direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C52⋊C6, C52⋊(C3×C6), C5⋊D5⋊C32, (C5×C15)⋊2C6, C52⋊C3⋊2C6, (C3×C5⋊D5)⋊C3, (C3×C52⋊C3)⋊4C2, SmallGroup(450,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C3×C52⋊C6 |
Generators and relations for C3×C52⋊C6
G = < a,b,c,d | a3=b5=c5=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2c3, dcd-1=b-1c-1 >
Character table of C3×C52⋊C6
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | |
| size | 1 | 25 | 1 | 1 | 25 | 25 | 25 | 25 | 25 | 25 | 6 | 6 | 6 | 6 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| 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 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | -1 | ζ6 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | linear of order 6 |
| ρ4 | 1 | -1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ6 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | -1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ6 | ζ6 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ65 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ8 | 1 | -1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | -1 | ζ65 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | linear of order 6 |
| ρ9 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ10 | 1 | -1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ65 | ζ65 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ6 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | linear of order 6 |
| ρ11 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | ζ65 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ13 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ14 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | ζ6 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | linear of order 6 |
| ρ15 | 1 | -1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ65 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ16 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ17 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ18 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ19 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -3-√5/2 | 1+√5 | 1-√5 | -3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5 | -3+√5/2 | -3-√5/2 | 1+√5 | -3+√5/2 | -3-√5/2 | 1+√5 | 1-√5 | orthogonal lifted from C52⋊C6 |
| ρ20 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -3+√5/2 | 1-√5 | 1+√5 | -3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√5 | -3-√5/2 | -3+√5/2 | 1-√5 | -3-√5/2 | -3+√5/2 | 1-√5 | 1+√5 | orthogonal lifted from C52⋊C6 |
| ρ21 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√5 | -3+√5/2 | -3-√5/2 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3-√5/2 | 1-√5 | 1+√5 | -3+√5/2 | 1-√5 | 1+√5 | -3+√5/2 | -3-√5/2 | orthogonal lifted from C52⋊C6 |
| ρ22 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5 | -3-√5/2 | -3+√5/2 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3+√5/2 | 1+√5 | 1-√5 | -3-√5/2 | 1+√5 | 1-√5 | -3-√5/2 | -3+√5/2 | orthogonal lifted from C52⋊C6 |
