direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C22⋊A4, C24⋊3C32, (C2×C6)⋊A4, C22⋊2(C3×A4), (C23×C6)⋊2C3, SmallGroup(144,194)
Series: Derived ►Chief ►Lower central ►Upper central
| C24 — C3×C22⋊A4 |
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 |
►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 C24⋊23- 1+2 C24⋊43- 1+2 C62⋊A4 C3×A42
C3×C22⋊A4 is a maximal quotient of
C24⋊43- 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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