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## G = C3×C32⋊C4order 108 = 22·33

### Direct product of C3 and C32⋊C4

Aliases: C3×C32⋊C4, C331C4, C322C12, C3⋊S3.C6, (C3×C3⋊S3).1C2, SmallGroup(108,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C3×C3⋊S3 — C3×C32⋊C4
 Lower central C32 — C3×C32⋊C4
 Upper central C1 — C3

Generators and relations for C3×C32⋊C4
G = < a,b,c,d | a3=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Character table of C3×C32⋊C4

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 4A 4B 6A 6B 12A 12B 12C 12D size 1 9 1 1 4 4 4 4 4 4 9 9 9 9 9 9 9 9 ρ1 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 linear of order 2 ρ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 ζ3 -1 -1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ4 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ5 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ6 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 ζ32 -1 -1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ7 1 -1 1 1 1 1 1 1 1 1 i -i -1 -1 -i -i i i linear of order 4 ρ8 1 -1 1 1 1 1 1 1 1 1 -i i -1 -1 i i -i -i linear of order 4 ρ9 1 -1 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 ζ32 -i i ζ65 ζ6 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 linear of order 12 ρ10 1 -1 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 ζ3 -i i ζ6 ζ65 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 linear of order 12 ρ11 1 -1 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 ζ32 i -i ζ65 ζ6 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 linear of order 12 ρ12 1 -1 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 ζ3 i -i ζ6 ζ65 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 linear of order 12 ρ13 4 0 4 4 -2 -2 1 -2 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ14 4 0 4 4 1 1 -2 1 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ15 4 0 -2-2√-3 -2+2√-3 1-√-3 1+√-3 ζ32 -2 1 ζ3 0 0 0 0 0 0 0 0 complex faithful ρ16 4 0 -2-2√-3 -2+2√-3 ζ3 ζ32 1+√-3 1 -2 1-√-3 0 0 0 0 0 0 0 0 complex faithful ρ17 4 0 -2+2√-3 -2-2√-3 ζ32 ζ3 1-√-3 1 -2 1+√-3 0 0 0 0 0 0 0 0 complex faithful ρ18 4 0 -2+2√-3 -2-2√-3 1+√-3 1-√-3 ζ3 -2 1 ζ32 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C3×C32⋊C4
On 12 points - transitive group 12T73
Generators in S12
(1 8 9)(2 5 10)(3 6 11)(4 7 12)
(2 5 10)(4 12 7)
(1 8 9)(2 5 10)(3 11 6)(4 12 7)
(1 2 3 4)(5 6 7 8)(9 10 11 12)

G:=sub<Sym(12)| (1,8,9)(2,5,10)(3,6,11)(4,7,12), (2,5,10)(4,12,7), (1,8,9)(2,5,10)(3,11,6)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12)>;

G:=Group( (1,8,9)(2,5,10)(3,6,11)(4,7,12), (2,5,10)(4,12,7), (1,8,9)(2,5,10)(3,11,6)(4,12,7), (1,2,3,4)(5,6,7,8)(9,10,11,12) );

G=PermutationGroup([(1,8,9),(2,5,10),(3,6,11),(4,7,12)], [(2,5,10),(4,12,7)], [(1,8,9),(2,5,10),(3,11,6),(4,12,7)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)])

G:=TransitiveGroup(12,73);

On 18 points - transitive group 18T44
Generators in S18
(1 3 6)(2 4 5)(7 12 18)(8 13 15)(9 14 16)(10 11 17)
(1 15 17)(2 18 16)(3 8 10)(4 7 9)(5 12 14)(6 13 11)
(2 16 18)(4 9 7)(5 14 12)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)

G:=sub<Sym(18)| (1,3,6)(2,4,5)(7,12,18)(8,13,15)(9,14,16)(10,11,17), (1,15,17)(2,18,16)(3,8,10)(4,7,9)(5,12,14)(6,13,11), (2,16,18)(4,9,7)(5,14,12), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18)>;

