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## G = C3×C3⋊D4order 72 = 23·32

### Direct product of C3 and C3⋊D4

Aliases: C3×C3⋊D4, C6C2, D62C6, Dic3⋊C6, C326D4, C622C2, C6.21D6, (C2×C6)⋊3S3, (C2×C6)⋊4C6, C32(C3×D4), (S3×C6)⋊4C2, C2.5(S3×C6), C6.5(C2×C6), C223(C3×S3), (C3×Dic3)⋊4C2, (C3×C6).10C22, SmallGroup(72,30)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×C3⋊D4
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — C3×C3⋊D4
 Lower central C3 — C6 — C3×C3⋊D4
 Upper central C1 — C6 — C2×C6

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

Permutation representations of C3×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  He36D4  Dic9⋊C6  He37D4  C32.S4  C62⋊S3  C32⋊S4  SL2(𝔽3).11D6
C3×C3⋊D4 is a maximal quotient of
He36D4  Dic9⋊C6

Polynomial with Galois group C3×C3⋊D4 over ℚ
actionf(x)Disc(f)
12T42x12-2x10-x8+6x6-9x5+12x4-12x3+8x2-3x+136·72·112·377

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

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