non-abelian, soluble, monomial
Aliases: (C4×C12)⋊1S3, C3⋊(C42⋊S3), C42⋊(C3⋊S3), C42⋊C3⋊2S3, (C2×C6).3S4, C22.(C3⋊S4), (C3×C42⋊C3)⋊4C2, SmallGroup(288,401)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C42⋊C3 — (C4×C12)⋊S3 |
Generators and relations for (C4×C12)⋊S3
G = < a,b,c,d | a4=b12=c3=d2=1, ab=ba, cac-1=dad=b9, cbc-1=a-1b7, dbd=ab8, dcd=c-1 >
Subgroups: 552 in 60 conjugacy classes, 11 normal (8 characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, C8, C2×C4, D4, Q8, C32, Dic3, C12, A4, D6, C2×C6, C42, M4(2), C4○D4, C3⋊S3, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, S4, C4≀C2, C3×A4, C42⋊C3, C4.Dic3, C4×C12, C4○D12, C3⋊S4, C42⋊4S3, C42⋊S3, C3×C42⋊C3, (C4×C12)⋊S3
Quotients: C1, C2, S3, C3⋊S3, S4, C3⋊S4, C42⋊S3, (C4×C12)⋊S3
Character table of (C4×C12)⋊S3
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 12A | 12B | 12C | 12D | |
| size | 1 | 3 | 36 | 2 | 32 | 32 | 32 | 3 | 3 | 6 | 36 | 6 | 36 | 36 | 6 | 6 | 6 | 6 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | orthogonal lifted from S3 |
| ρ5 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 3 | 3 | -1 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 3 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ8 | 3 | 3 | 1 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ9 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | 1 | -1 | -i | i | -1+2i | -1-2i | 1 | 1 | complex lifted from C42⋊S3 |
| ρ10 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | 1 | -1 | i | -i | -1-2i | -1+2i | 1 | 1 | complex lifted from C42⋊S3 |
| ρ11 | 3 | -1 | 1 | 3 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | -1 | -1 | i | -i | -1+2i | -1-2i | 1 | 1 | complex lifted from C42⋊S3 |
| ρ12 | 3 | -1 | 1 | 3 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | -1 | -1 | -i | i | -1-2i | -1+2i | 1 | 1 | complex lifted from C42⋊S3 |
| ρ13 | 6 | -2 | 0 | 6 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | orthogonal lifted from C42⋊S3 |
| ρ14 | 6 | 6 | 0 | -3 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | -3 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from C3⋊S4 |
| ρ15 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 1 | 0 | 0 | -1 | -1 | 1+2√3 | 1-2√3 | orthogonal faithful |
| ρ16 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 1 | 0 | 0 | -1 | -1 | 1-2√3 | 1+2√3 | orthogonal faithful |
| ρ17 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | -2+4i | -2-4i | 2 | 0 | 1 | 0 | 0 | 1+2i | 1-2i | -1 | -1 | complex faithful |
| ρ18 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | -2-4i | -2+4i | 2 | 0 | 1 | 0 | 0 | 1-2i | 1+2i | -1 | -1 | complex faithful |
►On 36 points
Generators in S36
(1 7 10 6)(2 8 11 4)(3 9 12 5)(25 28 31 34)(26 29 32 35)(27 30 33 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 21 30)(2 13 34)(3 17 26)(4 22 25)(5 14 29)(6 18 33)(7 24 27)(8 16 31)(9 20 35)(10 15 36)(11 19 28)(12 23 32)
(1 20)(2 16)(3 24)(4 19)(5 15)(6 23)(7 17)(8 13)(9 21)(10 14)(11 22)(12 18)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)
G:=sub<Sym(36)| (1,7,10,6)(2,8,11,4)(3,9,12,5)(25,28,31,34)(26,29,32,35)(27,30,33,36), (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), (1,21,30)(2,13,34)(3,17,26)(4,22,25)(5,14,29)(6,18,33)(7,24,27)(8,16,31)(9,20,35)(10,15,36)(11,19,28)(12,23,32), (1,20)(2,16)(3,24)(4,19)(5,15)(6,23)(7,17)(8,13)(9,21)(10,14)(11,22)(12,18)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)>;
G:=Group( (1,7,10,6)(2,8,11,4)(3,9,12,5)(25,28,31,34)(26,29,32,35)(27,30,33,36), (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), (1,21,30)(2,13,34)(3,17,26)(4,22,25)(5,14,29)(6,18,33)(7,24,27)(8,16,31)(9,20,35)(10,15,36)(11,19,28)(12,23,32), (1,20)(2,16)(3,24)(4,19)(5,15)(6,23)(7,17)(8,13)(9,21)(10,14)(11,22)(12,18)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33) );
G=PermutationGroup([[(1,7,10,6),(2,8,11,4),(3,9,12,5),(25,28,31,34),(26,29,32,35),(27,30,33,36)], [(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)], [(1,21,30),(2,13,34),(3,17,26),(4,22,25),(5,14,29),(6,18,33),(7,24,27),(8,16,31),(9,20,35),(10,15,36),(11,19,28),(12,23,32)], [(1,20),(2,16),(3,24),(4,19),(5,15),(6,23),(7,17),(8,13),(9,21),(10,14),(11,22),(12,18),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33)]])
Matrix representation of (C4×C12)⋊S3 ►in GL5(𝔽73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 46 |
| 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 46 |
| 72 | 1 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 27 |
| 0 | 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 47 |
| 0 | 0 | 72 | 72 | 47 |
| 0 | 0 | 0 | 45 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 72 | 47 |
| 0 | 0 | 45 | 0 | 1 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,46,0,0,0,46,0,46],[72,72,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,27,27,72],[1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,72,72,45,0,0,47,47,1],[0,1,0,0,0,1,0,0,0,0,0,0,72,72,45,0,0,0,72,0,0,0,0,47,1] >;
(C4×C12)⋊S3 in GAP, Magma, Sage, TeX
(C_4\times C_{12})\rtimes S_3 % in TeX
G:=Group("(C4xC12):S3"); // GroupNames label
G:=SmallGroup(288,401);
// by ID
G=gap.SmallGroup(288,401);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,57,254,1011,514,360,634,3476,102,9077,3036,5305]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^12=c^3=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=b^9,c*b*c^-1=a^-1*b^7,d*b*d=a*b^8,d*c*d=c^-1>;
// generators/relations
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