direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C52⋊C4, C15⋊4F5, C52⋊6C12, (C5×C15)⋊8C4, C5⋊2(C3×F5), C5⋊D5.3C6, (C3×C5⋊D5).4C2, SmallGroup(300,31)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C3×C52⋊C4 |
Generators and relations for C3×C52⋊C4
G = < a,b,c,d | a3=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2, dcd-1=c3 >
Character table of C3×C52⋊C4
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 6A | 6B | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | 15I | 15J | 15K | 15L | |
| size | 1 | 25 | 1 | 1 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | 25 | 25 | 25 | 25 | 25 | 25 | 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 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | 1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 12 |
| ρ11 | 1 | -1 | ζ3 | ζ32 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 12 |
| ρ12 | 1 | -1 | ζ32 | ζ3 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 12 |
| ρ13 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 4 | -1 | 4 | orthogonal lifted from F5 |
| ρ14 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 4 | -1 | 4 | -1 | orthogonal lifted from F5 |
| ρ15 | 4 | 0 | 4 | 4 | 0 | 0 | 3+√5/2 | -1 | -1 | 3-√5/2 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1-√5 | -1+√5 | 3+√5/2 | 3-√5/2 | -1-√5 | -1+√5 | -1 | -1 | -1 | -1 | orthogonal lifted from C52⋊C4 |
| ρ16 | 4 | 0 | 4 | 4 | 0 | 0 | -1-√5 | -1 | -1 | -1+√5 | 3-√5/2 | 3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 3-√5/2 | 3+√5/2 | -1-√5 | -1+√5 | 3-√5/2 | 3+√5/2 | -1 | -1 | -1 | -1 | orthogonal lifted from C52⋊C4 |
| ρ17 | 4 | 0 | 4 | 4 | 0 | 0 | -1+√5 | -1 | -1 | -1-√5 | 3+√5/2 | 3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 3+√5/2 | 3-√5/2 | -1+√5 | -1-√5 | 3+√5/2 | 3-√5/2 | -1 | -1 | -1 | -1 | orthogonal lifted from C52⋊C4 |
| ρ18 | 4 | 0 | 4 | 4 | 0 | 0 | 3-√5/2 | -1 | -1 | 3+√5/2 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1+√5 | -1-√5 | 3-√5/2 | 3+√5/2 | -1+√5 | -1-√5 | -1 | -1 | -1 | -1 | orthogonal lifted from C52⋊C4 |
| ρ19 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ65 | -2+2√-3 | ζ65 | -2-2√-3 | ζ6 | complex lifted from C3×F5 |
| ρ20 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ65 | ζ65 | -2+2√-3 | ζ6 | -2-2√-3 | complex lifted from C3×F5 |
| ρ21 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ6 | ζ6 | -2-2√-3 | ζ65 | -2+2√-3 | complex lifted from C3×F5 |
| ρ22 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ6 | -2-2√-3 | ζ6 | -2+2√-3 | ζ65 | complex lifted from C3×F5 |
| ρ23 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | 3-√5/2 | -1 | -1 | 3+√5/2 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3ζ53+ζ3ζ52+2ζ3 | ζ3ζ54+ζ3ζ5+2ζ3 | 2ζ3ζ54+2ζ3ζ5 | 2ζ3ζ53+2ζ3ζ52 | ζ32ζ53+ζ32ζ52+2ζ32 | ζ32ζ54+ζ32ζ5+2ζ32 | 2ζ32ζ54+2ζ32ζ5 | 2ζ32ζ53+2ζ32ζ52 | ζ6 | ζ6 | ζ65 | ζ65 | complex faithful |
| ρ24 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1+√5 | -1 | -1 | -1-√5 | 3+√5/2 | 3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ32ζ54+2ζ32ζ5 | 2ζ32ζ53+2ζ32ζ52 | ζ32ζ54+ζ32ζ5+2ζ32 | ζ32ζ53+ζ32ζ52+2ζ32 | 2ζ3ζ54+2ζ3ζ5 | 2ζ3ζ53+2ζ3ζ52 | ζ3ζ54+ζ3ζ5+2ζ3 | ζ3ζ53+ζ3ζ52+2ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | complex faithful |
| ρ25 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1+√5 | -1 | -1 | -1-√5 | 3+√5/2 | 3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ3ζ54+2ζ3ζ5 | 2ζ3ζ53+2ζ3ζ52 | ζ3ζ54+ζ3ζ5+2ζ3 | ζ3ζ53+ζ3ζ52+2ζ3 | 2ζ32ζ54+2ζ32ζ5 | 2ζ32ζ53+2ζ32ζ52 | ζ32ζ54+ζ32ζ5+2ζ32 | ζ32ζ53+ζ32ζ52+2ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | complex faithful |
| ρ26 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | 3-√5/2 | -1 | -1 | 3+√5/2 | -1+√5 | -1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32ζ53+ζ32ζ52+2ζ32 | ζ32ζ54+ζ32ζ5+2ζ32 | 2ζ32ζ54+2ζ32ζ5 | 2ζ32ζ53+2ζ32ζ52 | ζ3ζ53+ζ3ζ52+2ζ3 | ζ3ζ54+ζ3ζ5+2ζ3 | 2ζ3ζ54+2ζ3ζ5 | 2ζ3ζ53+2ζ3ζ52 | ζ65 | ζ65 | ζ6 | ζ6 | complex faithful |
