direct product, metabelian, supersoluble, monomial, A-group
Aliases: C4×C3⋊S3, C12⋊2S3, C6.13D6, C3⋊2(C4×S3), (C3×C12)⋊4C2, C32⋊5(C2×C4), C4○(C3⋊Dic3), C3⋊Dic3⋊4C2, (C3×C6).12C22, C2.1(C2×C3⋊S3), (C2×C3⋊S3).3C2, SmallGroup(72,32)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C4×C3⋊S3 |
Generators and relations for C4×C3⋊S3
G = < a,b,c,d | a4=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Character table of C4×C3⋊S3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | i | i | -i | -1 | -1 | -1 | -1 | -i | i | i | i | -i | -i | i | -i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | -1 | -i | i | i | i | -i | -i | i | -i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | -1 | i | -i | -i | -i | i | i | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | -i | i | -1 | -1 | -1 | -1 | i | -i | -i | -i | i | i | -i | i | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | -2 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | -2 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | orthogonal lifted from S3 |
| ρ17 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | -2i | 2i | 0 | 0 | 1 | 1 | -2 | 1 | i | -i | 2i | -i | i | -2i | -i | i | complex lifted from C4×S3 |
| ρ18 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 2i | -2i | 0 | 0 | 1 | 1 | -2 | 1 | -i | i | -2i | i | -i | 2i | i | -i | complex lifted from C4×S3 |
| ρ19 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 2i | -2i | 0 | 0 | 1 | -2 | 1 | 1 | 2i | -2i | i | i | -i | -i | i | -i | complex lifted from C4×S3 |
| ρ20 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | -2i | 2i | 0 | 0 | 1 | 1 | 1 | -2 | i | -i | -i | 2i | -2i | i | -i | i | complex lifted from C4×S3 |
| ρ21 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 2i | -2i | 0 | 0 | 1 | 1 | 1 | -2 | -i | i | i | -2i | 2i | -i | i | -i | complex lifted from C4×S3 |
| ρ22 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | -2i | 2i | 0 | 0 | -2 | 1 | 1 | 1 | i | -i | -i | -i | i | i | 2i | -2i | complex lifted from C4×S3 |
| ρ23 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 2i | -2i | 0 | 0 | -2 | 1 | 1 | 1 | -i | i | i | i | -i | -i | -2i | 2i | complex lifted from C4×S3 |
| ρ24 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2i | 2i | 0 | 0 | 1 | -2 | 1 | 1 | -2i | 2i | -i | -i | i | i | -i | i | complex lifted from C4×S3 |
►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 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(1 7 16)(2 8 13)(3 5 14)(4 6 15)(9 18 29)(10 19 30)(11 20 31)(12 17 32)(21 33 27)(22 34 28)(23 35 25)(24 36 26)
(5 14)(6 15)(7 16)(8 13)(9 27)(10 28)(11 25)(12 26)(17 36)(18 33)(19 34)(20 35)(21 29)(22 30)(23 31)(24 32)
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), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,7,16)(2,8,13)(3,5,14)(4,6,15)(9,18,29)(10,19,30)(11,20,31)(12,17,32)(21,33,27)(22,34,28)(23,35,25)(24,36,26), (5,14)(6,15)(7,16)(8,13)(9,27)(10,28)(11,25)(12,26)(17,36)(18,33)(19,34)(20,35)(21,29)(22,30)(23,31)(24,32)>;
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), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,7,16)(2,8,13)(3,5,14)(4,6,15)(9,18,29)(10,19,30)(11,20,31)(12,17,32)(21,33,27)(22,34,28)(23,35,25)(24,36,26), (5,14)(6,15)(7,16)(8,13)(9,27)(10,28)(11,25)(12,26)(17,36)(18,33)(19,34)(20,35)(21,29)(22,30)(23,31)(24,32) );
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)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,11),(6,36,12),(7,33,9),(8,34,10),(13,28,19),(14,25,20),(15,26,17),(16,27,18)], [(1,7,16),(2,8,13),(3,5,14),(4,6,15),(9,18,29),(10,19,30),(11,20,31),(12,17,32),(21,33,27),(22,34,28),(23,35,25),(24,36,26)], [(5,14),(6,15),(7,16),(8,13),(9,27),(10,28),(11,25),(12,26),(17,36),(18,33),(19,34),(20,35),(21,29),(22,30),(23,31),(24,32)]])
C4×C3⋊S3 is a maximal subgroup of
C12.29D6 C12.31D6 C24⋊S3 C3⋊S3⋊3C8 C32⋊M4(2) C4⋊(C32⋊C4) D12⋊S3 Dic3.D6 D6.D6 C4×S32 D6⋊D6 C12.59D6 C12.D6 C12.26D6 C33⋊8(C2×C4) C12.14S4 C30.D6
C4×C3⋊S3 is a maximal quotient of
C24⋊S3 C6.Dic6 C6.11D12 C33⋊8(C2×C4) C30.D6
Matrix representation of C4×C3⋊S3 ►in GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,8,0,0,0,0,8],[0,12,0,0,1,12,0,0,0,0,0,12,0,0,1,12],[1,0,0,0,0,1,0,0,0,0,12,1,0,0,12,0],[1,12,0,0,0,12,0,0,0,0,1,0,0,0,1,12] >;
C4×C3⋊S3 in GAP, Magma, Sage, TeX
C_4\times C_3\rtimes S_3
% in TeX
G:=Group("C4xC3:S3"); // GroupNames label
G:=SmallGroup(72,32);
// by ID
G=gap.SmallGroup(72,32);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,26,323,1204]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C4×C3⋊S3 in TeX
Character table of C4×C3⋊S3 in TeX