("C8.C2^2"); // GroupNames label
G:=SmallGroup(32,44);
// by ID
G=gap.SmallGroup(32,44);
# by ID
G:=PCGroup([5,-2,2,2,-2,-2,101,86,302,483,248,58]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=c^2=1,b*a*b=a^3,c*a*c=a^5,c*b*c=a^4*b>;
// generators/relations
Export
Subgroup lattice of C8.C22 in TeX
Character table of C8.C22 in TeX