| ρ27 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1-√5 | -1 | -1 | -1+√5 | 3-√5/2 | 3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ3ζ53+2ζ3ζ52 | 2ζ3ζ54+2ζ3ζ5 | ζ3ζ53+ζ3ζ52+2ζ3 | ζ3ζ54+ζ3ζ5+2ζ3 | 2ζ32ζ53+2ζ32ζ52 | 2ζ32ζ54+2ζ32ζ5 | ζ32ζ53+ζ32ζ52+2ζ32 | ζ32ζ54+ζ32ζ5+2ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | complex faithful |
| ρ28 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | 3+√5/2 | -1 | -1 | 3-√5/2 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32ζ54+ζ32ζ5+2ζ32 | ζ32ζ53+ζ32ζ52+2ζ32 | 2ζ32ζ53+2ζ32ζ52 | 2ζ32ζ54+2ζ32ζ5 | ζ3ζ54+ζ3ζ5+2ζ3 | ζ3ζ53+ζ3ζ52+2ζ3 | 2ζ3ζ53+2ζ3ζ52 | 2ζ3ζ54+2ζ3ζ5 | ζ65 | ζ65 | ζ6 | ζ6 | complex faithful |
| ρ29 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1-√5 | -1 | -1 | -1+√5 | 3-√5/2 | 3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ32ζ53+2ζ32ζ52 | 2ζ32ζ54+2ζ32ζ5 | ζ32ζ53+ζ32ζ52+2ζ32 | ζ32ζ54+ζ32ζ5+2ζ32 | 2ζ3ζ53+2ζ3ζ52 | 2ζ3ζ54+2ζ3ζ5 | ζ3ζ53+ζ3ζ52+2ζ3 | ζ3ζ54+ζ3ζ5+2ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | complex faithful |
| ρ30 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | 3+√5/2 | -1 | -1 | 3-√5/2 | -1-√5 | -1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3ζ54+ζ3ζ5+2ζ3 | ζ3ζ53+ζ3ζ52+2ζ3 | 2ζ3ζ53+2ζ3ζ52 | 2ζ3ζ54+2ζ3ζ5 | ζ32ζ54+ζ32ζ5+2ζ32 | ζ32ζ53+ζ32ζ52+2ζ32 | 2ζ32ζ53+2ζ32ζ52 | 2ζ32ζ54+2ζ32ζ5 | ζ6 | ζ6 | ζ65 | ζ65 | complex faithful |
►On 30 points - transitive group 30T73
Generators in S30
(1 13 8)(2 14 9)(3 15 10)(4 11 6)(5 12 7)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 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)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 20 19 18 17)(21 25 24 23 22)(26 30 29 28 27)
(1 16)(2 19 5 18)(3 17 4 20)(6 25 10 22)(7 23 9 24)(8 21)(11 30 15 27)(12 28 14 29)(13 26)
G:=sub<Sym(30)| (1,13,8)(2,14,9)(3,15,10)(4,11,6)(5,12,7)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (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), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27), (1,16)(2,19,5,18)(3,17,4,20)(6,25,10,22)(7,23,9,24)(8,21)(11,30,15,27)(12,28,14,29)(13,26)>;
G:=Group( (1,13,8)(2,14,9)(3,15,10)(4,11,6)(5,12,7)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (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), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,20,19,18,17)(21,25,24,23,22)(26,30,29,28,27), (1,16)(2,19,5,18)(3,17,4,20)(6,25,10,22)(7,23,9,24)(8,21)(11,30,15,27)(12,28,14,29)(13,26) );
G=PermutationGroup([[(1,13,8),(2,14,9),(3,15,10),(4,11,6),(5,12,7),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(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)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,20,19,18,17),(21,25,24,23,22),(26,30,29,28,27)], [(1,16),(2,19,5,18),(3,17,4,20),(6,25,10,22),(7,23,9,24),(8,21),(11,30,15,27),(12,28,14,29),(13,26)]])
G:=TransitiveGroup(30,73);
Matrix representation of C3×C52⋊C4 ►in GL4(𝔽61) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 60 | 1 | 0 | 0 |
| 16 | 44 | 0 | 0 |
| 59 | 44 | 18 | 18 |
| 17 | 1 | 43 | 60 |
| 0 | 18 | 0 | 0 |
| 44 | 17 | 0 | 0 |
| 44 | 17 | 0 | 1 |
| 1 | 18 | 60 | 43 |
| 0 | 0 | 60 | 1 |
| 18 | 1 | 59 | 43 |
| 0 | 0 | 60 | 0 |
| 17 | 1 | 60 | 0 |
G:=sub<GL(4,GF(61))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[60,16,59,17,1,44,44,1,0,0,18,43,0,0,18,60],[0,44,44,1,18,17,17,18,0,0,0,60,0,0,1,43],[0,18,0,17,0,1,0,1,60,59,60,60,1,43,0,0] >;
C3×C52⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_5^2\rtimes C_4
% in TeX
G:=Group("C3xC5^2:C4"); // GroupNames label
G:=SmallGroup(300,31);
// by ID
G=gap.SmallGroup(300,31);
# by ID
G:=PCGroup([5,-2,-3,-2,-5,-5,30,723,173,3004,1014]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^2,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of C3×C52⋊C4 in TeX
Character table of C3×C52⋊C4 in TeX