direct product, non-abelian, soluble, monomial
Aliases: S3×A4⋊C4, D6.5S4, (S3×A4)⋊C4, A4⋊3(C4×S3), C2.3(S3×S4), C23.5S32, C6.13(C2×S4), (C2×A4).5D6, (S3×C23).S3, C6.7S4⋊2C2, (C22×S3)⋊Dic3, (C22×C6).5D6, (C6×A4).5C22, C22⋊2(S3×Dic3), (C2×S3×A4).C2, C3⋊1(C2×A4⋊C4), (C2×C6)⋊(C2×Dic3), (C3×A4⋊C4)⋊3C2, (C3×A4)⋊3(C2×C4), SmallGroup(288,856)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×A4 — S3×A4⋊C4 |
Generators and relations for S3×A4⋊C4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e3=f4=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=e-1 >
Subgroups: 698 in 132 conjugacy classes, 25 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, S3, C6, C6, C2×C4, C23, C23, C32, Dic3, C12, A4, A4, D6, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3×S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×A4, C2×A4, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C3×Dic3, C3⋊Dic3, C3×A4, S3×C6, D6⋊C4, C6.D4, C3×C22⋊C4, A4⋊C4, A4⋊C4, S3×C2×C4, C22×A4, S3×C23, S3×Dic3, S3×A4, C6×A4, S3×C22⋊C4, C2×A4⋊C4, C3×A4⋊C4, C6.7S4, C2×S3×A4, S3×A4⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C4×S3, C2×Dic3, S4, S32, A4⋊C4, C2×S4, S3×Dic3, C2×A4⋊C4, S3×S4, S3×A4⋊C4
Character table of S3×A4⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 3 | 3 | 3 | 3 | 9 | 9 | 2 | 8 | 16 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | 6 | 6 | 8 | 16 | 24 | 24 | 12 | 12 | 12 | 12 | |
| 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 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -i | i | -i | i | -i | i | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | i | -i | i | -i | i | -i | -1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -i | i | i | -i | -i | i | -i | i | -1 | 1 | -1 | -1 | -1 | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | i | -i | -i | i | i | -i | i | -i | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | -1 | 2 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 2 | 0 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | -1 | 2 | -1 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | 1 | 0 | 0 | i | -i | -i | i | complex lifted from C4×S3 |
| ρ16 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | -1 | 2 | -1 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | 1 | 0 | 0 | -i | i | i | -i | complex lifted from C4×S3 |
| ρ17 | 3 | 3 | 3 | -1 | 3 | -1 | -1 | -1 | 3 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from S4 |
| ρ18 | 3 | 3 | 3 | -1 | 3 | -1 | -1 | -1 | 3 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from S4 |
| ρ19 | 3 | 3 | -3 | -1 | -3 | -1 | 1 | 1 | 3 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from C2×S4 |
| ρ20 | 3 | 3 | -3 | -1 | -3 | -1 | 1 | 1 | 3 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from C2×S4 |
| ρ21 | 3 | -3 | -3 | 1 | 3 | -1 | 1 | -1 | 3 | 0 | 0 | -i | i | -i | i | i | -i | -i | i | -3 | -1 | 1 | 0 | 0 | 0 | 0 | i | i | -i | -i | complex lifted from A4⋊C4 |
| ρ22 | 3 | -3 | -3 | 1 | 3 | -1 | 1 | -1 | 3 | 0 | 0 | i | -i | i | -i | -i | i | i | -i | -3 | -1 | 1 | 0 | 0 | 0 | 0 | -i | -i | i | i | complex lifted from A4⋊C4 |
| ρ23 | 3 | -3 | 3 | 1 | -3 | -1 | -1 | 1 | 3 | 0 | 0 | i | -i | i | -i | i | -i | -i | i | -3 | -1 | 1 | 0 | 0 | 0 | 0 | -i | -i | i | i | complex lifted from A4⋊C4 |
| ρ24 | 3 | -3 | 3 | 1 | -3 | -1 | -1 | 1 | 3 | 0 | 0 | -i | i | -i | i | -i | i | i | -i | -3 | -1 | 1 | 0 | 0 | 0 | 0 | i | i | -i | -i | complex lifted from A4⋊C4 |
| ρ25 | 4 | 4 | 0 | 4 | 0 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ26 | 4 | -4 | 0 | -4 | 0 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Dic3, Schur index 2 |
| ρ27 | 6 | 6 | 0 | -2 | 0 | -2 | 0 | 0 | -3 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from S3×S4 |
| ρ28 | 6 | 6 | 0 | -2 | 0 | -2 | 0 | 0 | -3 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from S3×S4 |
| ρ29 | 6 | -6 | 0 | 2 | 0 | -2 | 0 | 0 | -3 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | 0 | 0 | 0 | -i | -i | i | i | complex faithful |
