non-abelian, soluble, monomial
Aliases: C33⋊2D8, C6.15S3≀C2, C3⋊2(C32⋊D8), D6⋊S3⋊1S3, C33⋊4C8⋊1C2, C33⋊9D4⋊6C2, (C32×C6).9D4, C32⋊3(D4⋊S3), C3⋊Dic3.10D6, C2.4(C33⋊D4), (C3×D6⋊S3)⋊1C2, (C3×C6).15(C3⋊D4), (C3×C3⋊Dic3).7C22, SmallGroup(432,582)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32 — C3×C3⋊Dic3 — C33⋊D8 |
| C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊D8 |
Generators and relations for C33⋊D8
G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, dad-1=b-1, eae=b, bc=cb, dbd-1=ebe=a, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 684 in 96 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C32, Dic3, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, D12, C3⋊D4, C3×D4, C33, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C62, D4⋊S3, S3×C32, C3×C3⋊S3, C32×C6, C32⋊2C8, D6⋊S3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C3×C3⋊Dic3, S3×C3×C6, C6×C3⋊S3, C32⋊D8, C33⋊4C8, C3×D6⋊S3, C33⋊9D4, C33⋊D8
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊D4, D4⋊S3, S3≀C2, C32⋊D8, C33⋊D4, C33⋊D8
Character table of C33⋊D8
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 6P | 8A | 8B | 12 | |
| size | 1 | 1 | 12 | 36 | 2 | 4 | 4 | 4 | 4 | 8 | 18 | 2 | 4 | 4 | 4 | 4 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 36 | 36 | 54 | 54 | 36 | |
| 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 | 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 | -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 | -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 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | -2 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | 2 | -1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | 0 | orthogonal lifted from D8 |
| ρ9 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | 0 | orthogonal lifted from D8 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | -2 | -1 | 2 | -1 | -1 | 2 | -1 | -√-3 | 0 | √-3 | -√-3 | 0 | √-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | complex lifted from C3⋊D4 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | -2 | -1 | 2 | -1 | -1 | 2 | -1 | √-3 | 0 | -√-3 | √-3 | 0 | -√-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | complex lifted from C3⋊D4 |
| ρ12 | 4 | -4 | 0 | 0 | -2 | -2 | 4 | 4 | -2 | -2 | 0 | 2 | -4 | 2 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ13 | 4 | 4 | 0 | -2 | 4 | -2 | 1 | -2 | -2 | 1 | 0 | 4 | 1 | -2 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ14 | 4 | 4 | 0 | 2 | 4 | -2 | 1 | -2 | -2 | 1 | 0 | 4 | 1 | -2 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ15 | 4 | 4 | -2 | 0 | 4 | 1 | -2 | 1 | 1 | -2 | 0 | 4 | -2 | 1 | 1 | 1 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ16 | 4 | 4 | 2 | 0 | 4 | 1 | -2 | 1 | 1 | -2 | 0 | 4 | -2 | 1 | 1 | 1 | -2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ17 | 4 | -4 | 0 | 0 | -2 | -1+3√-3/2 | -2 | 1 | -1-3√-3/2 | 1 | 0 | 2 | 2 | 1+3√-3/2 | 1-3√-3/2 | -1 | -1 | 0 | -√-3 | -3+√-3/2 | 3+√-3/2 | √-3 | 3-√-3/2 | -3-√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ18 | 4 | -4 | 0 | 0 | 4 | 1 | -2 | 1 | 1 | -2 | 0 | -4 | 2 | -1 | -1 | -1 | 2 | 0 | -√-3 | -√-3 | -√-3 | √-3 | √-3 | √-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊D8 |
