metabelian, soluble, monomial, A-group
Aliases: C26⋊3C3, C24⋊8A4, C22⋊(C22⋊A4), SmallGroup(192,1541)
Series: Derived ►Chief ►Lower central ►Upper central
| C26 — C26⋊C3 |
Generators and relations for C26⋊C3
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g3=1, gag-1=ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, gbg-1=a, gcg-1=cd=dc, ce=ec, cf=fc, de=ed, df=fd, gdg-1=c, geg-1=ef=fe, gfg-1=e >
Subgroups: 3310 in 1015 conjugacy classes, 45 normal (3 characteristic)
C1, C2, C3, C22, C22, C23, A4, C24, C24, C25, C22⋊A4, C26, C26⋊C3
Quotients: C1, C3, A4, C22⋊A4, C26⋊C3
Character table of C26⋊C3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 2P | 2Q | 2R | 2S | 2T | 2U | 3A | 3B | |
| size | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 64 | 64 | |
| 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 | ζ3 | ζ32 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 3 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | 0 | 0 | orthogonal lifted from A4 |
| ρ5 | 3 | 3 | -1 | -1 | -1 | 3 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | orthogonal lifted from A4 |
| ρ6 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ7 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ8 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ9 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | 3 | 3 | 3 | 0 | 0 | orthogonal lifted from A4 |
| ρ10 | 3 | 3 | -1 | -1 | 3 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | 3 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ11 | 3 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ12 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | orthogonal lifted from A4 |
| ρ13 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ14 | 3 | 3 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ15 | 3 | -1 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ16 | 3 | -1 | -1 | -1 | -1 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ17 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | 3 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ18 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ19 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ20 | 3 | -1 | -1 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | orthogonal lifted from A4 |
| ρ21 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ22 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ23 | 3 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ24 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
►On 24 points - transitive group 24T390
Generators in S24
(1 14)(2 17)(3 7)(4 19)(5 10)(6 22)(8 16)(9 15)(11 21)(12 23)(13 18)(20 24)
(1 8)(2 15)(3 18)(4 23)(5 20)(6 11)(7 13)(9 17)(10 24)(12 19)(14 16)(21 22)
(1 14)(2 24)(3 11)(4 19)(5 9)(6 18)(7 21)(8 16)(10 15)(12 23)(13 22)(17 20)
(1 12)(2 15)(3 22)(4 16)(5 20)(6 7)(8 19)(9 17)(10 24)(11 13)(14 23)(18 21)
(1 23)(3 22)(4 8)(6 7)(11 13)(12 14)(16 19)(18 21)
(1 23)(2 24)(4 8)(5 9)(10 15)(12 14)(16 19)(17 20)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
G:=sub<Sym(24)| (1,14)(2,17)(3,7)(4,19)(5,10)(6,22)(8,16)(9,15)(11,21)(12,23)(13,18)(20,24), (1,8)(2,15)(3,18)(4,23)(5,20)(6,11)(7,13)(9,17)(10,24)(12,19)(14,16)(21,22), (1,14)(2,24)(3,11)(4,19)(5,9)(6,18)(7,21)(8,16)(10,15)(12,23)(13,22)(17,20), (1,12)(2,15)(3,22)(4,16)(5,20)(6,7)(8,19)(9,17)(10,24)(11,13)(14,23)(18,21), (1,23)(3,22)(4,8)(6,7)(11,13)(12,14)(16,19)(18,21), (1,23)(2,24)(4,8)(5,9)(10,15)(12,14)(16,19)(17,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)>;
G:=Group( (1,14)(2,17)(3,7)(4,19)(5,10)(6,22)(8,16)(9,15)(11,21)(12,23)(13,18)(20,24), (1,8)(2,15)(3,18)(4,23)(5,20)(6,11)(7,13)(9,17)(10,24)(12,19)(14,16)(21,22), (1,14)(2,24)(3,11)(4,19)(5,9)(6,18)(7,21)(8,16)(10,15)(12,23)(13,22)(17,20), (1,12)(2,15)(3,22)(4,16)(5,20)(6,7)(8,19)(9,17)(10,24)(11,13)(14,23)(18,21), (1,23)(3,22)(4,8)(6,7)(11,13)(12,14)(16,19)(18,21), (1,23)(2,24)(4,8)(5,9)(10,15)(12,14)(16,19)(17,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24) );
G=PermutationGroup([[(1,14),(2,17),(3,7),(4,19),(5,10),(6,22),(8,16),(9,15),(11,21),(12,23),(13,18),(20,24)], [(1,8),(2,15),(3,18),(4,23),(5,20),(6,11),(7,13),(9,17),(10,24),(12,19),(14,16),(21,22)], [(1,14),(2,24),(3,11),(4,19),(5,9),(6,18),(7,21),(8,16),(10,15),(12,23),(13,22),(17,20)], [(1,12),(2,15),(3,22),(4,16),(5,20),(6,7),(8,19),(9,17),(10,24),(11,13),(14,23),(18,21)], [(1,23),(3,22),(4,8),(6,7),(11,13),(12,14),(16,19),(18,21)], [(1,23),(2,24),(4,8),(5,9),(10,15),(12,14),(16,19),(17,20)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)]])
G:=TransitiveGroup(24,390);
Matrix representation of C26⋊C3 ►in GL9(𝔽7)
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 |
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 |
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(9,GF(7))| [6,0,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6],[1,0,3,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1],[1,0,3,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6],[6,0,4,0,0,0,0,0,0,0,6,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6],[6,0,4,0,0,0,0,0,0,0,6,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6],[0,3,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;
C26⋊C3 in GAP, Magma, Sage, TeX
C_2^6\rtimes C_3
% in TeX
G:=Group("C2^6:C3"); // GroupNames label
G:=SmallGroup(192,1541);
// by ID
G=gap.SmallGroup(192,1541);
# by ID
G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,191,675,1264,4037,7062]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^3=1,g*a*g^-1=a*b=b*a,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,g*b*g^-1=a,g*c*g^-1=c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,g*d*g^-1=c,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
// generators/relations
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