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G = C3×C42⋊C3order 144 = 24·32

Direct product of C3 and C42⋊C3

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

Aliases: C3×C42⋊C3, C42⋊C32, (C4×C12)⋊C3, (C2×C6).2A4, C22.(C3×A4), SmallGroup(144,68)

Series: Derived Chief Lower central Upper central

C1C42 — C3×C42⋊C3
C1C22C42C42⋊C3 — C3×C42⋊C3
C42 — C3×C42⋊C3
C1C3

Generators and relations for C3×C42⋊C3
 G = < a,b,c,d | a3=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >

3C2
16C3
16C3
16C3
3C4
3C4
3C6
16C32
3C2×C4
3C12
3C12
4A4
4A4
4A4
3C2×C12
4C3×A4

Character table of C3×C42⋊C3

 class 123A3B3C3D3E3F3G3H4A4B4C4D6A6B12A12B12C12D12E12F12G12H
 size 131116161616161633333333333333
ρ1111111111111111111111111    trivial
ρ211ζ32ζ3ζ321ζ3ζ321ζ31111ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ311ζ32ζ31ζ321ζ3ζ3ζ321111ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ411ζ3ζ321ζ31ζ32ζ32ζ31111ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ51111ζ32ζ3ζ3ζ3ζ32ζ3211111111111111    linear of order 3
ρ611ζ32ζ3ζ3ζ3ζ321ζ3211111ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ711ζ3ζ32ζ32ζ32ζ31ζ311111ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ81111ζ3ζ32ζ32ζ32ζ3ζ311111111111111    linear of order 3
ρ911ζ3ζ32ζ31ζ32ζ31ζ321111ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ103333000000-1-1-1-133-1-1-1-1-1-1-1-1    orthogonal lifted from A4
ρ1133-3-3-3/2-3+3-3/2000000-1-1-1-1-3-3-3/2-3+3-3/2ζ65ζ6ζ65ζ6ζ65ζ65ζ6ζ6    complex lifted from C3×A4
ρ1233-3+3-3/2-3-3-3/2000000-1-1-1-1-3+3-3/2-3-3-3/2ζ6ζ65ζ6ζ65ζ6ζ6ζ65ζ65    complex lifted from C3×A4
ρ133-1330000001-1+2i-1-2i1-1-11-1+2i-1-2i-1-2i1-1+2i11    complex lifted from C42⋊C3
ρ143-133000000-1-2i11-1+2i-1-1-1+2i111-1-2i1-1+2i-1-2i    complex lifted from C42⋊C3
ρ153-133000000-1+2i11-1-2i-1-1-1-2i111-1+2i1-1-2i-1+2i    complex lifted from C42⋊C3
ρ163-1330000001-1-2i-1+2i1-1-11-1-2i-1+2i-1+2i1-1-2i11    complex lifted from C42⋊C3
ρ173-1-3-3-3/2-3+3-3/20000001-1-2i-1+2i1ζ6ζ65ζ343ζ32324ζ334ζ3232ζ343ζ33ζ32ζ32    complex faithful
ρ183-1-3-3-3/2-3+3-3/20000001-1+2i-1-2i1ζ6ζ65ζ34ζ323243ζ3343ζ3232ζ34ζ33ζ32ζ32    complex faithful
ρ193-1-3+3-3/2-3-3-3/2000000-1+2i11-1-2iζ65ζ643ζ3232ζ3ζ32ζ34ζ3232ζ3243ζ334ζ33    complex faithful
ρ203-1-3-3-3/2-3+3-3/2000000-1-2i11-1+2iζ6ζ654ζ33ζ32ζ3ζ3243ζ33ζ34ζ323243ζ3232    complex faithful
ρ213-1-3+3-3/2-3-3-3/2000000-1-2i11-1+2iζ65ζ64ζ3232ζ3ζ32ζ343ζ3232ζ324ζ3343ζ33    complex faithful
ρ223-1-3-3-3/2-3+3-3/2000000-1+2i11-1-2iζ6ζ6543ζ33ζ32ζ3ζ324ζ33ζ343ζ32324ζ3232    complex faithful
ρ233-1-3+3-3/2-3-3-3/20000001-1+2i-1-2i1ζ65ζ6ζ324ζ3343ζ323243ζ33ζ324ζ3232ζ3ζ3    complex faithful
ρ243-1-3+3-3/2-3-3-3/20000001-1-2i-1+2i1ζ65ζ6ζ3243ζ334ζ32324ζ33ζ3243ζ3232ζ3ζ3    complex faithful

