metabelian, soluble, monomial, A-group
Aliases: C52⋊C18, C5⋊D5⋊C9, (C5×C15).C6, C52⋊C9⋊2C2, C3.(C52⋊C6), (C3×C5⋊D5).C3, SmallGroup(450,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊C18 |
Generators and relations for C52⋊C18
G = < a,b,c | a5=b5=c18=1, ab=ba, cac-1=a2b3, cbc-1=a-1b-1 >
Character table of C52⋊C18
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 25 | 1 | 1 | 6 | 6 | 6 | 6 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 25 | 25 | 25 | 25 | 25 | 25 | |
| 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 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ5 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | ζ95 | ζ97 | ζ94 | ζ92 | ζ98 | ζ9 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | -ζ95 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | -ζ94 | linear of order 18 |
| ρ8 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | ζ92 | ζ9 | ζ97 | ζ98 | ζ95 | ζ94 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | -ζ92 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | -ζ97 | linear of order 18 |
| ρ9 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ92 | ζ9 | ζ97 | ζ98 | ζ95 | ζ94 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ92 | ζ98 | ζ95 | ζ94 | ζ9 | ζ97 | linear of order 9 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | ζ97 | ζ98 | ζ92 | ζ9 | ζ94 | ζ95 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | -ζ97 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | -ζ92 | linear of order 18 |
| ρ11 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | ζ94 | ζ92 | ζ95 | ζ97 | ζ9 | ζ98 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | -ζ94 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | -ζ95 | linear of order 18 |
| ρ12 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ95 | ζ97 | ζ94 | ζ92 | ζ98 | ζ9 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ95 | ζ92 | ζ98 | ζ9 | ζ97 | ζ94 | linear of order 9 |
| ρ13 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ94 | ζ92 | ζ95 | ζ97 | ζ9 | ζ98 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ94 | ζ97 | ζ9 | ζ98 | ζ92 | ζ95 | linear of order 9 |
| ρ14 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ98 | ζ94 | ζ9 | ζ95 | ζ92 | ζ97 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ98 | ζ95 | ζ92 | ζ97 | ζ94 | ζ9 | linear of order 9 |
| ρ15 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | ζ9 | ζ95 | ζ98 | ζ94 | ζ97 | ζ92 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | -ζ9 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | -ζ98 | linear of order 18 |
| ρ16 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | ζ98 | ζ94 | ζ9 | ζ95 | ζ92 | ζ97 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | -ζ98 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | -ζ9 | linear of order 18 |
| ρ17 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ97 | ζ98 | ζ92 | ζ9 | ζ94 | ζ95 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ97 | ζ9 | ζ94 | ζ95 | ζ98 | ζ92 | linear of order 9 |
