metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊1C4, C12⋊1F5, D5.1D12, C20⋊1Dic3, D10.13D6, D5.2Dic6, Dic5⋊3Dic3, C4⋊(C3⋊F5), C5⋊(C4⋊Dic3), C3⋊1(C4⋊F5), C15⋊2(C4⋊C4), (C4×D5).4S3, (C3×D5).3D4, C6.11(C2×F5), (C3×D5).2Q8, C30.11(C2×C4), (C3×Dic5)⋊5C4, (D5×C12).6C2, C10.4(C2×Dic3), (C6×D5).20C22, C2.5(C2×C3⋊F5), (C2×C3⋊F5).3C2, SmallGroup(240,121)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60⋊C4
G = < a,b | a60=b4=1, bab-1=a47 >
Character table of C60⋊C4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 10 | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 30A | 30B | 60A | 60B | 60C | 60D | |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 10 | 30 | 30 | 30 | 30 | 4 | 2 | 10 | 10 | 4 | 2 | 2 | 10 | 10 | 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 | 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 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | i | -i | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | -i | i | i | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | i | i | -i | -i | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | -i | i | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 1 | 1 | 1 | 1 | -1 | -1 | -2 | -2 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | -1 | -2 | √3 | -√3 | √3 | -√3 | -1 | -1 | 0 | 0 | 1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ13 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 1 | -1 | -2 | -√3 | √3 | -√3 | √3 | -1 | -1 | 0 | 0 | 1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ14 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | 1 | 1 | 2 | -1 | -1 | 1 | 1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | 1 | 1 | 2 | 1 | 1 | -1 | -1 | -1 | -1 | -2 | -2 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | -1 | 1 | -2 | √3 | -√3 | -√3 | √3 | -1 | -1 | 0 | 0 | 1 | 1 | √3 | -√3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1 | -1 | 1 | -2 | -√3 | √3 | √3 | -√3 | -1 | -1 | 0 | 0 | 1 | 1 | -√3 | √3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 4 | 4 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ20 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ21 | 4 | 4 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | -1 | -1 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ22 | 4 | 4 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | 1 | 1 | 1-√-15/2 | 1+√-15/2 | -1-√-15/2 | -1+√-15/2 | -1+√-15/2 | -1-√-15/2 | complex lifted from C2×C3⋊F5 |
| ρ23 | 4 | 4 | 0 | 0 | -2 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | -1 | -1 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ24 | 4 | 4 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | 1 | 1 | 1+√-15/2 | 1-√-15/2 | -1+√-15/2 | -1-√-15/2 | -1-√-15/2 | -1+√-15/2 | complex lifted from C2×C3⋊F5 |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | √-5 | -√-5 | 1 | 1 | √-5 | √-5 | -√-5 | -√-5 | complex lifted from C4⋊F5 |
| ρ26 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -4 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -√-5 | √-5 | 1 | 1 | -√-5 | -√-5 | √-5 | √-5 | complex lifted from C4⋊F5 |