| ρ19 | 4 | 4 | -2 | 0 | -2 | -1-3√-3/2 | -2 | 1 | -1+3√-3/2 | 1 | 0 | -2 | -2 | -1+3√-3/2 | -1-3√-3/2 | 1 | 1 | 1+√-3 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊D4 |
| ρ20 | 4 | 4 | 2 | 0 | -2 | -1-3√-3/2 | -2 | 1 | -1+3√-3/2 | 1 | 0 | -2 | -2 | -1+3√-3/2 | -1-3√-3/2 | 1 | 1 | -1-√-3 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊D4 |
| ρ21 | 4 | -4 | 0 | 0 | -2 | -1-3√-3/2 | -2 | 1 | -1+3√-3/2 | 1 | 0 | 2 | 2 | 1-3√-3/2 | 1+3√-3/2 | -1 | -1 | 0 | √-3 | -3-√-3/2 | 3-√-3/2 | -√-3 | 3+√-3/2 | -3+√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | -2 | -1-3√-3/2 | -2 | 1 | -1+3√-3/2 | 1 | 0 | 2 | 2 | 1-3√-3/2 | 1+3√-3/2 | -1 | -1 | 0 | -√-3 | 3+√-3/2 | -3+√-3/2 | √-3 | -3-√-3/2 | 3-√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 4 | 4 | 2 | 0 | -2 | -1+3√-3/2 | -2 | 1 | -1-3√-3/2 | 1 | 0 | -2 | -2 | -1-3√-3/2 | -1+3√-3/2 | 1 | 1 | -1+√-3 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊D4 |
| ρ24 | 4 | -4 | 0 | 0 | 4 | -2 | 1 | -2 | -2 | 1 | 0 | -4 | -1 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-3 | √-3 | 0 | 0 | 0 | complex lifted from C32⋊D8 |
| ρ25 | 4 | -4 | 0 | 0 | 4 | -2 | 1 | -2 | -2 | 1 | 0 | -4 | -1 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-3 | -√-3 | 0 | 0 | 0 | complex lifted from C32⋊D8 |
| ρ26 | 4 | -4 | 0 | 0 | -2 | -1+3√-3/2 | -2 | 1 | -1-3√-3/2 | 1 | 0 | 2 | 2 | 1+3√-3/2 | 1-3√-3/2 | -1 | -1 | 0 | √-3 | 3-√-3/2 | -3-√-3/2 | -√-3 | -3+√-3/2 | 3+√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 4 | -4 | 0 | 0 | 4 | 1 | -2 | 1 | 1 | -2 | 0 | -4 | 2 | -1 | -1 | -1 | 2 | 0 | √-3 | √-3 | √-3 | -√-3 | -√-3 | -√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊D8 |
| ρ28 | 4 | 4 | -2 | 0 | -2 | -1+3√-3/2 | -2 | 1 | -1-3√-3/2 | 1 | 0 | -2 | -2 | -1-3√-3/2 | -1+3√-3/2 | 1 | 1 | 1-√-3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊D4 |
| ρ29 | 8 | 8 | 0 | 0 | -4 | 2 | 2 | -4 | 2 | -1 | 0 | -4 | 2 | 2 | 2 | -4 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C33⋊D4 |
| ρ30 | 8 | -8 | 0 | 0 | -4 | 2 | 2 | -4 | 2 | -1 | 0 | 4 | -2 | -2 | -2 | 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 24 points - transitive group 24T1290
Generators in S24
(2 22 11)(4 13 24)(6 18 15)(8 9 20)
(1 10 21)(3 23 12)(5 14 17)(7 19 16)
(1 21 10)(2 11 22)(3 23 12)(4 13 24)(5 17 14)(6 15 18)(7 19 16)(8 9 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)
(1 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 24)(18 23)(19 22)(20 21)
G:=sub<Sym(24)| (2,22,11)(4,13,24)(6,18,15)(8,9,20), (1,10,21)(3,23,12)(5,14,17)(7,19,16), (1,21,10)(2,11,22)(3,23,12)(4,13,24)(5,17,14)(6,15,18)(7,19,16)(8,9,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,24)(18,23)(19,22)(20,21)>;
