direct product, metabelian, supersoluble, monomial, rational
Aliases: D4xC3:S3, C12:3D6, C62:4C22, C3:4(S3xD4), (C2xC6):6D6, (C3xD4):2S3, C12:S3:6C2, C32:11(C2xD4), (C3xC12):4C22, (D4xC32):5C2, C32:7D4:3C2, C6.34(C22xS3), (C3xC6).33C23, C3:Dic3:6C22, C4:1(C2xC3:S3), (C4xC3:S3):3C2, C22:2(C2xC3:S3), (C2xC3:S3):6C22, (C22xC3:S3):4C2, C2.6(C22xC3:S3), SmallGroup(144,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4xC3:S3
G = < a,b,c,d,e | a4=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 610 in 162 conjugacy classes, 49 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, D4, D4, C23, C32, Dic3, C12, D6, C2xC6, C2xD4, C3:S3, C3:S3, C3xC6, C3xC6, C4xS3, D12, C3:D4, C3xD4, C22xS3, C3:Dic3, C3xC12, C2xC3:S3, C2xC3:S3, C2xC3:S3, C62, S3xD4, C4xC3:S3, C12:S3, C32:7D4, D4xC32, C22xC3:S3, D4xC3:S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:S3, C22xS3, C2xC3:S3, S3xD4, C22xC3:S3, D4xC3:S3
Character table of D4xC3:S3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 2 | 2 | 9 | 9 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | 0 | -1 | -1 | 2 | -1 | 1 | -1 | -1 | -1 | 2 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | -1 | -1 | -1 | 2 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | -1 | 1 | -2 | 1 | 1 | 2 | -1 | -1 | 1 | 1 | 1 | -2 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | 1 | -1 | 2 | -1 | -1 | -2 | 1 | 1 | 1 | 1 | 1 | -2 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | 2 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 2 | -1 | orthogonal lifted from D6 |
| ρ17 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | 0 | -1 | -1 | -1 | 2 | -2 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | 0 | -1 | -1 | -1 | 2 | 2 | -2 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -2 | 1 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | 0 | -1 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | -2 | -1 | -1 | 2 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | 0 | -1 | -1 | 2 | -1 | 1 | 1 | 1 | 1 | -2 | 1 | 1 | -2 | -1 | 2 | -1 | -1 | orthogonal lifted from D6 |
| ρ22 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 0 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 2 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ25 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | 0 | -1 | 2 | -1 | -1 | 1 | -1 | -1 | 2 | -1 | 1 | -2 | 1 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ26 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | 0 | -1 | 2 | -1 | -1 | -1 | 1 | 1 | -2 | 1 | -1 | 2 | -1 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | 2 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | 2 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | -4 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
►On 36 points
(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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)
(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)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 25)(10 26)(11 27)(12 28)(17 34)(18 35)(19 36)(20 33)(21 31)(22 32)(23 29)(24 30)
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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35), (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), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,25)(10,26)(11,27)(12,28)(17,34)(18,35)(19,36)(20,33)(21,31)(22,32)(23,29)(24,30)>;
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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35), (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), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,25)(10,26)(11,27)(12,28)(17,34)(18,35)(19,36)(20,33)(21,31)(22,32)(23,29)(24,30) );
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)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,14),(15,16),(17,18),(19,20),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35)], [(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)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15),(9,25),(10,26),(11,27),(12,28),(17,34),(18,35),(19,36),(20,33),(21,31),(22,32),(23,29),(24,30)]])
D4xC3:S3 is a maximal subgroup of
C3:S3.5D8 D12:D6 Dic6:D6 D12:5D6 C24:8D6 C24:7D6 S32xD4 Dic6:12D6 D12:13D6 C32:82+ 1+4 C62.154C23 C12:S32 C62:23D6
D4xC3:S3 is a maximal quotient of
C62:6Q8 C62.225C23 C62:12D4 C62.227C23 C62.228C23 C62.229C23 C12:2Dic6 C62.237C23 C62.238C23 C12:3D12 C62.240C23 C24:8D6 C24.26D6 C24:7D6 C24.32D6 C24.40D6 C24.35D6 C24.28D6 C62:13D4 C62.256C23 C62:14D4 C62.258C23 C12:S32 C62:23D6
Matrix representation of D4xC3:S3 ►in GL6(Z)
| -1 | 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 | -2 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| -1 | 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 | -2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-2,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-2,1],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;
D4xC3:S3 in GAP, Magma, Sage, TeX
D_4\times C_3\rtimes S_3
% in TeX
G:=Group("D4xC3:S3"); // GroupNames label
G:=SmallGroup(144,172);
// by ID
G=gap.SmallGroup(144,172);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,116,964,3461]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^3=d^3=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations
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