metabelian, soluble, monomial, A-group
Aliases: C33⋊4C8, C6.(C32⋊C4), C3⋊(C32⋊2C8), C32⋊4(C3⋊C8), C2.(C33⋊C4), C3⋊Dic3.2S3, (C32×C6).2C4, (C3×C6).5Dic3, (C3×C3⋊Dic3).4C2, SmallGroup(216,118)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊4C8 |
| C33 — C33⋊4C8 |
Generators and relations for C33⋊4C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, dbd-1=a-1b-1, dcd-1=c-1 >
Character table of C33⋊4C8
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 9 | 9 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 27 | 27 | 27 | 27 | 18 | 18 | |
| ρ1 | 1 | 1 | 1 | 1 | 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 | -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 | -i | i | i | -i | -1 | -1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | -i | i | -1 | -1 | linear of order 4 |
| ρ5 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ83 | ζ85 | ζ8 | ζ87 | -i | i | linear of order 8 |
| ρ6 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ87 | ζ8 | ζ85 | ζ83 | -i | i | linear of order 8 |
| ρ7 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ8 | ζ87 | ζ83 | ζ85 | i | -i | linear of order 8 |
| ρ8 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ85 | ζ83 | ζ87 | ζ8 | i | -i | linear of order 8 |
| ρ9 | 2 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -2 | -2 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ11 | 2 | -2 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2i | -2i | 1 | 1 | -2 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | -i | i | complex lifted from C3⋊C8 |
| ρ12 | 2 | -2 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -2i | 2i | 1 | 1 | -2 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | i | -i | complex lifted from C3⋊C8 |
| ρ13 | 4 | 4 | 4 | 1 | -2 | -2 | 1 | 1 | -2 | 0 | 0 | 4 | -2 | 1 | 1 | 1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ14 | 4 | 4 | 4 | -2 | 1 | 1 | -2 | -2 | 1 | 0 | 0 | 4 | 1 | -2 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
| ρ15 | 4 | -4 | 4 | -2 | 1 | 1 | -2 | -2 | 1 | 0 | 0 | -4 | -1 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊2C8, Schur index 2 |
| ρ16 | 4 | -4 | 4 | 1 | -2 | -2 | 1 | 1 | -2 | 0 | 0 | -4 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊2C8, Schur index 2 |
| ρ17 | 4 | 4 | -2 | 1 | 1 | -2 | -1+3√-3/2 | -1-3√-3/2 | 1 | 0 | 0 | -2 | 1 | 1 | -1+3√-3/2 | -1-3√-3/2 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ18 | 4 | 4 | -2 | 1 | 1 | -2 | -1-3√-3/2 | -1+3√-3/2 | 1 | 0 | 0 | -2 | 1 | 1 | -1-3√-3/2 | -1+3√-3/2 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ19 | 4 | -4 | -2 | -2 | -1+3√-3/2 | 1 | 1 | 1 | -1-3√-3/2 | 0 | 0 | 2 | 1-3√-3/2 | 2 | -1 | -1 | 1+3√-3/2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 4 | 4 | -2 | -2 | -1-3√-3/2 | 1 | 1 | 1 | -1+3√-3/2 | 0 | 0 | -2 | -1-3√-3/2 | -2 | 1 | 1 | -1+3√-3/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
| ρ21 | 4 | -4 | -2 | -2 | -1-3√-3/2 | 1 | 1 | 1 | -1+3√-3/2 | 0 | 0 | 2 | 1+3√-3/2 | 2 | -1 | -1 | 1-3√-3/2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | -2 | 1 | 1 | -2 | -1+3√-3/2 | -1-3√-3/2 | 1 | 0 | 0 | 2 | -1 | -1 | 1-3√-3/2 | 1+3√-3/2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 4 | -4 | -2 | 1 | 1 | -2 | -1-3√-3/2 | -1+3√-3/2 | 1 | 0 | 0 | 2 | -1 | -1 | 1+3√-3/2 | 1-3√-3/2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | 4 | -2 | -2 | -1+3√-3/2 | 1 | 1 | 1 | -1-3√-3/2 | 0 | 0 | -2 | -1+3√-3/2 | -2 | 1 | 1 | -1-3√-3/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
►On 24 points - transitive group 24T552
Generators in S24
(2 22 13)(4 15 24)(6 18 9)(8 11 20)
(1 12 21)(2 22 13)(3 23 14)(4 15 24)(5 16 17)(6 18 9)(7 19 10)(8 11 20)
(1 21 12)(2 13 22)(3 23 14)(4 15 24)(5 17 16)(6 9 18)(7 19 10)(8 11 20)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (2,22,13)(4,15,24)(6,18,9)(8,11,20), (1,12,21)(2,22,13)(3,23,14)(4,15,24)(5,16,17)(6,18,9)(7,19,10)(8,11,20), (1,21,12)(2,13,22)(3,23,14)(4,15,24)(5,17,16)(6,9,18)(7,19,10)(8,11,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
G:=Group( (2,22,13)(4,15,24)(6,18,9)(8,11,20), (1,12,21)(2,22,13)(3,23,14)(4,15,24)(5,16,17)(6,18,9)(7,19,10)(8,11,20), (1,21,12)(2,13,22)(3,23,14)(4,15,24)(5,17,16)(6,9,18)(7,19,10)(8,11,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(2,22,13),(4,15,24),(6,18,9),(8,11,20)], [(1,12,21),(2,22,13),(3,23,14),(4,15,24),(5,16,17),(6,18,9),(7,19,10),(8,11,20)], [(1,21,12),(2,13,22),(3,23,14),(4,15,24),(5,17,16),(6,9,18),(7,19,10),(8,11,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,552);
C33⋊4C8 is a maximal subgroup of
S3×C32⋊2C8 C33⋊5(C2×C8) C33⋊M4(2) C33⋊2M4(2) C33⋊D8 C33⋊6SD16 C33⋊7SD16 C33⋊Q16 C33⋊7(C2×C8) C33⋊4M4(2) C33⋊12M4(2)
C33⋊4C8 is a maximal quotient of
C33⋊4C16
Matrix representation of C33⋊4C8 ►in GL4(𝔽7) generated by
| 3 | 2 | 4 | 3 |
| 4 | 5 | 5 | 6 |
| 3 | 3 | 6 | 1 |
| 0 | 0 | 0 | 1 |
| 0 | 5 | 2 | 6 |
| 0 | 2 | 0 | 2 |
| 3 | 3 | 6 | 1 |
| 0 | 0 | 0 | 4 |
| 3 | 6 | 3 | 2 |
| 6 | 3 | 4 | 2 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 5 | 3 | 3 | 1 |
| 2 | 2 | 1 | 6 |
| 2 | 5 | 6 | 5 |
| 3 | 3 | 4 | 1 |
G:=sub<GL(4,GF(7))| [3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[0,0,3,0,5,2,3,0,2,0,6,0,6,2,1,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[5,2,2,3,3,2,5,3,3,1,6,4,1,6,5,1] >;
C33⋊4C8 in GAP, Magma, Sage, TeX
C_3^3\rtimes_4C_8
% in TeX
G:=Group("C3^3:4C8"); // GroupNames label
G:=SmallGroup(216,118);
// by ID
G=gap.SmallGroup(216,118);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,3,-3,12,31,963,201,964,730,5189]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,b*c=c*b,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C33⋊4C8 in TeX
Character table of C33⋊4C8 in TeX