G:=Group( (2,22,11)(4,13,24)(6,18,15)(8,9,20), (1,10,21)(3,23,12)(5,14,17)(7,19,16), (1,21,10)(2,11,22)(3,23,12)(4,13,24)(5,17,14)(6,15,18)(7,19,16)(8,9,20), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,24)(18,23)(19,22)(20,21) );
G=PermutationGroup([[(2,22,11),(4,13,24),(6,18,15),(8,9,20)], [(1,10,21),(3,23,12),(5,14,17),(7,19,16)], [(1,21,10),(2,11,22),(3,23,12),(4,13,24),(5,17,14),(6,15,18),(7,19,16),(8,9,20)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,24),(18,23),(19,22),(20,21)]])
G:=TransitiveGroup(24,1290);
►On 24 points - transitive group 24T1314
Generators in S24
(1 23 10)(2 24 11)(3 12 17)(4 13 18)(5 19 14)(6 20 15)(7 16 21)(8 9 22)
(1 10 23)(2 24 11)(3 17 12)(4 13 18)(5 14 19)(6 20 15)(7 21 16)(8 9 22)
(1 23 10)(2 11 24)(3 17 12)(4 13 18)(5 19 14)(6 15 20)(7 21 16)(8 9 22)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(16 17)
G:=sub<Sym(24)| (1,23,10)(2,24,11)(3,12,17)(4,13,18)(5,19,14)(6,20,15)(7,16,21)(8,9,22), (1,10,23)(2,24,11)(3,17,12)(4,13,18)(5,14,19)(6,20,15)(7,21,16)(8,9,22), (1,23,10)(2,11,24)(3,17,12)(4,13,18)(5,19,14)(6,15,20)(7,21,16)(8,9,22), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)>;
G:=Group( (1,23,10)(2,24,11)(3,12,17)(4,13,18)(5,19,14)(6,20,15)(7,16,21)(8,9,22), (1,10,23)(2,24,11)(3,17,12)(4,13,18)(5,14,19)(6,20,15)(7,21,16)(8,9,22), (1,23,10)(2,11,24)(3,17,12)(4,13,18)(5,19,14)(6,15,20)(7,21,16)(8,9,22), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17) );
G=PermutationGroup([[(1,23,10),(2,24,11),(3,12,17),(4,13,18),(5,19,14),(6,20,15),(7,16,21),(8,9,22)], [(1,10,23),(2,24,11),(3,17,12),(4,13,18),(5,14,19),(6,20,15),(7,21,16),(8,9,22)], [(1,23,10),(2,11,24),(3,17,12),(4,13,18),(5,19,14),(6,15,20),(7,21,16),(8,9,22)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,8),(3,7),(4,6),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(16,17)]])
G:=TransitiveGroup(24,1314);
Matrix representation of C33⋊D8 ►in GL4(𝔽7) generated by
| 1 | 0 | 4 | 0 |
| 5 | 6 | 1 | 4 |
| 4 | 4 | 0 | 6 |
| 0 | 0 | 0 | 1 |
| 6 | 2 | 1 | 1 |
| 2 | 6 | 6 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 |
| 3 | 1 | 4 | 5 |
| 1 | 3 | 3 | 5 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 2 |
| 6 | 5 | 0 | 1 |
| 0 | 0 | 3 | 4 |
| 1 | 6 | 3 | 2 |
| 1 | 1 | 6 | 5 |
| 6 | 1 | 1 | 1 |
| 0 | 3 | 4 | 4 |
| 1 | 6 | 3 | 2 |
| 6 | 6 | 1 | 2 |
G:=sub<GL(4,GF(7))| [1,5,4,0,0,6,4,0,4,1,0,0,0,4,6,1],[6,2,0,0,2,6,0,0,1,6,1,0,1,1,0,2],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[6,0,1,1,5,0,6,1,0,3,3,6,1,4,2,5],[6,0,1,6,1,3,6,6,1,4,3,1,1,4,2,2] >;
C33⋊D8 in GAP, Magma, Sage, TeX
C_3^3\rtimes D_8
% in TeX
G:=Group("C3^3:D8"); // GroupNames label
G:=SmallGroup(432,582);
// by ID
G=gap.SmallGroup(432,582);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,254,135,58,1684,571,298,677,1027,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=b^-1,e*a*e=b,b*c=c*b,d*b*d^-1=e*b*e=a,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations
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