| ρ27 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 1 | -2√3 | 2√3 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | √-5 | -√-5 | -1+√-15/2 | -1-√-15/2 | -ζ4ζ3+ζ4ζ53+ζ4ζ52 | -ζ43ζ3+ζ43ζ54+ζ43ζ5 | -ζ43ζ32+ζ43ζ53+ζ43ζ52 | -ζ4ζ32+ζ4ζ54+ζ4ζ5 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 1 | -2√3 | 2√3 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | -√-5 | √-5 | -1-√-15/2 | -1+√-15/2 | -ζ43ζ32+ζ43ζ53+ζ43ζ52 | -ζ4ζ32+ζ4ζ54+ζ4ζ5 | -ζ4ζ3+ζ4ζ53+ζ4ζ52 | -ζ43ζ3+ζ43ζ54+ζ43ζ5 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 1 | 2√3 | -2√3 | 0 | 0 | 1+√-15/2 | 1-√-15/2 | -√-5 | √-5 | -1+√-15/2 | -1-√-15/2 | -ζ4ζ32+ζ4ζ54+ζ4ζ5 | -ζ43ζ32+ζ43ζ53+ζ43ζ52 | -ζ43ζ3+ζ43ζ54+ζ43ζ5 | -ζ4ζ3+ζ4ζ53+ζ4ζ52 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 1 | 2√3 | -2√3 | 0 | 0 | 1-√-15/2 | 1+√-15/2 | √-5 | -√-5 | -1-√-15/2 | -1+√-15/2 | -ζ43ζ3+ζ43ζ54+ζ43ζ5 | -ζ4ζ3+ζ4ζ53+ζ4ζ52 | -ζ4ζ32+ζ4ζ54+ζ4ζ5 | -ζ43ζ32+ζ43ζ53+ζ43ζ52 | complex faithful |
►On 60 points
Generators in S60
(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 55 56 57 58 59 60)
(1 16)(2 39 50 3)(4 25 28 37)(5 48 17 24)(6 11)(7 34 55 58)(8 57 44 45)(9 20 33 32)(10 43 22 19)(12 29 60 53)(13 52 49 40)(14 15 38 27)(18 47 54 35)(21 56)(23 42 59 30)(26 51)(31 46)(36 41)
G:=sub<Sym(60)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,16)(2,39,50,3)(4,25,28,37)(5,48,17,24)(6,11)(7,34,55,58)(8,57,44,45)(9,20,33,32)(10,43,22,19)(12,29,60,53)(13,52,49,40)(14,15,38,27)(18,47,54,35)(21,56)(23,42,59,30)(26,51)(31,46)(36,41)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,16)(2,39,50,3)(4,25,28,37)(5,48,17,24)(6,11)(7,34,55,58)(8,57,44,45)(9,20,33,32)(10,43,22,19)(12,29,60,53)(13,52,49,40)(14,15,38,27)(18,47,54,35)(21,56)(23,42,59,30)(26,51)(31,46)(36,41) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)], [(1,16),(2,39,50,3),(4,25,28,37),(5,48,17,24),(6,11),(7,34,55,58),(8,57,44,45),(9,20,33,32),(10,43,22,19),(12,29,60,53),(13,52,49,40),(14,15,38,27),(18,47,54,35),(21,56),(23,42,59,30),(26,51),(31,46),(36,41)]])
C60⋊C4 is a maximal subgroup of
D12⋊F5 Dic30⋊C4 Dic5.Dic6 Dic5.4Dic6 C120⋊C4 D5.D24 D20⋊Dic3 Dic10⋊2Dic3 F5×Dic6 C4⋊F5⋊3S3 F5×D12 S3×C4⋊F5 (C2×C12)⋊6F5 D4×C3⋊F5 Q8×C3⋊F5
C60⋊C4 is a maximal quotient of
C120⋊C4 D5.D24 C40.Dic3 C24.1F5 C60⋊C8 Dic5.13D12 D10.10D12
Matrix representation of C60⋊C4 ►in GL6(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 29 | 16 | 45 |
| 0 | 0 | 32 | 29 | 45 | 0 |
| 0 | 0 | 16 | 0 | 45 | 29 |
| 0 | 0 | 32 | 45 | 16 | 29 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 32 | 29 |
| 0 | 0 | 29 | 0 | 32 | 45 |
| 0 | 0 | 45 | 32 | 0 | 29 |
| 0 | 0 | 29 | 32 | 16 | 0 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,1,0,0,0,0,0,0,0,32,16,32,0,0,29,29,0,45,0,0,16,45,45,16,0,0,45,0,29,29],[50,11,0,0,0,0,0,11,0,0,0,0,0,0,0,29,45,29,0,0,16,0,32,32,0,0,32,32,0,16,0,0,29,45,29,0] >;
C60⋊C4 in GAP, Magma, Sage, TeX
C_{60}\rtimes C_4 % in TeX
G:=Group("C60:C4"); // GroupNames label
G:=SmallGroup(240,121);
// by ID
G=gap.SmallGroup(240,121);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,55,964,5189,1745]);
// Polycyclic
G:=Group<a,b|a^60=b^4=1,b*a*b^-1=a^47>;
// generators/relations
Export
Subgroup lattice of C60⋊C4 in TeX
Character table of C60⋊C4 in TeX