Aliases: C22.2S5, C2.CSU2(𝔽5), SL2(𝔽5)⋊2C4, C2.3(A5⋊C4), C2.(C2.S5), (C2×SL2(𝔽5)).1C2, SmallGroup(480,219)
Series: Chief►Derived ►Lower central ►Upper central
| C1 — C2 — C22 — C2×SL2(𝔽5) — C22.2S5 |
| SL2(𝔽5) — C22.2S5 |
| SL2(𝔽5) — C22.2S5 |
10C3
6C5
15C4
15C4
20C4
10C6
10C6
10C6
6C10
6C10
6C10
5Q8
10C2×C4
15Q8
15C2×C4
30C8
10Dic3
10C2×C6
10Dic3
20Dic3
20C12
6Dic5
6Dic5
15C2×C8
15C4⋊C4
10C2×Dic3
10C2×C12
10C2×Dic3
15Q8⋊C4
10Dic3⋊C4
Character table of C22.2S5
| 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 | i | -i | -1 | 1 | 1 | -1 | 1 | -1 | i | -i | i | -i | 1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ4 | 1 | -1 | 1 | -1 | 1 | -i | i | -1 | 1 | 1 | -1 | 1 | -1 | -i | i | -i | i | 1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| 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 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 | -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 |
| ρ8 | 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 C2.S5, Schur index 2 |
| ρ9 | 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 |
| ρ10 | 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 |
| ρ11 | 4 | -4 | 4 | -4 | 1 | 2i | -2i | 0 | 0 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | i | i | -i | -i | complex lifted from A5⋊C4 |
| ρ12 | 4 | -4 | 4 | -4 | 1 | -2i | 2i | 0 | 0 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -i | -i | i | i | complex lifted from A5⋊C4 |
| 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 | complex lifted from C2.S5 |
| 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 | complex lifted from C2.S5 |
| 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 S5 |
| ρ17 | 5 | -5 | 5 | -5 | -1 | -i | i | -1 | 1 | 0 | 1 | -1 | 1 | i | -i | i | -i | 0 | 0 | 0 | i | i | -i | -i | complex lifted from A5⋊C4 |
| ρ18 | 5 | -5 | 5 | -5 | -1 | i | -i | -1 | 1 | 0 | 1 | -1 | 1 | -i | i | -i | i | 0 | 0 | 0 | -i | -i | i | i | complex lifted from A5⋊C4 |
| 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 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 A5⋊C4 |
| 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 | complex lifted from C2.S5 |
| 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 | complex lifted from C2.S5 |
►On 96 points
Generators in S96
(1 92 29 7 86 35)(2 78 17 8 84 23)(3 75 44 9 81 38)(4 87 55 10 93 49)(5 58 16 11 52 22)(6 37 26 12 43 32)(13 28 57 19 34 51)(14 83 66 20 77 72)(15 39 63 21 45 69)(18 46 31 24 40 25)(27 60 62 33 54 68)(30 89 61 36 95 67)(41 91 82 47 85 76)(42 88 64 48 94 70)(50 79 96 56 73 90)(53 74 65 59 80 71)
(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)
G:=sub<Sym(96)| (1,92,29,7,86,35)(2,78,17,8,84,23)(3,75,44,9,81,38)(4,87,55,10,93,49)(5,58,16,11,52,22)(6,37,26,12,43,32)(13,28,57,19,34,51)(14,83,66,20,77,72)(15,39,63,21,45,69)(18,46,31,24,40,25)(27,60,62,33,54,68)(30,89,61,36,95,67)(41,91,82,47,85,76)(42,88,64,48,94,70)(50,79,96,56,73,90)(53,74,65,59,80,71), (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)>;
G:=Group( (1,92,29,7,86,35)(2,78,17,8,84,23)(3,75,44,9,81,38)(4,87,55,10,93,49)(5,58,16,11,52,22)(6,37,26,12,43,32)(13,28,57,19,34,51)(14,83,66,20,77,72)(15,39,63,21,45,69)(18,46,31,24,40,25)(27,60,62,33,54,68)(30,89,61,36,95,67)(41,91,82,47,85,76)(42,88,64,48,94,70)(50,79,96,56,73,90)(53,74,65,59,80,71), (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) );
G=PermutationGroup([[(1,92,29,7,86,35),(2,78,17,8,84,23),(3,75,44,9,81,38),(4,87,55,10,93,49),(5,58,16,11,52,22),(6,37,26,12,43,32),(13,28,57,19,34,51),(14,83,66,20,77,72),(15,39,63,21,45,69),(18,46,31,24,40,25),(27,60,62,33,54,68),(30,89,61,36,95,67),(41,91,82,47,85,76),(42,88,64,48,94,70),(50,79,96,56,73,90),(53,74,65,59,80,71)], [(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)]])
Matrix representation of C22.2S5 ►in GL8(𝔽241)
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 88 | 34 | 49 | 224 |
| 0 | 0 | 0 | 0 | 22 | 22 | 161 | 194 |
| 0 | 0 | 0 | 0 | 128 | 144 | 50 | 181 |
| 0 | 0 | 0 | 0 | 194 | 196 | 121 | 82 |
| 177 | 177 | 177 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 101 | 216 | 92 | 110 |
| 0 | 0 | 0 | 0 | 88 | 118 | 189 | 119 |
| 0 | 0 | 0 | 0 | 57 | 63 | 225 | 87 |
| 0 | 0 | 0 | 0 | 209 | 180 | 171 | 69 |
G:=sub<GL(8,GF(241))| [0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,88,22,128,194,0,0,0,0,34,22,144,196,0,0,0,0,49,161,50,121,0,0,0,0,224,194,181,82],[177,0,0,0,0,0,0,0,177,0,0,64,0,0,0,0,177,64,0,0,0,0,0,0,177,0,64,0,0,0,0,0,0,0,0,0,101,88,57,209,0,0,0,0,216,118,63,180,0,0,0,0,92,189,225,171,0,0,0,0,110,119,87,69] >;
C22.2S5 in GAP, Magma, Sage, TeX
C_2^2._2S_5
% in TeX
G:=Group("C2^2.2S5"); // GroupNames label
G:=SmallGroup(480,219);
// by ID
G=gap.SmallGroup(480,219);
# by ID
Export
Subgroup lattice of C22.2S5 in TeX
Character table of C22.2S5 in TeX