direct product, non-abelian, not soluble
Aliases: C22×SL2(𝔽5), C23.A5, C22.5(C2×A5), C2.7(C22×A5), SmallGroup(480,960)
Series: Chief►Derived ►Lower central ►Upper central
| SL2(𝔽5) — C22×SL2(𝔽5) |
| SL2(𝔽5) — C22×SL2(𝔽5) |
Subgroups: 788 in 104 conjugacy classes, 21 normal (5 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C2×C4, Q8, C23, C10, Dic3, C2×C6, C22×C4, C2×Q8, Dic5, C2×C10, SL2(𝔽3), C2×Dic3, C22×C6, C22×Q8, C2×Dic5, C22×C10, C2×SL2(𝔽3), C22×Dic3, C22×Dic5, C22×SL2(𝔽3), SL2(𝔽5), C2×SL2(𝔽5), C22×SL2(𝔽5)
Quotients: C1, C2, C22, A5, SL2(𝔽5), C2×A5, C2×SL2(𝔽5), C22×A5, C22×SL2(𝔽5)
►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 53 60 24 83 5 87 28 46 72)(2 48 65 19 78 6 92 33 41 67)(3 23 94 75 61 10 45 50 86 29)(4 18 89 70 66 9 40 55 81 34)(7 69 37 27 90 12 80 25 59 56)(8 74 42 32 95 11 85 20 64 51)(13 36 77 49 38 15 58 76 93 26)(14 31 82 54 43 16 63 71 88 21)(17 39)(22 44)(30 62)(35 57)(47 91)(52 96)(68 79)(73 84)
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), (1,53,60,24,83,5,87,28,46,72)(2,48,65,19,78,6,92,33,41,67)(3,23,94,75,61,10,45,50,86,29)(4,18,89,70,66,9,40,55,81,34)(7,69,37,27,90,12,80,25,59,56)(8,74,42,32,95,11,85,20,64,51)(13,36,77,49,38,15,58,76,93,26)(14,31,82,54,43,16,63,71,88,21)(17,39)(22,44)(30,62)(35,57)(47,91)(52,96)(68,79)(73,84)>;
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), (1,53,60,24,83,5,87,28,46,72)(2,48,65,19,78,6,92,33,41,67)(3,23,94,75,61,10,45,50,86,29)(4,18,89,70,66,9,40,55,81,34)(7,69,37,27,90,12,80,25,59,56)(8,74,42,32,95,11,85,20,64,51)(13,36,77,49,38,15,58,76,93,26)(14,31,82,54,43,16,63,71,88,21)(17,39)(22,44)(30,62)(35,57)(47,91)(52,96)(68,79)(73,84) );
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)], [(1,53,60,24,83,5,87,28,46,72),(2,48,65,19,78,6,92,33,41,67),(3,23,94,75,61,10,45,50,86,29),(4,18,89,70,66,9,40,55,81,34),(7,69,37,27,90,12,80,25,59,56),(8,74,42,32,95,11,85,20,64,51),(13,36,77,49,38,15,58,76,93,26),(14,31,82,54,43,16,63,71,88,21),(17,39),(22,44),(30,62),(35,57),(47,91),(52,96),(68,79),(73,84)]])
36 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 |
| size | 1 | 1 | ··· | 1 | 20 | 30 | 30 | 30 | 30 | 12 | 12 | 20 | ··· | 20 | 12 | ··· | 12 |
36 irreducible representations
| dim | 1 | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 |
| type | + | + | - | + | + | + | - | + | + | + | - |
| image | C1 | C2 | SL2(𝔽5) | A5 | C2×A5 | A5 | SL2(𝔽5) | C2×A5 | A5 | C2×A5 | SL2(𝔽5) |
| kernel | C22×SL2(𝔽5) | C2×SL2(𝔽5) | C22 | C23 | C22 | C23 | C22 | C22 | C23 | C22 | C22 |
| # reps | 1 | 3 | 8 | 2 | 6 | 1 | 4 | 3 | 1 | 3 | 4 |
Matrix representation of C22×SL2(𝔽5) ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 28 | 49 |
| 0 | 0 | 59 | 51 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 34 |
| 0 | 0 | 40 | 8 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,60,0,0,0,0,28,59,0,0,49,51],[60,0,0,0,0,1,0,0,0,0,10,40,0,0,34,8] >;
C22×SL2(𝔽5) in GAP, Magma, Sage, TeX
C_2^2\times {\rm SL}_2({\mathbb F}_5) % in TeX
G:=Group("C2^2xSL(2,5)"); // GroupNames label
G:=SmallGroup(480,960);
// by ID
G=gap.SmallGroup(480,960);
# by ID