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## G = C3×C22⋊A4order 144 = 24·32

### Direct product of C3 and C22⋊A4

Aliases: C3×C22⋊A4, C243C32, (C2×C6)⋊A4, C222(C3×A4), (C23×C6)⋊2C3, SmallGroup(144,194)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3×C22⋊A4
 Chief series C1 — C22 — C24 — C22⋊A4 — C3×C22⋊A4
 Lower central C24 — C3×C22⋊A4
 Upper central C1 — C3

Generators and relations for C3×C22⋊A4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf-1=b, fdf-1=de=ed, fef-1=d >

Subgroups: 282 in 82 conjugacy classes, 18 normal (5 characteristic)
C1, C2, C3, C3, C22, C22, C6, C23, C32, A4, C2×C6, C2×C6, C24, C22×C6, C3×A4, C22⋊A4, C23×C6, C3×C22⋊A4
Quotients: C1, C3, C32, A4, C3×A4, C22⋊A4, C3×C22⋊A4

Character table of C3×C22⋊A4

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J size 1 3 3 3 3 3 1 1 16 16 16 16 16 16 3 3 3 3 3 3 3 3 3 3 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 trivial ρ2 1 1 1 1 1 1 ζ32 ζ3 1 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ3 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ4 1 1 1 1 1 1 ζ3 ζ32 1 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ5 1 1 1 1 1 1 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ6 1 1 1 1 1 1 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ7 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ8 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ9 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ10 3 -1 -1 -1 3 -1 3 3 0 0 0 0 0 0 3 -1 -1 -1 -1 3 -1 -1 -1 -1 orthogonal lifted from A4 ρ11 3 -1 -1 -1 -1 3 3 3 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 3 3 -1 orthogonal lifted from A4 ρ12 3 -1 3 -1 -1 -1 3 3 0 0 0 0 0 0 -1 -1 3 -1 -1 -1 -1 -1 -1 3 orthogonal lifted from A4 ρ13 3 -1 -1 3 -1 -1 3 3 0 0 0 0 0 0 -1 3 -1 -1 3 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ14 3 3 -1 -1 -1 -1 3 3 0 0 0 0 0 0 -1 -1 -1 3 -1 -1 3 -1 -1 -1 orthogonal lifted from A4 ρ15 3 -1 3 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ6 ζ6 -3+3√-3/2 ζ6 ζ65 ζ65 ζ65 ζ65 ζ6 -3-3√-3/2 complex lifted from C3×A4 ρ16 3 -1 -1 -1 -1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ65 ζ65 ζ6 ζ65 ζ6 ζ6 ζ6 -3-3√-3/2 -3+3√-3/2 ζ65 complex lifted from C3×A4 ρ17 3 -1 -1 -1 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -3+3√-3/2 ζ65 ζ6 ζ65 ζ6 -3-3√-3/2 ζ6 ζ6 ζ65 ζ65 complex lifted from C3×A4 ρ18 3 -1 -1 -1 -1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ6 ζ6 ζ65 ζ6 ζ65 ζ65 ζ65 -3+3√-3/2 -3-3√-3/2 ζ6 complex lifted from C3×A4 ρ19 3 3 -1 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ6 ζ6 ζ65 -3-3√-3/2 ζ65 ζ65 -3+3√-3/2 ζ65 ζ6 ζ6 complex lifted from C3×A4 ρ20 3 3 -1 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ65 ζ65 ζ6 -3+3√-3/2 ζ6 ζ6 -3-3√-3/2 ζ6 ζ65 ζ65 complex lifted from C3×A4 ρ21 3 -1 3 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ65 ζ65 -3-3√-3/2 ζ65 ζ6 ζ6 ζ6 ζ6 ζ65 -3+3√-3/2 complex lifted from C3×A4 ρ22 3 -1 -1 3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ6 -3-3√-3/2 ζ65 ζ6 -3+3√-3/2 ζ65 ζ65 ζ65 ζ6 ζ6 complex lifted from C3×A4 ρ23 3 -1 -1 -1 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -3-3√-3/2 ζ6 ζ65 ζ6 ζ65 -3+3√-3/2 ζ65 ζ65 ζ6 ζ6 complex lifted from C3×A4 ρ24 3 -1 -1 3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ65 -3+3√-3/2 ζ6 ζ65 -3-3√-3/2 ζ6 ζ6 ζ6 ζ65 ζ65 complex lifted from C3×A4

Smallest permutation representation of C3×C22⋊A4
On 36 points
Generators in S36
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(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 18)(2 16)(3 17)(4 34)(5 35)(6 36)(7 31)(8 32)(9 33)(10 15)(11 13)(12 14)(19 26)(20 27)(21 25)(22 30)(23 28)(24 29)
(1 15)(2 13)(3 14)(4 8)(5 9)(6 7)(10 18)(11 16)(12 17)(19 29)(20 30)(21 28)(22 27)(23 25)(24 26)(31 36)(32 34)(33 35)
(1 18)(2 16)(3 17)(10 15)(11 13)(12 14)(19 24)(20 22)(21 23)(25 28)(26 29)(27 30)
(1 18)(2 16)(3 17)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(31 36)(32 34)(33 35)
(1 5 26)(2 6 27)(3 4 25)(7 30 16)(8 28 17)(9 29 18)(10 35 24)(11 36 22)(12 34 23)(13 31 20)(14 32 21)(15 33 19)

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

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

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

C3×C22⋊A4 is a maximal subgroup of
(C22×S3)⋊A4  (C2×C6)⋊S4  C3.A42  C24⋊He3  C2423- 1+2  C2443- 1+2  C62⋊A4  C3×A42
C3×C22⋊A4 is a maximal quotient of
C2443- 1+2  C62.A4  C62⋊A4

Matrix representation of C3×C22⋊A4 in GL6(𝔽7)

 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 3 1 0 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 6 0 0 0 0 0 0 6
,
 1 0 0 0 0 0 4 6 0 0 0 0 2 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 0 6 0 0 0 0 5 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 3 1 0 0 0 0 0 0 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 3 0 0 0 0 0 6 2 0 0 0 0 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0

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

C3×C22⋊A4 in GAP, Magma, Sage, TeX

C_3\times C_2^2\rtimes A_4
% in TeX

G:=Group("C3xC2^2:A4");
// GroupNames label

G:=SmallGroup(144,194);
// by ID

G=gap.SmallGroup(144,194);
# by ID

G:=PCGroup([6,-3,-3,-2,2,-2,2,326,651,2164,3893]);
// Polycyclic

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

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