| ρ23 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√5 | -3+√5/2 | -3-√5/2 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3ζ53+ζ3ζ52-ζ3 | -2ζ3ζ54-2ζ3ζ5 | -2ζ3ζ53-2ζ3ζ52 | ζ3ζ54+ζ3ζ5-ζ3 | -2ζ32ζ54-2ζ32ζ5 | -2ζ32ζ53-2ζ32ζ52 | ζ32ζ54+ζ32ζ5-ζ32 | ζ32ζ53+ζ32ζ52-ζ32 | complex faithful |
| ρ24 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5 | -3-√5/2 | -3+√5/2 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3ζ54+ζ3ζ5-ζ3 | -2ζ3ζ53-2ζ3ζ52 | -2ζ3ζ54-2ζ3ζ5 | ζ3ζ53+ζ3ζ52-ζ3 | -2ζ32ζ53-2ζ32ζ52 | -2ζ32ζ54-2ζ32ζ5 | ζ32ζ53+ζ32ζ52-ζ32 | ζ32ζ54+ζ32ζ5-ζ32 | complex faithful |
| ρ25 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | -3+√5/2 | 1-√5 | 1+√5 | -3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2ζ3ζ53-2ζ3ζ52 | ζ3ζ53+ζ3ζ52-ζ3 | ζ3ζ54+ζ3ζ5-ζ3 | -2ζ3ζ54-2ζ3ζ5 | ζ32ζ53+ζ32ζ52-ζ32 | ζ32ζ54+ζ32ζ5-ζ32 | -2ζ32ζ54-2ζ32ζ5 | -2ζ32ζ53-2ζ32ζ52 | complex faithful |
| ρ26 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | -3+√5/2 | 1-√5 | 1+√5 | -3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2ζ32ζ53-2ζ32ζ52 | ζ32ζ53+ζ32ζ52-ζ32 | ζ32ζ54+ζ32ζ5-ζ32 | -2ζ32ζ54-2ζ32ζ5 | ζ3ζ53+ζ3ζ52-ζ3 | ζ3ζ54+ζ3ζ5-ζ3 | -2ζ3ζ54-2ζ3ζ5 | -2ζ3ζ53-2ζ3ζ52 | complex faithful |
| ρ27 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5 | -3-√5/2 | -3+√5/2 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32ζ54+ζ32ζ5-ζ32 | -2ζ32ζ53-2ζ32ζ52 | -2ζ32ζ54-2ζ32ζ5 | ζ32ζ53+ζ32ζ52-ζ32 | -2ζ3ζ53-2ζ3ζ52 | -2ζ3ζ54-2ζ3ζ5 | ζ3ζ53+ζ3ζ52-ζ3 | ζ3ζ54+ζ3ζ5-ζ3 | complex faithful |
| ρ28 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | -3-√5/2 | 1+√5 | 1-√5 | -3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2ζ3ζ54-2ζ3ζ5 | ζ3ζ54+ζ3ζ5-ζ3 | ζ3ζ53+ζ3ζ52-ζ3 | -2ζ3ζ53-2ζ3ζ52 | ζ32ζ54+ζ32ζ5-ζ32 | ζ32ζ53+ζ32ζ52-ζ32 | -2ζ32ζ53-2ζ32ζ52 | -2ζ32ζ54-2ζ32ζ5 | complex faithful |
| ρ29 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | -3-√5/2 | 1+√5 | 1-√5 | -3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2ζ32ζ54-2ζ32ζ5 | ζ32ζ54+ζ32ζ5-ζ32 | ζ32ζ53+ζ32ζ52-ζ32 | -2ζ32ζ53-2ζ32ζ52 | ζ3ζ54+ζ3ζ5-ζ3 | ζ3ζ53+ζ3ζ52-ζ3 | -2ζ3ζ53-2ζ3ζ52 | -2ζ3ζ54-2ζ3ζ5 | complex faithful |
| ρ30 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√5 | -3+√5/2 | -3-√5/2 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32ζ53+ζ32ζ52-ζ32 | -2ζ32ζ54-2ζ32ζ5 | -2ζ32ζ53-2ζ32ζ52 | ζ32ζ54+ζ32ζ5-ζ32 | -2ζ3ζ54-2ζ3ζ5 | -2ζ3ζ53-2ζ3ζ52 | ζ3ζ54+ζ3ζ5-ζ3 | ζ3ζ53+ζ3ζ52-ζ3 | complex faithful |
►On 45 points
Generators in S45
(1 6 7)(2 4 8)(3 5 9)(10 24 38)(11 25 39)(12 26 34)(13 27 35)(14 22 36)(15 23 37)(16 32 40)(17 33 41)(18 28 42)(19 29 43)(20 30 44)(21 31 45)
(1 18 25 22 21)(2 26 16 19 23)(3 20 27 24 17)(4 34 32 29 37)(5 30 35 38 33)(6 28 39 36 31)(7 42 11 14 45)(8 12 40 43 15)(9 44 13 10 41)
(2 16 23 26 19)(3 17 24 27 20)(4 32 37 34 29)(5 33 38 35 30)(8 40 15 12 43)(9 41 10 13 44)
