Copied to
clipboard

## G = C3×C42⋊C3order 144 = 24·32

### Direct product of C3 and C42⋊C3

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

 Derived series C1 — C42 — C3×C42⋊C3
 Chief series C1 — C22 — C42 — C42⋊C3 — C3×C42⋊C3
 Lower central C42 — C3×C42⋊C3
 Upper central C1 — C3

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 >

Character table of C3×C42⋊C3

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 4A 4B 4C 4D 6A 6B 12A 12B 12C 12D 12E 12F 12G 12H size 1 3 1 1 16 16 16 16 16 16 3 3 3 3 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 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ3 1 1 ζ32 ζ3 1 ζ32 1 ζ3 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ4 1 1 ζ3 ζ32 1 ζ3 1 ζ32 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ5 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 1 ζ32 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ7 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 1 ζ3 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ8 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ9 1 1 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ10 3 3 3 3 0 0 0 0 0 0 -1 -1 -1 -1 3 3 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from A4 ρ11 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -1 -1 -1 -3-3√-3/2 -3+3√-3/2 ζ65 ζ6 ζ65 ζ6 ζ65 ζ65 ζ6 ζ6 complex lifted from C3×A4 ρ12 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -1 -1 -1 -3+3√-3/2 -3-3√-3/2 ζ6 ζ65 ζ6 ζ65 ζ6 ζ6 ζ65 ζ65 complex lifted from C3×A4 ρ13 3 -1 3 3 0 0 0 0 0 0 1 -1+2i -1-2i 1 -1 -1 1 -1+2i -1-2i -1-2i 1 -1+2i 1 1 complex lifted from C42⋊C3 ρ14 3 -1 3 3 0 0 0 0 0 0 -1-2i 1 1 -1+2i -1 -1 -1+2i 1 1 1 -1-2i 1 -1+2i -1-2i complex lifted from C42⋊C3 ρ15 3 -1 3 3 0 0 0 0 0 0 -1+2i 1 1 -1-2i -1 -1 -1-2i 1 1 1 -1+2i 1 -1-2i -1+2i complex lifted from C42⋊C3 ρ16 3 -1 3 3 0 0 0 0 0 0 1 -1-2i -1+2i 1 -1 -1 1 -1-2i -1+2i -1+2i 1 -1-2i 1 1 complex lifted from C42⋊C3 ρ17 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 1 -1-2i -1+2i 1 ζ6 ζ65 ζ3 2ζ43ζ32-ζ32 2ζ4ζ3-ζ3 2ζ4ζ32-ζ32 ζ3 2ζ43ζ3-ζ3 ζ32 ζ32 complex faithful ρ18 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 1 -1+2i -1-2i 1 ζ6 ζ65 ζ3 2ζ4ζ32-ζ32 2ζ43ζ3-ζ3 2ζ43ζ32-ζ32 ζ3 2ζ4ζ3-ζ3 ζ32 ζ32 complex faithful ρ19 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1+2i 1 1 -1-2i ζ65 ζ6 2ζ43ζ32-ζ32 ζ3 ζ32 ζ3 2ζ4ζ32-ζ32 ζ32 2ζ43ζ3-ζ3 2ζ4ζ3-ζ3 complex faithful ρ20 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1-2i 1 1 -1+2i ζ6 ζ65 2ζ4ζ3-ζ3 ζ32 ζ3 ζ32 2ζ43ζ3-ζ3 ζ3 2ζ4ζ32-ζ32 2ζ43ζ32-ζ32 complex faithful ρ21 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1-2i 1 1 -1+2i ζ65 ζ6 2ζ4ζ32-ζ32 ζ3 ζ32 ζ3 2ζ43ζ32-ζ32 ζ32 2ζ4ζ3-ζ3 2ζ43ζ3-ζ3 complex faithful ρ22 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1+2i 1 1 -1-2i ζ6 ζ65 2ζ43ζ3-ζ3 ζ32 ζ3 ζ32 2ζ4ζ3-ζ3 ζ3 2ζ43ζ32-ζ32 2ζ4ζ32-ζ32 complex faithful ρ23 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 1 -1+2i -1-2i 1 ζ65 ζ6 ζ32 2ζ4ζ3-ζ3 2ζ43ζ32-ζ32 2ζ43ζ3-ζ3 ζ32 2ζ4ζ32-ζ32 ζ3 ζ3 complex faithful ρ24 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 1 -1-2i -1+2i 1 ζ65 ζ6 ζ32 2ζ43ζ3-ζ3 2ζ4ζ32-ζ32 2ζ4ζ3-ζ3 ζ32 2ζ43ζ32-ζ32 ζ3 ζ3 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

 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 1 0 0 0 1 1 0 0
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

׿
×
𝔽