Aliases: C4.2S5, CSU2(𝔽5)⋊1C2, SL2(𝔽5).2C22, C2.6(C2×S5), C4.A5.1C2, SmallGroup(480,947)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(𝔽5) — C4.S5 |
| SL2(𝔽5) — C4.S5 |
Subgroups: 614 in 64 conjugacy classes, 8 normal (6 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, D6, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, SL2(𝔽3), Dic6, C4×S3, C3×Q8, C8.C22, C5⋊C8, C4×D5, CSU2(𝔽3), C4.A4, S3×Q8, C4.F5, C4.S4, SL2(𝔽5), CSU2(𝔽5), C4.A5, C4.S5
Quotients: C1, C2, C22, S5, C2×S5, C4.S5
Character table of C4.S5
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 5 | 6 | 8A | 8B | 10 | 12A | 12B | 12C | 20A | 20B | |
| size | 1 | 1 | 30 | 20 | 2 | 20 | 20 | 30 | 24 | 20 | 60 | 60 | 24 | 40 | 40 | 40 | 24 | 24 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 4 | 4 | 0 | 1 | -4 | 2 | -2 | 0 | -1 | 1 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from C2×S5 |
| ρ6 | 4 | 4 | 0 | 1 | -4 | -2 | 2 | 0 | -1 | 1 | 0 | 0 | -1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×S5 |
| ρ7 | 4 | 4 | 0 | 1 | 4 | 2 | 2 | 0 | -1 | 1 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | -1 | orthogonal lifted from S5 |
| ρ8 | 4 | 4 | 0 | 1 | 4 | -2 | -2 | 0 | -1 | 1 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from S5 |
| ρ9 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 1 | 0 | 0 | 0 | -√-5 | √-5 | complex faithful |
| ρ10 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 1 | 0 | 0 | 0 | √-5 | -√-5 | complex faithful |
| ρ11 | 5 | 5 | 1 | -1 | 5 | 1 | 1 | 1 | 0 | -1 | -1 | -1 | 0 | 1 | 1 | -1 | 0 | 0 | orthogonal lifted from S5 |
| ρ12 | 5 | 5 | -1 | -1 | -5 | -1 | 1 | 1 | 0 | -1 | 1 | -1 | 0 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from C2×S5 |
| ρ13 | 5 | 5 | -1 | -1 | -5 | 1 | -1 | 1 | 0 | -1 | -1 | 1 | 0 | 1 | -1 | 1 | 0 | 0 | orthogonal lifted from C2×S5 |
| ρ14 | 5 | 5 | 1 | -1 | 5 | -1 | -1 | 1 | 0 | -1 | 1 | 1 | 0 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S5 |
| ρ15 | 6 | 6 | 2 | 0 | -6 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from C2×S5 |
| ρ16 | 6 | 6 | -2 | 0 | 6 | 0 | 0 | -2 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S5 |
| ρ17 | 8 | -8 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ18 | 12 | -12 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(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)
(1 35 5 39)(2 28 6 32)(3 8 7 4)(9 14 13 10)(11 38 15 34)(12 25 16 29)(17 22 21 18)(19 36 23 40)(20 27 24 31)(26 41 30 45)(33 44 37 48)(42 47 46 43)
(1 44 5 48)(2 41 6 45)(3 46 7 42)(4 43 8 47)(9 17 13 21)(10 22 14 18)(11 19 15 23)(12 24 16 20)(25 27 29 31)(26 32 30 28)(33 39 37 35)(34 36 38 40)
G:=sub<Sym(48)| (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), (1,35,5,39)(2,28,6,32)(3,8,7,4)(9,14,13,10)(11,38,15,34)(12,25,16,29)(17,22,21,18)(19,36,23,40)(20,27,24,31)(26,41,30,45)(33,44,37,48)(42,47,46,43), (1,44,5,48)(2,41,6,45)(3,46,7,42)(4,43,8,47)(9,17,13,21)(10,22,14,18)(11,19,15,23)(12,24,16,20)(25,27,29,31)(26,32,30,28)(33,39,37,35)(34,36,38,40)>;
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), (1,35,5,39)(2,28,6,32)(3,8,7,4)(9,14,13,10)(11,38,15,34)(12,25,16,29)(17,22,21,18)(19,36,23,40)(20,27,24,31)(26,41,30,45)(33,44,37,48)(42,47,46,43), (1,44,5,48)(2,41,6,45)(3,46,7,42)(4,43,8,47)(9,17,13,21)(10,22,14,18)(11,19,15,23)(12,24,16,20)(25,27,29,31)(26,32,30,28)(33,39,37,35)(34,36,38,40) );
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)], [(1,35,5,39),(2,28,6,32),(3,8,7,4),(9,14,13,10),(11,38,15,34),(12,25,16,29),(17,22,21,18),(19,36,23,40),(20,27,24,31),(26,41,30,45),(33,44,37,48),(42,47,46,43)], [(1,44,5,48),(2,41,6,45),(3,46,7,42),(4,43,8,47),(9,17,13,21),(10,22,14,18),(11,19,15,23),(12,24,16,20),(25,27,29,31),(26,32,30,28),(33,39,37,35),(34,36,38,40)]])
Matrix representation of C4.S5 ►in GL4(𝔽3) generated by
| 0 | 0 | 1 | 2 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 2 | 2 |
| 0 | 2 | 1 | 1 |
| 2 | 2 | 0 | 2 |
| 0 | 2 | 1 | 2 |
| 2 | 0 | 0 | 1 |
| 2 | 2 | 2 | 2 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 2 |
| 1 | 1 | 0 | 0 |
| 1 | 2 | 0 | 0 |
G:=sub<GL(4,GF(3))| [0,2,0,0,0,0,2,2,1,0,2,1,2,0,2,1],[2,0,2,2,2,2,0,2,0,1,0,2,2,2,1,2],[0,0,1,1,0,0,1,2,1,1,0,0,1,2,0,0] >;
C4.S5 in GAP, Magma, Sage, TeX
C_4.S_5
% in TeX
G:=Group("C4.S5"); // GroupNames label
G:=SmallGroup(480,947);
// by ID
G=gap.SmallGroup(480,947);
# by ID
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