G:=Group( (1,3,6)(2,4,5)(7,12,18)(8,13,15)(9,14,16)(10,11,17), (1,15,17)(2,18,16)(3,8,10)(4,7,9)(5,12,14)(6,13,11), (2,16,18)(4,9,7)(5,14,12), (1,2)(3,4)(5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18) );

G=PermutationGroup([(1,3,6),(2,4,5),(7,12,18),(8,13,15),(9,14,16),(10,11,17)], [(1,15,17),(2,18,16),(3,8,10),(4,7,9),(5,12,14),(6,13,11)], [(2,16,18),(4,9,7),(5,14,12)], [(1,2),(3,4),(5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)])

G:=TransitiveGroup(18,44);

On 27 points - transitive group 27T33
Generators in S27
(1 2 3)(4 19 9)(5 16 10)(6 17 11)(7 18 8)(12 20 27)(13 21 24)(14 22 25)(15 23 26)
(1 20 22)(2 27 25)(3 12 14)(4 24 7)(5 26 6)(8 9 21)(10 23 11)(13 18 19)(15 17 16)
(1 11 9)(2 6 4)(3 17 19)(5 24 27)(7 25 26)(8 22 23)(10 21 20)(12 16 13)(14 15 18)
(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)

G:=sub<Sym(27)| (1,2,3)(4,19,9)(5,16,10)(6,17,11)(7,18,8)(12,20,27)(13,21,24)(14,22,25)(15,23,26), (1,20,22)(2,27,25)(3,12,14)(4,24,7)(5,26,6)(8,9,21)(10,23,11)(13,18,19)(15,17,16), (1,11,9)(2,6,4)(3,17,19)(5,24,27)(7,25,26)(8,22,23)(10,21,20)(12,16,13)(14,15,18), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27)>;

G:=Group( (1,2,3)(4,19,9)(5,16,10)(6,17,11)(7,18,8)(12,20,27)(13,21,24)(14,22,25)(15,23,26), (1,20,22)(2,27,25)(3,12,14)(4,24,7)(5,26,6)(8,9,21)(10,23,11)(13,18,19)(15,17,16), (1,11,9)(2,6,4)(3,17,19)(5,24,27)(7,25,26)(8,22,23)(10,21,20)(12,16,13)(14,15,18), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27) );

G=PermutationGroup([(1,2,3),(4,19,9),(5,16,10),(6,17,11),(7,18,8),(12,20,27),(13,21,24),(14,22,25),(15,23,26)], [(1,20,22),(2,27,25),(3,12,14),(4,24,7),(5,26,6),(8,9,21),(10,23,11),(13,18,19),(15,17,16)], [(1,11,9),(2,6,4),(3,17,19),(5,24,27),(7,25,26),(8,22,23),(10,21,20),(12,16,13),(14,15,18)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)])

G:=TransitiveGroup(27,33);

C3×C32⋊C4 is a maximal subgroup of   C3⋊F9  C322D12  C33⋊Q8
C3×C32⋊C4 is a maximal quotient of   He3.3C12

Polynomial with Galois group C3×C32⋊C4 over ℚ
actionf(x)Disc(f)
12T73x12-4x11+38x10-163x9+1073x8-3580x7+16135x6-41958x5+122109x4-219057x3+358551x2-335799x+164511232·332·139·532·614·5032·329392·334976032

Matrix representation of C3×C32⋊C4 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 4 2 5 0 2 3 5 2 2 2 6 2 2 4 3 2
,
 6 0 3 1 5 3 6 0 1 6 4 5 3 3 2 6
,
 4 5 1 4 3 1 2 1 3 2 1 4 0 0 0 1
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[4,2,2,2,2,3,2,4,5,5,6,3,0,2,2,2],[6,5,1,3,0,3,6,3,3,6,4,2,1,0,5,6],[4,3,3,0,5,1,2,0,1,2,1,0,4,1,4,1] >;

C3×C32⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes C_4
% in TeX

G:=Group("C3xC3^2:C4");
// GroupNames label

G:=SmallGroup(108,36);
// by ID

G=gap.SmallGroup(108,36);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,3,30,1683,93,2404,314]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

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