| ρ30 | 6 | -6 | 0 | 2 | 0 | -2 | 0 | 0 | -3 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | 0 | 0 | 0 | i | i | -i | -i | complex faithful |
►On 36 points
Generators in S36
(1 6 10)(2 7 11)(3 8 12)(4 5 9)(13 27 17)(14 28 18)(15 25 19)(16 26 20)(21 35 31)(22 36 32)(23 33 29)(24 34 30)
(5 9)(6 10)(7 11)(8 12)(17 27)(18 28)(19 25)(20 26)(21 35)(22 36)(23 33)(24 34)
(1 2)(3 4)(5 8)(6 7)(9 12)(10 11)(13 15)(14 31)(16 29)(17 19)(18 35)(20 33)(21 28)(22 24)(23 26)(25 27)(30 32)(34 36)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 32)(14 29)(15 30)(16 31)(17 36)(18 33)(19 34)(20 35)(21 26)(22 27)(23 28)(24 25)
(1 16 30)(2 31 13)(3 14 32)(4 29 15)(5 23 25)(6 26 24)(7 21 27)(8 28 22)(9 33 19)(10 20 34)(11 35 17)(12 18 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)
G:=sub<Sym(36)| (1,6,10)(2,7,11)(3,8,12)(4,5,9)(13,27,17)(14,28,18)(15,25,19)(16,26,20)(21,35,31)(22,36,32)(23,33,29)(24,34,30), (5,9)(6,10)(7,11)(8,12)(17,27)(18,28)(19,25)(20,26)(21,35)(22,36)(23,33)(24,34), (1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,31)(16,29)(17,19)(18,35)(20,33)(21,28)(22,24)(23,26)(25,27)(30,32)(34,36), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,32)(14,29)(15,30)(16,31)(17,36)(18,33)(19,34)(20,35)(21,26)(22,27)(23,28)(24,25), (1,16,30)(2,31,13)(3,14,32)(4,29,15)(5,23,25)(6,26,24)(7,21,27)(8,28,22)(9,33,19)(10,20,34)(11,35,17)(12,18,36), (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)>;
G:=Group( (1,6,10)(2,7,11)(3,8,12)(4,5,9)(13,27,17)(14,28,18)(15,25,19)(16,26,20)(21,35,31)(22,36,32)(23,33,29)(24,34,30), (5,9)(6,10)(7,11)(8,12)(17,27)(18,28)(19,25)(20,26)(21,35)(22,36)(23,33)(24,34), (1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,31)(16,29)(17,19)(18,35)(20,33)(21,28)(22,24)(23,26)(25,27)(30,32)(34,36), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,32)(14,29)(15,30)(16,31)(17,36)(18,33)(19,34)(20,35)(21,26)(22,27)(23,28)(24,25), (1,16,30)(2,31,13)(3,14,32)(4,29,15)(5,23,25)(6,26,24)(7,21,27)(8,28,22)(9,33,19)(10,20,34)(11,35,17)(12,18,36), (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) );
G=PermutationGroup([[(1,6,10),(2,7,11),(3,8,12),(4,5,9),(13,27,17),(14,28,18),(15,25,19),(16,26,20),(21,35,31),(22,36,32),(23,33,29),(24,34,30)], [(5,9),(6,10),(7,11),(8,12),(17,27),(18,28),(19,25),(20,26),(21,35),(22,36),(23,33),(24,34)], [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11),(13,15),(14,31),(16,29),(17,19),(18,35),(20,33),(21,28),(22,24),(23,26),(25,27),(30,32),(34,36)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,32),(14,29),(15,30),(16,31),(17,36),(18,33),(19,34),(20,35),(21,26),(22,27),(23,28),(24,25)], [(1,16,30),(2,31,13),(3,14,32),(4,29,15),(5,23,25),(6,26,24),(7,21,27),(8,28,22),(9,33,19),(10,20,34),(11,35,17),(12,18,36)], [(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)]])
Matrix representation of S3×A4⋊C4 ►in GL5(𝔽13)
| 11 | 11 | 0 | 0 | 0 |
| 8 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 8 | 5 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 5 | 8 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 8 |
| 0 | 0 | 11 | 10 | 8 |
| 0 | 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 11 | 10 | 8 |
G:=sub<GL(5,GF(13))| [11,8,0,0,0,11,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,5,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,12,8,0,0,12,0,5,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,5,0,0,0,12,8,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,3,11,1,0,0,2,10,0,0,0,8,8,0],[12,0,0,0,0,0,12,0,0,0,0,0,0,12,11,0,0,1,0,10,0,0,0,0,8] >;
S3×A4⋊C4 in GAP, Magma, Sage, TeX
S_3\times A_4\rtimes C_4
% in TeX
G:=Group("S3xA4:C4"); // GroupNames label
G:=SmallGroup(288,856);
// by ID
G=gap.SmallGroup(288,856);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,36,234,1684,3036,782,1777,1350]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^3=f^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations
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