direct product, non-abelian, not soluble
Aliases: C2×CSU2(𝔽5), C22.3S5, SL2(𝔽5).3C22, C2.8(C2×S5), (C2×SL2(𝔽5)).2C2, SmallGroup(480,949)
Series: Chief►Derived ►Lower central ►Upper central
| C1 — C2 — C22 — C2×SL2(𝔽5) — C2×CSU2(𝔽5) |
| SL2(𝔽5) — C2×CSU2(𝔽5) |
| SL2(𝔽5) — C2×CSU2(𝔽5) |
Subgroups: 586 in 66 conjugacy classes, 10 normal (6 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C8, C2×C4, Q8, C10, Dic3, C12, C2×C6, C2×C8, Q16, C2×Q8, Dic5, C2×C10, SL2(𝔽3), Dic6, C2×Dic3, C2×C12, C2×Q16, C5⋊C8, C2×Dic5, CSU2(𝔽3), C2×SL2(𝔽3), C2×Dic6, C2×C5⋊C8, C2×CSU2(𝔽3), SL2(𝔽5), CSU2(𝔽5), C2×SL2(𝔽5), C2×CSU2(𝔽5)
Quotients: C1, C2, C22, S5, CSU2(𝔽5), C2×S5, C2×CSU2(𝔽5)
Character table of C2×CSU2(𝔽5)
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 30 | 30 | 24 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 24 | 24 | 24 | 20 | 20 | 20 | 20 | |
| 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 | -4 | -4 | 1 | 2 | -2 | 0 | 0 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×S5 |
| ρ6 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 0 | 0 | -1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S5 |
| ρ7 | 4 | 4 | -4 | -4 | 1 | -2 | 2 | 0 | 0 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×S5 |
| ρ8 | 4 | 4 | 4 | 4 | 1 | -2 | -2 | 0 | 0 | -1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from S5 |
| ρ9 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽5), Schur index 2 |
| ρ10 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽5), Schur index 2 |
| ρ11 | 4 | -4 | 4 | -4 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√3 | √3 | -√3 | √3 | symplectic lifted from CSU2(𝔽5), Schur index 2 |
| ρ12 | 4 | -4 | 4 | -4 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √3 | -√3 | √3 | -√3 | symplectic lifted from CSU2(𝔽5), Schur index 2 |
| ρ13 | 4 | -4 | -4 | 4 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | √3 | √3 | -√3 | symplectic lifted from CSU2(𝔽5), Schur index 2 |
| ρ14 | 4 | -4 | -4 | 4 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | -√3 | -√3 | √3 | symplectic lifted from CSU2(𝔽5), Schur index 2 |
| ρ15 | 5 | 5 | 5 | 5 | -1 | -1 | -1 | 1 | 1 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S5 |
| ρ16 | 5 | 5 | -5 | -5 | -1 | -1 | 1 | 1 | -1 | 0 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×S5 |
| ρ17 | 5 | 5 | 5 | 5 | -1 | 1 | 1 | 1 | 1 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S5 |
| ρ18 | 5 | 5 | -5 | -5 | -1 | 1 | -1 | 1 | -1 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×S5 |
| ρ19 | 6 | 6 | -6 | -6 | 0 | 0 | 0 | -2 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S5 |
| ρ20 | 6 | 6 | 6 | 6 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S5 |
| ρ21 | 6 | -6 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽5), Schur index 2 |
| ρ22 | 6 | -6 | -6 | 6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽5), Schur index 2 |
| ρ23 | 6 | -6 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽5), Schur index 2 |
| ρ24 | 6 | -6 | -6 | 6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽5), Schur index 2 |
►On 96 points
Generators in S96
(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)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 21 3 66 7 15 9 72)(2 16 57 20 8 22 51 14)(4 74 34 65 10 80 28 71)(5 83 48 73 11 77 42 79)(6 61 30 82 12 67 36 76)(13 58 85 46 19 52 91 40)(17 44 95 56 23 38 89 50)(18 41 84 43 24 47 78 37)(25 93 27 75 31 87 33 81)(26 88 45 92 32 94 39 86)(29 68 54 70 35 62 60 64)(49 96 53 63 55 90 59 69)
G:=sub<Sym(96)| (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)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,21,3,66,7,15,9,72)(2,16,57,20,8,22,51,14)(4,74,34,65,10,80,28,71)(5,83,48,73,11,77,42,79)(6,61,30,82,12,67,36,76)(13,58,85,46,19,52,91,40)(17,44,95,56,23,38,89,50)(18,41,84,43,24,47,78,37)(25,93,27,75,31,87,33,81)(26,88,45,92,32,94,39,86)(29,68,54,70,35,62,60,64)(49,96,53,63,55,90,59,69)>;
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)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,21,3,66,7,15,9,72)(2,16,57,20,8,22,51,14)(4,74,34,65,10,80,28,71)(5,83,48,73,11,77,42,79)(6,61,30,82,12,67,36,76)(13,58,85,46,19,52,91,40)(17,44,95,56,23,38,89,50)(18,41,84,43,24,47,78,37)(25,93,27,75,31,87,33,81)(26,88,45,92,32,94,39,86)(29,68,54,70,35,62,60,64)(49,96,53,63,55,90,59,69) );
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),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,21,3,66,7,15,9,72),(2,16,57,20,8,22,51,14),(4,74,34,65,10,80,28,71),(5,83,48,73,11,77,42,79),(6,61,30,82,12,67,36,76),(13,58,85,46,19,52,91,40),(17,44,95,56,23,38,89,50),(18,41,84,43,24,47,78,37),(25,93,27,75,31,87,33,81),(26,88,45,92,32,94,39,86),(29,68,54,70,35,62,60,64),(49,96,53,63,55,90,59,69)]])
Matrix representation of C2×CSU2(𝔽5) ►in GL8(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 158 | 215 | 166 | 26 |
| 0 | 0 | 0 | 0 | 80 | 94 | 41 | 37 |
| 0 | 0 | 0 | 0 | 113 | 140 | 228 | 206 |
| 0 | 0 | 0 | 0 | 65 | 158 | 174 | 2 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 183 | 223 | 28 | 49 |
| 0 | 0 | 0 | 0 | 221 | 193 | 10 | 60 |
| 0 | 0 | 0 | 0 | 41 | 74 | 220 | 9 |
| 0 | 0 | 0 | 0 | 218 | 196 | 145 | 127 |
G:=sub<GL(8,GF(241))| [240,240,240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,158,80,113,65,0,0,0,0,215,94,140,158,0,0,0,0,166,41,228,174,0,0,0,0,26,37,206,2],[0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,183,221,41,218,0,0,0,0,223,193,74,196,0,0,0,0,28,10,220,145,0,0,0,0,49,60,9,127] >;
C2×CSU2(𝔽5) in GAP, Magma, Sage, TeX
C_2\times {\rm CSU}_2({\mathbb F}_5) % in TeX
G:=Group("C2xCSU(2,5)"); // GroupNames label
G:=SmallGroup(480,949);
// by ID
G=gap.SmallGroup(480,949);
# by ID
Export