(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)
G:=sub<Sym(45)| (1,6,7)(2,4,8)(3,5,9)(10,24,38)(11,25,39)(12,26,34)(13,27,35)(14,22,36)(15,23,37)(16,32,40)(17,33,41)(18,28,42)(19,29,43)(20,30,44)(21,31,45), (1,18,25,22,21)(2,26,16,19,23)(3,20,27,24,17)(4,34,32,29,37)(5,30,35,38,33)(6,28,39,36,31)(7,42,11,14,45)(8,12,40,43,15)(9,44,13,10,41), (2,16,23,26,19)(3,17,24,27,20)(4,32,37,34,29)(5,33,38,35,30)(8,40,15,12,43)(9,41,10,13,44), (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)>;
G:=Group( (1,6,7)(2,4,8)(3,5,9)(10,24,38)(11,25,39)(12,26,34)(13,27,35)(14,22,36)(15,23,37)(16,32,40)(17,33,41)(18,28,42)(19,29,43)(20,30,44)(21,31,45), (1,18,25,22,21)(2,26,16,19,23)(3,20,27,24,17)(4,34,32,29,37)(5,30,35,38,33)(6,28,39,36,31)(7,42,11,14,45)(8,12,40,43,15)(9,44,13,10,41), (2,16,23,26,19)(3,17,24,27,20)(4,32,37,34,29)(5,33,38,35,30)(8,40,15,12,43)(9,41,10,13,44), (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) );
G=PermutationGroup([[(1,6,7),(2,4,8),(3,5,9),(10,24,38),(11,25,39),(12,26,34),(13,27,35),(14,22,36),(15,23,37),(16,32,40),(17,33,41),(18,28,42),(19,29,43),(20,30,44),(21,31,45)], [(1,18,25,22,21),(2,26,16,19,23),(3,20,27,24,17),(4,34,32,29,37),(5,30,35,38,33),(6,28,39,36,31),(7,42,11,14,45),(8,12,40,43,15),(9,44,13,10,41)], [(2,16,23,26,19),(3,17,24,27,20),(4,32,37,34,29),(5,33,38,35,30),(8,40,15,12,43),(9,41,10,13,44)], [(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)]])
Matrix representation of C3×C52⋊C6 ►in GL6(𝔽31)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 13 | 30 | 0 | 0 | 0 | 0 |
| 14 | 30 | 0 | 0 | 0 | 0 |
| 25 | 0 | 30 | 12 | 0 | 0 |
| 21 | 21 | 19 | 19 | 0 | 0 |
| 5 | 0 | 0 | 0 | 12 | 30 |
| 3 | 26 | 0 | 0 | 1 | 0 |
| 30 | 1 | 0 | 0 | 0 | 0 |
| 17 | 13 | 0 | 0 | 0 | 0 |
| 16 | 25 | 12 | 30 | 0 | 0 |
| 6 | 0 | 1 | 0 | 0 | 0 |
| 26 | 0 | 0 | 0 | 1 | 0 |
| 26 | 0 | 0 | 0 | 0 | 1 |
| 19 | 1 | 24 | 26 | 0 | 0 |
| 30 | 13 | 28 | 2 | 0 | 0 |
| 0 | 0 | 30 | 0 | 5 | 0 |
| 10 | 25 | 30 | 0 | 29 | 26 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 28 | 5 | 6 | 0 | 0 | 0 |
G:=sub<GL(6,GF(31))| [5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[13,14,25,21,5,3,30,30,0,21,0,26,0,0,30,19,0,0,0,0,12,19,0,0,0,0,0,0,12,1,0,0,0,0,30,0],[30,17,16,6,26,26,1,13,25,0,0,0,0,0,12,1,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[19,30,0,10,0,28,1,13,0,25,0,5,24,28,30,30,6,6,26,2,0,0,0,0,0,0,5,29,0,0,0,0,0,26,0,0] >;
C3×C52⋊C6 in GAP, Magma, Sage, TeX
C_3\times C_5^2\rtimes C_6
% in TeX
G:=Group("C3xC5^2:C6"); // GroupNames label
G:=SmallGroup(450,22);
// by ID
G=gap.SmallGroup(450,22);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,5,1443,2348,9004,1359]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^5=c^5=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^2*c^3,d*c*d^-1=b^-1*c^-1>;
// generators/relations
Export
Subgroup lattice of C3×C52⋊C6 in TeX
Character table of C3×C52⋊C6 in TeX