direct product, metabelian, supersoluble, monomial
Aliases: C3×C3⋊D4, C6≀C2, D6⋊2C6, Dic3⋊C6, C32⋊6D4, C62⋊2C2, C6.21D6, (C2×C6)⋊3S3, (C2×C6)⋊4C6, C3⋊2(C3×D4), (S3×C6)⋊4C2, C2.5(S3×C6), C6.5(C2×C6), C22⋊3(C3×S3), (C3×Dic3)⋊4C2, (C3×C6).10C22, SmallGroup(72,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C3⋊D4
G = < a,b,c,d | a3=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Character table of C3×C3⋊D4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 12A | 12B | |
| size | 1 | 1 | 2 | 6 | 1 | 1 | 2 | 2 | 2 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | -1 | 1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ32 | ζ3 | ζ6 | ζ65 | ζ3 | ζ32 | linear of order 6 |
| ρ8 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | -1 | ζ3 | ζ32 | -1 | 1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ3 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ9 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | -1 | ζ32 | ζ3 | -1 | 1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ32 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ11 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | -1 | 1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ3 | ζ32 | ζ65 | ζ6 | ζ32 | ζ3 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ13 | 2 | 2 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 2 | 2 | 1 | -1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ16 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | 2 | 0 | 1+√-3 | 1-√-3 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ17 | 2 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | -2 | -2 | -√-3 | 1 | √-3 | 0 | -√-3 | √-3 | 0 | -√-3 | √-3 | 1 | 1 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ18 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 1+√-3 | 1-√-3 | -√-3 | 1 | √-3 | 0 | 3+√-3/2 | -3-√-3/2 | 0 | -3+√-3/2 | 3-√-3/2 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 2 | 2 | -2 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | -1-√-3 | -1+√-3 | 1 | -1 | 1 | 1-√-3 | ζ3 | ζ3 | 1+√-3 | ζ32 | ζ32 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | complex lifted from S3×C6 |
| ρ20 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 1-√-3 | 1+√-3 | -√-3 | 1 | √-3 | 0 | -3+√-3/2 | 3-√-3/2 | 0 | 3+√-3/2 | -3-√-3/2 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | 1+√-3 | 1-√-3 | √-3 | 1 | -√-3 | 0 | -3-√-3/2 | 3+√-3/2 | 0 | 3-√-3/2 | -3+√-3/2 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | 1-√-3 | 1+√-3 | √-3 | 1 | -√-3 | 0 | 3-√-3/2 | -3+√-3/2 | 0 | -3-√-3/2 | 3+√-3/2 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | 2 | 0 | 1-√-3 | 1+√-3 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ24 | 2 | 2 | -2 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | -1+√-3 | -1-√-3 | 1 | -1 | 1 | 1+√-3 | ζ32 | ζ32 | 1-√-3 | ζ3 | ζ3 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | complex lifted from S3×C6 |
| ρ25 | 2 | 2 | 2 | 0 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | 0 | -1+√-3 | -1-√-3 | -1 | -1 | -1 | -1-√-3 | ζ6 | ζ6 | -1+√-3 | ζ65 | ζ65 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ26 | 2 | 2 | 2 | 0 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | 0 | -1-√-3 | -1+√-3 | -1 | -1 | -1 | -1+√-3 | ζ65 | ζ65 | -1-√-3 | ζ6 | ζ6 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ27 | 2 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | -2 | -2 | √-3 | 1 | -√-3 | 0 | √-3 | -√-3 | 0 | √-3 | -√-3 | 1 | 1 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
►On 12 points - transitive group 12T42
Generators in S12
(1 8 9)(2 5 10)(3 6 11)(4 7 12)
(1 8 9)(2 10 5)(3 6 11)(4 12 7)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)
G:=sub<Sym(12)| (1,8,9)(2,5,10)(3,6,11)(4,7,12), (1,8,9)(2,10,5)(3,6,11)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11)>;
G:=Group( (1,8,9)(2,5,10)(3,6,11)(4,7,12), (1,8,9)(2,10,5)(3,6,11)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,6)(7,8)(9,12)(10,11) );
G=PermutationGroup([[(1,8,9),(2,5,10),(3,6,11),(4,7,12)], [(1,8,9),(2,10,5),(3,6,11),(4,12,7)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]])
G:=TransitiveGroup(12,42);
►On 24 points - transitive group 24T77
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 20 21)(2 22 17)(3 18 23)(4 24 19)(5 10 14)(6 15 11)(7 12 16)(8 13 9)
(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 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
G:=sub<Sym(24)| (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,10,14)(6,15,11)(7,12,16)(8,13,9), (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)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22)>;
G:=Group( (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (1,20,21)(2,22,17)(3,18,23)(4,24,19)(5,10,14)(6,15,11)(7,12,16)(8,13,9), (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)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22) );
G=PermutationGroup([[(1,20,21),(2,17,22),(3,18,23),(4,19,24),(5,10,14),(6,11,15),(7,12,16),(8,9,13)], [(1,20,21),(2,22,17),(3,18,23),(4,24,19),(5,10,14),(6,15,11),(7,12,16),(8,13,9)], [(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),(9,20),(10,19),(11,18),(12,17),(13,21),(14,24),(15,23),(16,22)]])
G:=TransitiveGroup(24,77);
C3×C3⋊D4 is a maximal subgroup of
D6.3D6 D6.4D6 Dic3⋊D6 C3×S3×D4 He3⋊6D4 Dic9⋊C6 He3⋊7D4 C32.S4 C62⋊S3 C32⋊S4 SL2(𝔽3).11D6
C3×C3⋊D4 is a maximal quotient of
He3⋊6D4 Dic9⋊C6
| action | f(x) | Disc(f) |
|---|---|---|
| 12T42 | x12-2x10-x8+6x6-9x5+12x4-12x3+8x2-3x+1 | 36·72·112·377 |
Matrix representation of C3×C3⋊D4 ►in GL2(𝔽7) generated by
| 4 | 0 |
| 0 | 4 |
| 2 | 0 |
| 0 | 4 |
| 0 | 6 |
| 1 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(7))| [4,0,0,4],[2,0,0,4],[0,1,6,0],[0,1,1,0] >;
C3×C3⋊D4 in GAP, Magma, Sage, TeX
C_3\times C_3\rtimes D_4
% in TeX
G:=Group("C3xC3:D4"); // GroupNames label
G:=SmallGroup(72,30);
// by ID
G=gap.SmallGroup(72,30);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-3,141,1204]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C3×C3⋊D4 in TeX
Character table of C3×C3⋊D4 in TeX