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G = C3×C3⋊D4order 72 = 23·32

Direct product of C3 and C3⋊D4

direct product, metabelian, supersoluble, monomial

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

C1C6 — C3×C3⋊D4
C1C3C6C3×C6S3×C6 — C3×C3⋊D4
C3C6 — C3×C3⋊D4
C1C6C2×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 >

2C2
6C2
2C3
3C4
3C22
2C6
2C6
2S3
2C6
2C6
2C6
6C6
3D4
2C2×C6
3C2×C6
3C12
2C3×C6
2C3×S3
3C3×D4

Character table of C3×C3⋊D4

 class 12A2B2C3A3B3C3D3E46A6B6C6D6E6F6G6H6I6J6K6L6M6N6O12A12B
 size 112611222611222222222226666
ρ1111111111111111111111111111    trivial
ρ211-1-111111111-11-1-1-1-1-1-1-111-1-111    linear of order 2
ρ311-1111111-111-11-1-1-1-1-1-1-11111-1-1    linear of order 2
ρ4111-111111-11111111111111-1-1-1-1    linear of order 2
ρ51111ζ3ζ32ζ32ζ311ζ3ζ32111ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ6111-1ζ32ζ3ζ3ζ321-1ζ32ζ3111ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ6ζ65ζ65ζ6    linear of order 6
ρ711-1-1ζ32ζ3ζ3ζ3211ζ32ζ3-11-1ζ65ζ65ζ65ζ6ζ6ζ6ζ32ζ3ζ6ζ65ζ3ζ32    linear of order 6
ρ811-11ζ3ζ32ζ32ζ31-1ζ3ζ32-11-1ζ6ζ6ζ6ζ65ζ65ζ65ζ3ζ32ζ3ζ32ζ6ζ65    linear of order 6
ρ911-11ζ32ζ3ζ3ζ321-1ζ32ζ3-11-1ζ65ζ65ζ65ζ6ζ6ζ6ζ32ζ3ζ32ζ3ζ65ζ6    linear of order 6
ρ10111-1ζ3ζ32ζ32ζ31-1ζ3ζ32111ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ65ζ6ζ6ζ65    linear of order 6
ρ1111-1-1ζ3ζ32ζ32ζ311ζ3ζ32-11-1ζ6ζ6ζ6ζ65ζ65ζ65ζ3ζ32ζ65ζ6ζ32ζ3    linear of order 6
ρ121111ζ32ζ3ζ3ζ3211ζ32ζ3111ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ13222022-1-1-1022-1-1-12-1-12-1-1-1-10000    orthogonal lifted from S3
ρ142-200222220-2-20-20000000-2-20000    orthogonal lifted from D4
ρ1522-2022-1-1-10221-11-211-211-1-10000    orthogonal lifted from D6
ρ162-200-1--3-1+-3-1+-3-1--3201+-31--30-200000001+-31--30000    complex lifted from C3×D4
ρ172-20022-1-1-10-2-2--31-30--3-30--3-3110000    complex lifted from C3⋊D4
ρ182-200-1--3-1+-3ζ65ζ6-101+-31--3--31-303+-3/2-3--3/20-3+-3/23--3/2ζ32ζ30000    complex faithful
ρ1922-20-1--3-1+-3ζ65ζ6-10-1--3-1+-31-111--3ζ3ζ31+-3ζ32ζ32ζ6ζ650000    complex lifted from S3×C6
ρ202-200-1+-3-1--3ζ6ζ65-101--31+-3--31-30-3+-3/23--3/203+-3/2-3--3/2ζ3ζ320000    complex faithful
ρ212-200-1--3-1+-3ζ65ζ6-101+-31--3-31--30-3--3/23+-3/203--3/2-3+-3/2ζ32ζ30000    complex faithful
ρ222-200-1+-3-1--3ζ6ζ65-101--31+-3-31--303--3/2-3+-3/20-3--3/23+-3/2ζ3ζ320000    complex faithful
ρ232-200-1+-3-1--3-1--3-1+-3201--31+-30-200000001--31+-30000    complex lifted from C3×D4
ρ2422-20-1+-3-1--3ζ6ζ65-10-1+-3-1--31-111+-3ζ32ζ321--3ζ3ζ3ζ65ζ60000    complex lifted from S3×C6
ρ252220-1+-3-1--3ζ6ζ65-10-1+-3-1--3-1-1-1-1--3ζ6ζ6-1+-3ζ65ζ65ζ65ζ60000    complex lifted from C3×S3
ρ262220-1--3-1+-3ζ65ζ6-10-1--3-1+-3-1-1-1-1+-3ζ65ζ65-1--3ζ6ζ6ζ6ζ650000    complex lifted from C3×S3
ρ272-20022-1-1-10-2-2-31--30-3--30-3--3110000    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

40
04
,
20
04
,
06
10
,
01
10
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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Subgroup lattice of C3×C3⋊D4 in TeX
Character table of C3×C3⋊D4 in TeX

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