Smallest permutation representation of C3×C42⋊C3
On 36 points
Generators in S36
(1 4 12)(2 5 11)(3 8 7)(6 9 10)(13 31 18)(14 32 19)(15 29 20)(16 30 17)(21 36 28)(22 33 25)(23 34 26)(24 35 27)
(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)
(1 2 10 3)(4 5 6 8)(7 12 11 9)(13 14 15 16)(17 18 19 20)(21 23)(22 24)(25 27)(26 28)(29 30 31 32)(33 35)(34 36)
(1 28 32)(2 27 29)(3 25 31)(4 21 19)(5 24 20)(6 23 17)(7 33 13)(8 22 18)(9 34 16)(10 26 30)(11 35 15)(12 36 14)

G:=sub<Sym(36)| (1,4,12)(2,5,11)(3,8,7)(6,9,10)(13,31,18)(14,32,19)(15,29,20)(16,30,17)(21,36,28)(22,33,25)(23,34,26)(24,35,27), (13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,2,10,3)(4,5,6,8)(7,12,11,9)(13,14,15,16)(17,18,19,20)(21,23)(22,24)(25,27)(26,28)(29,30,31,32)(33,35)(34,36), (1,28,32)(2,27,29)(3,25,31)(4,21,19)(5,24,20)(6,23,17)(7,33,13)(8,22,18)(9,34,16)(10,26,30)(11,35,15)(12,36,14)>;

G:=Group( (1,4,12)(2,5,11)(3,8,7)(6,9,10)(13,31,18)(14,32,19)(15,29,20)(16,30,17)(21,36,28)(22,33,25)(23,34,26)(24,35,27), (13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,2,10,3)(4,5,6,8)(7,12,11,9)(13,14,15,16)(17,18,19,20)(21,23)(22,24)(25,27)(26,28)(29,30,31,32)(33,35)(34,36), (1,28,32)(2,27,29)(3,25,31)(4,21,19)(5,24,20)(6,23,17)(7,33,13)(8,22,18)(9,34,16)(10,26,30)(11,35,15)(12,36,14) );

G=PermutationGroup([(1,4,12),(2,5,11),(3,8,7),(6,9,10),(13,31,18),(14,32,19),(15,29,20),(16,30,17),(21,36,28),(22,33,25),(23,34,26),(24,35,27)], [(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)], [(1,2,10,3),(4,5,6,8),(7,12,11,9),(13,14,15,16),(17,18,19,20),(21,23),(22,24),(25,27),(26,28),(29,30,31,32),(33,35),(34,36)], [(1,28,32),(2,27,29),(3,25,31),(4,21,19),(5,24,20),(6,23,17),(7,33,13),(8,22,18),(9,34,16),(10,26,30),(11,35,15),(12,36,14)])

C3×C42⋊C3 is a maximal subgroup of   (C4×C12)⋊S3  (C4×C12)⋊C6  C42⋊C3⋊S3  C42⋊3- 1+2  C42⋊He3
C3×C42⋊C3 is a maximal quotient of   C42⋊3- 1+2  C122.C3  C42⋊He3

Matrix representation of C3×C42⋊C3 in GL3(𝔽13) generated by

300
030
003
,
800
050
001
,
800
0120
008
,
010
001
100
G:=sub<GL(3,GF(13))| [3,0,0,0,3,0,0,0,3],[8,0,0,0,5,0,0,0,1],[8,0,0,0,12,0,0,0,8],[0,0,1,1,0,0,0,1,0] >;

C3×C42⋊C3 in GAP, Magma, Sage, TeX

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

G:=Group("C3xC4^2:C3");
// GroupNames label

G:=SmallGroup(144,68);
// by ID

G=gap.SmallGroup(144,68);
# by ID

G:=PCGroup([6,-3,-3,-2,2,-2,2,326,230,2379,69,2164,3893]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=c^4=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations

Export

Subgroup lattice of C3×C42⋊C3 in TeX
Character table of C3×C42⋊C3 in TeX

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