Aliases: C4.6S5, CSU2(𝔽5)⋊3C2, SL2(𝔽5).1C22, C4.A5⋊4C2, C2.5(C2×S5), C2.S5⋊3C2, SmallGroup(480,946)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(𝔽5) — C4.6S5 |
| SL2(𝔽5) — C4.6S5 |
Subgroups: 674 in 67 conjugacy classes, 8 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, D5, C10, Dic3, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, D10, SL2(𝔽3), Dic6, C4×S3, D12, C3⋊D4, C2×C12, C4○D8, C5⋊C8, C4×D5, CSU2(𝔽3), GL2(𝔽3), C4.A4, C4○D12, D5⋊C8, C4.6S4, SL2(𝔽5), CSU2(𝔽5), C2.S5, C4.A5, C4.6S5
Quotients: C1, C2, C22, S5, C2×S5, C4.6S5
Character table of C4.6S5
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 12C | 12D | 20A | 20B | |
| size | 1 | 1 | 20 | 30 | 20 | 1 | 1 | 20 | 30 | 24 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 24 | 20 | 20 | 20 | 20 | 24 | 24 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 4 | 4 | 2 | 0 | 1 | -4 | -4 | -2 | 0 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×S5 |
| ρ6 | 4 | 4 | -2 | 0 | 1 | -4 | -4 | 2 | 0 | -1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×S5 |
| ρ7 | 4 | 4 | -2 | 0 | 1 | 4 | 4 | -2 | 0 | -1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from S5 |
| ρ8 | 4 | 4 | 2 | 0 | 1 | 4 | 4 | 2 | 0 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from S5 |
| ρ9 | 4 | -4 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | -1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -2i | 2i | i | -i | complex faithful, Schur index 2 |
| ρ10 | 4 | -4 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | -1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 2i | -2i | -i | i | complex faithful, Schur index 2 |
| ρ11 | 4 | -4 | 0 | 0 | 1 | 4i | -4i | 0 | 0 | -1 | -√-3 | √-3 | -1 | 0 | 0 | 0 | 0 | 1 | √3 | -√3 | -i | i | -i | i | complex faithful |
| ρ12 | 4 | -4 | 0 | 0 | 1 | -4i | 4i | 0 | 0 | -1 | √-3 | -√-3 | -1 | 0 | 0 | 0 | 0 | 1 | √3 | -√3 | i | -i | i | -i | complex faithful |
| ρ13 | 4 | -4 | 0 | 0 | 1 | 4i | -4i | 0 | 0 | -1 | √-3 | -√-3 | -1 | 0 | 0 | 0 | 0 | 1 | -√3 | √3 | -i | i | -i | i | complex faithful |
| ρ14 | 4 | -4 | 0 | 0 | 1 | -4i | 4i | 0 | 0 | -1 | -√-3 | √-3 | -1 | 0 | 0 | 0 | 0 | 1 | -√3 | √3 | i | -i | i | -i | complex faithful |
| ρ15 | 5 | 5 | 1 | 1 | -1 | 5 | 5 | 1 | 1 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | -1 | -1 | 0 | 0 | orthogonal lifted from S5 |
| ρ16 | 5 | 5 | 1 | -1 | -1 | -5 | -5 | -1 | 1 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from C2×S5 |
| ρ17 | 5 | 5 | -1 | -1 | -1 | -5 | -5 | 1 | 1 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from C2×S5 |
| ρ18 | 5 | 5 | -1 | 1 | -1 | 5 | 5 | -1 | 1 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S5 |
| ρ19 | 6 | 6 | 0 | 2 | 0 | -6 | -6 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from C2×S5 |
| ρ20 | 6 | 6 | 0 | -2 | 0 | 6 | 6 | 0 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S5 |
| ρ21 | 6 | -6 | 0 | 0 | 0 | 6i | -6i | 0 | 0 | 1 | 0 | 0 | 0 | -√2 | √2 | -√-2 | √-2 | -1 | 0 | 0 | 0 | 0 | i | -i | complex faithful |
| ρ22 | 6 | -6 | 0 | 0 | 0 | -6i | 6i | 0 | 0 | 1 | 0 | 0 | 0 | -√2 | √2 | √-2 | -√-2 | -1 | 0 | 0 | 0 | 0 | -i | i | complex faithful |
| ρ23 | 6 | -6 | 0 | 0 | 0 | -6i | 6i | 0 | 0 | 1 | 0 | 0 | 0 | √2 | -√2 | -√-2 | √-2 | -1 | 0 | 0 | 0 | 0 | -i | i | complex faithful |
| ρ24 | 6 | -6 | 0 | 0 | 0 | 6i | -6i | 0 | 0 | 1 | 0 | 0 | 0 | √2 | -√2 | √-2 | -√-2 | -1 | 0 | 0 | 0 | 0 | i | -i | complex faithful |
►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 14 29 15 43 26)(2 19 34 20 48 11)(3 24 39 25 33 16)(4 9 44 10 38 21)(5 30 18 31 12 42)(6 35 23 36 17 47)(7 40 28 41 22 32)(8 45 13 46 27 37)
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), (1,14,29,15,43,26)(2,19,34,20,48,11)(3,24,39,25,33,16)(4,9,44,10,38,21)(5,30,18,31,12,42)(6,35,23,36,17,47)(7,40,28,41,22,32)(8,45,13,46,27,37)>;
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), (1,14,29,15,43,26)(2,19,34,20,48,11)(3,24,39,25,33,16)(4,9,44,10,38,21)(5,30,18,31,12,42)(6,35,23,36,17,47)(7,40,28,41,22,32)(8,45,13,46,27,37) );
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)], [(1,14,29,15,43,26),(2,19,34,20,48,11),(3,24,39,25,33,16),(4,9,44,10,38,21),(5,30,18,31,12,42),(6,35,23,36,17,47),(7,40,28,41,22,32),(8,45,13,46,27,37)]])
Matrix representation of C4.6S5 ►in GL4(𝔽5) generated by
| 2 | 4 | 4 | 0 |
| 4 | 4 | 2 | 1 |
| 2 | 0 | 2 | 4 |
| 1 | 3 | 4 | 4 |
| 2 | 3 | 1 | 0 |
| 3 | 3 | 0 | 1 |
| 2 | 1 | 3 | 4 |
| 1 | 0 | 0 | 2 |
G:=sub<GL(4,GF(5))| [2,4,2,1,4,4,0,3,4,2,2,4,0,1,4,4],[2,3,2,1,3,3,1,0,1,0,3,0,0,1,4,2] >;
C4.6S5 in GAP, Magma, Sage, TeX
C_4._6S_5
% in TeX
G:=Group("C4.6S5"); // GroupNames label
G:=SmallGroup(480,946);
// by ID
G=gap.SmallGroup(480,946);
# by ID
Export