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G = C3×C32⋊C4order 108 = 22·33

Direct product of C3 and C32⋊C4

direct product, metabelian, soluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C32 — C3×C32⋊C4
C1C32C3⋊S3C3×C3⋊S3 — C3×C32⋊C4
C32 — C3×C32⋊C4
C1C3

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 >

9C2
2C3
2C3
4C3
4C3
9C4
6S3
6S3
9C6
2C32
2C32
4C32
4C32
9C12
6C3×S3
6C3×S3

Character table of C3×C32⋊C4

 class 123A3B3C3D3E3F3G3H4A4B6A6B12A12B12C12D
 size 191144444499999999
ρ1111111111111111111    trivial
ρ21111111111-1-111-1-1-1-1    linear of order 2
ρ311ζ32ζ3ζ3ζ32ζ3211ζ3-1-1ζ32ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ411ζ3ζ32ζ32ζ3ζ311ζ3211ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ511ζ32ζ3ζ3ζ32ζ3211ζ311ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ611ζ3ζ32ζ32ζ3ζ311ζ32-1-1ζ3ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ71-111111111i-i-1-1-i-iii    linear of order 4
ρ81-111111111-ii-1-1ii-i-i    linear of order 4
ρ91-1ζ3ζ32ζ32ζ3ζ311ζ32-iiζ65ζ6ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3    linear of order 12
ρ101-1ζ32ζ3ζ3ζ32ζ3211ζ3-iiζ6ζ65ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32    linear of order 12
ρ111-1ζ3ζ32ζ32ζ3ζ311ζ32i-iζ65ζ6ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3    linear of order 12
ρ121-1ζ32ζ3ζ3ζ32ζ3211ζ3i-iζ6ζ65ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32    linear of order 12
ρ134044-2-21-21100000000    orthogonal lifted from C32⋊C4
ρ14404411-21-2-200000000    orthogonal lifted from C32⋊C4
ρ1540-2-2-3-2+2-31--31+-3ζ32-21ζ300000000    complex faithful
ρ1640-2-2-3-2+2-3ζ3ζ321+-31-21--300000000    complex faithful
ρ1740-2+2-3-2-2-3ζ32ζ31--31-21+-300000000    complex faithful
ρ1840-2+2-3-2-2-31+-31--3ζ3-21ζ3200000000    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

4000
0400
0040
0004
,
4250
2352
2262
2432
,
6031
5360
1645
3326
,
4514
3121
3214
0001
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

Export

Subgroup lattice of C3×C32⋊C4 in TeX
Character table of C3×C32⋊C4 in TeX

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