direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C9⋊S3, C6⋊D9, C18⋊S3, C9⋊2D6, C3⋊2D18, C32.4D6, (C3×C18)⋊3C2, (C3×C6).8S3, (C3×C9)⋊4C22, C6.2(C3⋊S3), C3.(C2×C3⋊S3), SmallGroup(108,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C2×C9⋊S3 |
Generators and relations for C2×C9⋊S3
G = < a,b,c,d | a2=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
27C2
27C2
27C22
9S3
9S3
9S3
9S3
9S3
9S3
9S3
9S3
9D6
9D6
9D6
9D6
3D9
3D9
3D9
3D9
3D9
3D9
3D18
3D18
3D18
Character table of C2×C9⋊S3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | 18A | 18B | 18C | 18D | 18E | 18F | 18G | 18H | 18I | |
| size | 1 | 1 | 27 | 27 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 1 | -2 | 1 | 1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | -2 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 1 | -2 | 1 | 1 | -1 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 1 | 1 | 1 | 1 | -2 | -2 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | orthogonal lifted from S3 |
| ρ10 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 1 | -2 | 1 | 1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 1 | -2 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | 1 | 1 | 1 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ14 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ15 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 1 | 1 | -2 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ16 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ17 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ18 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | 1 | 1 | 1 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ19 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ20 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | -2 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ21 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | 1 | 1 | 1 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ22 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ23 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ24 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 1 | 1 | -2 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ25 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | -2 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ26 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | -2 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ27 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ28 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ29 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ30 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 1 | 1 | -2 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
►On 54 points
Generators in S54
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 10)(9 11)(19 44)(20 45)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 54)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)
(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)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)
(1 35 37)(2 36 38)(3 28 39)(4 29 40)(5 30 41)(6 31 42)(7 32 43)(8 33 44)(9 34 45)(10 50 19)(11 51 20)(12 52 21)(13 53 22)(14 54 23)(15 46 24)(16 47 25)(17 48 26)(18 49 27)
(2 9)(3 8)(4 7)(5 6)(10 14)(11 13)(15 18)(16 17)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 45)
G:=sub<Sym(54)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11)(19,44)(20,45)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,54)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,35,37)(2,36,38)(3,28,39)(4,29,40)(5,30,41)(6,31,42)(7,32,43)(8,33,44)(9,34,45)(10,50,19)(11,51,20)(12,52,21)(13,53,22)(14,54,23)(15,46,24)(16,47,25)(17,48,26)(18,49,27), (2,9)(3,8)(4,7)(5,6)(10,14)(11,13)(15,18)(16,17)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(36,45)>;
G:=Group( (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11)(19,44)(20,45)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,54)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54), (1,35,37)(2,36,38)(3,28,39)(4,29,40)(5,30,41)(6,31,42)(7,32,43)(8,33,44)(9,34,45)(10,50,19)(11,51,20)(12,52,21)(13,53,22)(14,54,23)(15,46,24)(16,47,25)(17,48,26)(18,49,27), (2,9)(3,8)(4,7)(5,6)(10,14)(11,13)(15,18)(16,17)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,44)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(36,45) );
G=PermutationGroup([[(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,10),(9,11),(19,44),(20,45),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,54),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,53)], [(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),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)], [(1,35,37),(2,36,38),(3,28,39),(4,29,40),(5,30,41),(6,31,42),(7,32,43),(8,33,44),(9,34,45),(10,50,19),(11,51,20),(12,52,21),(13,53,22),(14,54,23),(15,46,24),(16,47,25),(17,48,26),(18,49,27)], [(2,9),(3,8),(4,7),(5,6),(10,14),(11,13),(15,18),(16,17),(19,54),(20,53),(21,52),(22,51),(23,50),(24,49),(25,48),(26,47),(27,46),(28,44),(29,43),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(36,45)]])
C2×C9⋊S3 is a maximal subgroup of
C18.D6 C3⋊D36 C9⋊D12 C36⋊S3 C6.D18 C2×S3×D9 C18.6S4 C32.3GL2(𝔽3)
C2×C9⋊S3 is a maximal quotient of
C12.D9 C36⋊S3 C6.D18
Matrix representation of C2×C9⋊S3 ►in GL4(𝔽19) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 5 | 7 | 0 | 0 |
| 12 | 17 | 0 | 0 |
| 0 | 0 | 12 | 17 |
| 0 | 0 | 2 | 14 |
| 0 | 1 | 0 | 0 |
| 18 | 18 | 0 | 0 |
| 0 | 0 | 18 | 18 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 18 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(19))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[5,12,0,0,7,17,0,0,0,0,12,2,0,0,17,14],[0,18,0,0,1,18,0,0,0,0,18,1,0,0,18,0],[1,18,0,0,0,18,0,0,0,0,18,1,0,0,0,1] >;
C2×C9⋊S3 in GAP, Magma, Sage, TeX
C_2\times C_9\rtimes S_3
% in TeX
G:=Group("C2xC9:S3"); // GroupNames label
G:=SmallGroup(108,27);
// by ID
G=gap.SmallGroup(108,27);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-3,662,282,483,1804]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^9=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 C2×C9⋊S3 in TeX
Character table of C2×C9⋊S3 in TeX