| ρ18 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ9 | ζ95 | ζ98 | ζ94 | ζ97 | ζ92 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ9 | ζ94 | ζ97 | ζ92 | ζ95 | ζ98 | linear of order 9 |
| ρ19 | 6 | 0 | 6 | 6 | -3+√5/2 | 1-√5 | 1+√5 | -3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5 | 1+√5 | -3-√5/2 | -3-√5/2 | -3+√5/2 | 1-√5 | 1+√5 | -3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C6 |
| ρ20 | 6 | 0 | 6 | 6 | 1-√5 | -3-√5/2 | -3+√5/2 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3-√5/2 | -3+√5/2 | 1+√5 | 1+√5 | 1-√5 | -3-√5/2 | -3+√5/2 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C6 |
| ρ21 | 6 | 0 | 6 | 6 | 1+√5 | -3+√5/2 | -3-√5/2 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3+√5/2 | -3-√5/2 | 1-√5 | 1-√5 | 1+√5 | -3+√5/2 | -3-√5/2 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C6 |
| ρ22 | 6 | 0 | 6 | 6 | -3-√5/2 | 1+√5 | 1-√5 | -3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√5 | 1-√5 | -3+√5/2 | -3+√5/2 | -3-√5/2 | 1+√5 | 1-√5 | -3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C6 |
| ρ23 | 6 | 0 | -3-3√-3 | -3+3√-3 | 1+√5 | -3+√5/2 | -3-√5/2 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3ζ54+ζ3ζ5-ζ3 | ζ3ζ53+ζ3ζ52-ζ3 | -2ζ3ζ54-2ζ3ζ5 | -2ζ32ζ54-2ζ32ζ5 | -2ζ32ζ53-2ζ32ζ52 | ζ32ζ54+ζ32ζ5-ζ32 | ζ32ζ53+ζ32ζ52-ζ32 | -2ζ3ζ53-2ζ3ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 6 | 0 | -3-3√-3 | -3+3√-3 | 1-√5 | -3-√5/2 | -3+√5/2 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3ζ53+ζ3ζ52-ζ3 | ζ3ζ54+ζ3ζ5-ζ3 | -2ζ3ζ53-2ζ3ζ52 | -2ζ32ζ53-2ζ32ζ52 | -2ζ32ζ54-2ζ32ζ5 | ζ32ζ53+ζ32ζ52-ζ32 | ζ32ζ54+ζ32ζ5-ζ32 | -2ζ3ζ54-2ζ3ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 6 | 0 | -3+3√-3 | -3-3√-3 | 1+√5 | -3+√5/2 | -3-√5/2 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32ζ54+ζ32ζ5-ζ32 | ζ32ζ53+ζ32ζ52-ζ32 | -2ζ32ζ54-2ζ32ζ5 | -2ζ3ζ54-2ζ3ζ5 | -2ζ3ζ53-2ζ3ζ52 | ζ3ζ54+ζ3ζ5-ζ3 | ζ3ζ53+ζ3ζ52-ζ3 | -2ζ32ζ53-2ζ32ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ26 | 6 | 0 | -3-3√-3 | -3+3√-3 | -3-√5/2 | 1+√5 | 1-√5 | -3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2ζ3ζ53-2ζ3ζ52 | -2ζ3ζ54-2ζ3ζ5 | ζ3ζ54+ζ3ζ5-ζ3 | ζ32ζ54+ζ32ζ5-ζ32 | ζ32ζ53+ζ32ζ52-ζ32 | -2ζ32ζ53-2ζ32ζ52 | -2ζ32ζ54-2ζ32ζ5 | ζ3ζ53+ζ3ζ52-ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 6 | 0 | -3+3√-3 | -3-3√-3 | -3-√5/2 | 1+√5 | 1-√5 | -3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2ζ32ζ53-2ζ32ζ52 | -2ζ32ζ54-2ζ32ζ5 | ζ32ζ54+ζ32ζ5-ζ32 | ζ3ζ54+ζ3ζ5-ζ3 | ζ3ζ53+ζ3ζ52-ζ3 | -2ζ3ζ53-2ζ3ζ52 | -2ζ3ζ54-2ζ3ζ5 | ζ32ζ53+ζ32ζ52-ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ28 | 6 | 0 | -3+3√-3 | -3-3√-3 | 1-√5 | -3-√5/2 | -3+√5/2 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32ζ53+ζ32ζ52-ζ32 | ζ32ζ54+ζ32ζ5-ζ32 | -2ζ32ζ53-2ζ32ζ52 | -2ζ3ζ53-2ζ3ζ52 | -2ζ3ζ54-2ζ3ζ5 | ζ3ζ53+ζ3ζ52-ζ3 | ζ3ζ54+ζ3ζ5-ζ3 | -2ζ32ζ54-2ζ32ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ29 | 6 | 0 | -3-3√-3 | -3+3√-3 | -3+√5/2 | 1-√5 | 1+√5 | -3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2ζ3ζ54-2ζ3ζ5 | -2ζ3ζ53-2ζ3ζ52 | ζ3ζ53+ζ3ζ52-ζ3 | ζ32ζ53+ζ32ζ52-ζ32 | ζ32ζ54+ζ32ζ5-ζ32 | -2ζ32ζ54-2ζ32ζ5 | -2ζ32ζ53-2ζ32ζ52 | ζ3ζ54+ζ3ζ5-ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 6 | 0 | -3+3√-3 | -3-3√-3 | -3+√5/2 | 1-√5 | 1+√5 | -3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2ζ32ζ54-2ζ32ζ5 | -2ζ32ζ53-2ζ32ζ52 | ζ32ζ53+ζ32ζ52-ζ32 | ζ3ζ53+ζ3ζ52-ζ3 | ζ3ζ54+ζ3ζ5-ζ3 | -2ζ3ζ54-2ζ3ζ5 | -2ζ3ζ53-2ζ3ζ52 | ζ32ζ54+ζ32ζ5-ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 45 points
Generators in S45
(2 27 29 38 18)(3 10 30 39 19)(5 21 41 32 12)(6 22 42 33 13)(8 15 35 44 24)(9 16 36 45 25)
(1 28 17 26 37)(2 38 27 18 29)(3 10 30 39 19)(4 40 11 20 31)(5 32 21 12 41)(6 22 42 33 13)(7 34 23 14 43)(8 44 15 24 35)(9 16 36 45 25)
(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)| (2,27,29,38,18)(3,10,30,39,19)(5,21,41,32,12)(6,22,42,33,13)(8,15,35,44,24)(9,16,36,45,25), (1,28,17,26,37)(2,38,27,18,29)(3,10,30,39,19)(4,40,11,20,31)(5,32,21,12,41)(6,22,42,33,13)(7,34,23,14,43)(8,44,15,24,35)(9,16,36,45,25), (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( (2,27,29,38,18)(3,10,30,39,19)(5,21,41,32,12)(6,22,42,33,13)(8,15,35,44,24)(9,16,36,45,25), (1,28,17,26,37)(2,38,27,18,29)(3,10,30,39,19)(4,40,11,20,31)(5,32,21,12,41)(6,22,42,33,13)(7,34,23,14,43)(8,44,15,24,35)(9,16,36,45,25), (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([[(2,27,29,38,18),(3,10,30,39,19),(5,21,41,32,12),(6,22,42,33,13),(8,15,35,44,24),(9,16,36,45,25)], [(1,28,17,26,37),(2,38,27,18,29),(3,10,30,39,19),(4,40,11,20,31),(5,32,21,12,41),(6,22,42,33,13),(7,34,23,14,43),(8,44,15,24,35),(9,16,36,45,25)], [(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 C52⋊C18 ►in GL6(𝔽181)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 167 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 62 | 82 | 65 | 5 | 14 | 14 |
| 37 | 119 | 176 | 0 | 167 | 180 |
| 180 | 1 | 0 | 0 | 0 | 0 |
| 12 | 168 | 0 | 0 | 0 | 0 |
| 0 | 0 | 180 | 1 | 0 | 0 |
| 0 | 0 | 12 | 168 | 0 | 0 |
| 124 | 82 | 130 | 121 | 14 | 14 |
| 99 | 119 | 60 | 116 | 167 | 180 |
| 147 | 101 | 122 | 138 | 4 | 133 |
| 80 | 135 | 43 | 16 | 81 | 129 |
| 87 | 132 | 0 | 0 | 0 | 0 |
| 87 | 94 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 80 | 0 |
| 0 | 0 | 87 | 132 | 80 | 0 |
G:=sub<GL(6,GF(181))| [1,0,0,0,62,37,0,1,0,0,82,119,0,0,13,13,65,176,0,0,167,0,5,0,0,0,0,0,14,167,0,0,0,0,14,180],[180,12,0,0,124,99,1,168,0,0,82,119,0,0,180,12,130,60,0,0,1,168,121,116,0,0,0,0,14,167,0,0,0,0,14,180],[147,80,87,87,0,0,101,135,132,94,0,0,122,43,0,0,0,87,138,16,0,0,0,132,4,81,0,0,80,80,133,129,0,0,0,0] >;
C52⋊C18 in GAP, Magma, Sage, TeX
C_5^2\rtimes C_{18} % in TeX
G:=Group("C5^2:C18"); // GroupNames label
G:=SmallGroup(450,12);
// by ID
G=gap.SmallGroup(450,12);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,5,36,1443,2348,9004,1359]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^18=1,a*b=b*a,c*a*c^-1=a^2*b^3,c*b*c^-1=a^-1*b^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊C18 in TeX
Character table of C52⋊C18 in TeX