direct product, non-abelian, not soluble
Aliases: C2×C2.S5, C22.4S5, SL2(𝔽5)⋊2C22, C2.9(C2×S5), (C2×SL2(𝔽5))⋊2C2, SmallGroup(480,950)
Series: Chief►Derived ►Lower central ►Upper central
| C1 — C2 — C22 — C2×SL2(𝔽5) — C2×C2.S5 |
| SL2(𝔽5) — C2×C2.S5 |
| SL2(𝔽5) — C2×C2.S5 |
Subgroups: 826 in 78 conjugacy classes, 10 normal (6 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C8, C2×C4, D4, Q8, C23, C10, Dic3, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, Dic5, C2×C10, SL2(𝔽3), C2×Dic3, C3⋊D4, C22×S3, C22×C6, C2×SD16, C5⋊C8, C2×Dic5, GL2(𝔽3), C2×SL2(𝔽3), C2×C3⋊D4, C2×C5⋊C8, C2×GL2(𝔽3), SL2(𝔽5), C2.S5, C2×SL2(𝔽5), C2×C2.S5
Quotients: C1, C2, C22, S5, C2.S5, C2×S5, C2×C2.S5
Character table of C2×C2.S5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 10A | 10B | 10C | |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 30 | 30 | 24 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 24 | 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 | 4 | 4 | -2 | -2 | 1 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S5 |
| ρ6 | 4 | -4 | 4 | -4 | -2 | 2 | 1 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | orthogonal lifted from C2×S5 |
| ρ7 | 4 | 4 | 4 | 4 | 2 | 2 | 1 | 0 | 0 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S5 |
| ρ8 | 4 | -4 | 4 | -4 | 2 | -2 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | orthogonal lifted from C2×S5 |
| ρ9 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | symplectic lifted from C2.S5, Schur index 2 |
| ρ10 | 4 | 4 | -4 | -4 | 0 | 0 | -2 | 0 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | symplectic lifted from C2.S5, Schur index 2 |
| ρ11 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | -1 | -1 | √-3 | -√-3 | -√-3 | √-3 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | complex lifted from C2.S5 |
| ρ12 | 4 | -4 | -4 | 4 | 0 | 0 | 1 | 0 | 0 | -1 | -1 | -√-3 | √-3 | √-3 | -√-3 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | complex lifted from C2.S5 |
| ρ13 | 4 | 4 | -4 | -4 | 0 | 0 | 1 | 0 | 0 | -1 | 1 | √-3 | -√-3 | √-3 | -√-3 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | complex lifted from C2.S5 |
| ρ14 | 4 | 4 | -4 | -4 | 0 | 0 | 1 | 0 | 0 | -1 | 1 | -√-3 | √-3 | -√-3 | √-3 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | complex lifted from C2.S5 |
| ρ15 | 5 | 5 | 5 | 5 | -1 | -1 | -1 | 1 | 1 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from S5 |
| ρ16 | 5 | -5 | 5 | -5 | -1 | 1 | -1 | 1 | -1 | 0 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from C2×S5 |
| ρ17 | 5 | -5 | 5 | -5 | 1 | -1 | -1 | 1 | -1 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from C2×S5 |
| ρ18 | 5 | 5 | 5 | 5 | 1 | 1 | -1 | 1 | 1 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S5 |
| ρ19 | 6 | 6 | 6 | 6 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | orthogonal lifted from S5 |
| ρ20 | 6 | -6 | 6 | -6 | 0 | 0 | 0 | -2 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | orthogonal lifted from C2×S5 |
| ρ21 | 6 | -6 | -6 | 6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | 1 | -1 | -1 | complex lifted from C2.S5 |
| ρ22 | 6 | 6 | -6 | -6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | -1 | 1 | -1 | complex lifted from C2.S5 |
| ρ23 | 6 | -6 | -6 | 6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | 1 | -1 | -1 | complex lifted from C2.S5 |
| ρ24 | 6 | 6 | -6 | -6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | -1 | 1 | -1 | complex lifted from C2.S5 |
►On 80 points
Generators in S80
(1 47)(2 72 74 48 61 34)(3 18 21 41 50 53)(4 77 71 42 37 60)(5 43)(6 68 78 44 57 38)(7 22 17 45 54 49)(8 73 67 46 33 64)(9 29)(10 66 24 30 63 56)(11 76 75 31 36 35)(12 23 69 32 55 58)(13 25)(14 70 20 26 59 52)(15 80 79 27 40 39)(16 19 65 28 51 62)
(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)
G:=sub<Sym(80)| (1,47)(2,72,74,48,61,34)(3,18,21,41,50,53)(4,77,71,42,37,60)(5,43)(6,68,78,44,57,38)(7,22,17,45,54,49)(8,73,67,46,33,64)(9,29)(10,66,24,30,63,56)(11,76,75,31,36,35)(12,23,69,32,55,58)(13,25)(14,70,20,26,59,52)(15,80,79,27,40,39)(16,19,65,28,51,62), (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)>;
G:=Group( (1,47)(2,72,74,48,61,34)(3,18,21,41,50,53)(4,77,71,42,37,60)(5,43)(6,68,78,44,57,38)(7,22,17,45,54,49)(8,73,67,46,33,64)(9,29)(10,66,24,30,63,56)(11,76,75,31,36,35)(12,23,69,32,55,58)(13,25)(14,70,20,26,59,52)(15,80,79,27,40,39)(16,19,65,28,51,62), (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) );
G=PermutationGroup([[(1,47),(2,72,74,48,61,34),(3,18,21,41,50,53),(4,77,71,42,37,60),(5,43),(6,68,78,44,57,38),(7,22,17,45,54,49),(8,73,67,46,33,64),(9,29),(10,66,24,30,63,56),(11,76,75,31,36,35),(12,23,69,32,55,58),(13,25),(14,70,20,26,59,52),(15,80,79,27,40,39),(16,19,65,28,51,62)], [(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)]])
Matrix representation of C2×C2.S5 ►in GL5(𝔽241)
| 240 | 0 | 0 | 0 | 0 |
| 0 | 1 | 178 | 211 | 181 |
| 0 | 0 | 183 | 3 | 5 |
| 0 | 0 | 3 | 32 | 33 |
| 0 | 0 | 90 | 238 | 26 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 |
| 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 | 240 |
| 0 | 0 | 2 | 0 | 1 |
G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,1,0,0,0,0,178,183,3,90,0,211,3,32,238,0,181,5,33,26],[1,0,0,0,0,0,0,1,0,0,0,240,240,240,2,0,1,0,0,0,0,0,0,240,1] >;
C2×C2.S5 in GAP, Magma, Sage, TeX
C_2\times C_2.S_5
% in TeX
G:=Group("C2xC2.S5"); // GroupNames label
G:=SmallGroup(480,950);
// by ID
G=gap.SmallGroup(480,950);
# by ID
Export