non-abelian, soluble, monomial
Aliases: C42⋊D15, C5⋊(C42⋊S3), (C4×C20)⋊1S3, C42⋊C3⋊2D5, C22.(C5⋊S4), (C2×C10).2S4, (C5×C42⋊C3)⋊4C2, SmallGroup(480,258)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C42⋊C3 — C42⋊D15 |
Generators and relations for C42⋊D15
G = < a,b,c,d | a4=b4=c15=d2=1, ab=ba, cac-1=dbd=a-1b-1, ad=da, cbc-1=a, dcd=c-1 >
3C2
60C2
16C3
3C4
3C4
30C4
30C22
80S3
3C10
12D5
16C15
15Q8
15D4
30C8
30D4
30C2×C4
4A4
3C20
3C20
6D10
6Dic5
16D15
15M4(2)
15C4○D4
20S4
3D20
15C4≀C2
Character table of C42⋊D15
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 10A | 10B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 3 | 60 | 32 | 3 | 3 | 6 | 60 | 2 | 2 | 60 | 60 | 6 | 6 | 32 | 32 | 32 | 32 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| 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 | 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 | 1 | 1 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ5 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ6 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D15 |
| ρ7 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D15 |
| ρ8 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D15 |
| ρ9 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D15 |
| ρ10 | 3 | 3 | 1 | 0 | -1 | -1 | -1 | 1 | 3 | 3 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ11 | 3 | 3 | -1 | 0 | -1 | -1 | -1 | -1 | 3 | 3 | 1 | 1 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ12 | 3 | -1 | 1 | 0 | -1-2i | -1+2i | 1 | -1 | 3 | 3 | -i | i | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1-2i | -1+2i | 1 | 1 | 1 | -1-2i | -1+2i | complex lifted from C42⋊S3 |
| ρ13 | 3 | -1 | 1 | 0 | -1+2i | -1-2i | 1 | -1 | 3 | 3 | i | -i | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1+2i | -1-2i | 1 | 1 | 1 | -1+2i | -1-2i | complex lifted from C42⋊S3 |
| ρ14 | 3 | -1 | -1 | 0 | -1+2i | -1-2i | 1 | 1 | 3 | 3 | -i | i | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1+2i | -1-2i | 1 | 1 | 1 | -1+2i | -1-2i | complex lifted from C42⋊S3 |
| ρ15 | 3 | -1 | -1 | 0 | -1-2i | -1+2i | 1 | 1 | 3 | 3 | i | -i | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1-2i | -1+2i | 1 | 1 | 1 | -1-2i | -1+2i | complex lifted from C42⋊S3 |
| ρ16 | 6 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 6 | 6 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | orthogonal lifted from C42⋊S3 |
| ρ17 | 6 | 6 | 0 | 0 | -2 | -2 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from C5⋊S4 |
| ρ18 | 6 | 6 | 0 | 0 | -2 | -2 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from C5⋊S4 |
| ρ19 | 6 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | -1+√5/2 | -1-√5/2 | 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 | -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 | -1-√5/2 | -1+√5/2 | orthogonal faithful |
| ρ20 | 6 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | -1-√5/2 | -1+√5/2 | 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 | 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 | -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | -1+√5/2 | -1-√5/2 | orthogonal faithful |
| ρ21 | 6 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 | -1-√5/2 | -1+√5/2 | -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 | -1+√5/2 | -1-√5/2 | orthogonal faithful |
| ρ22 | 6 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 | -1+√5/2 | -1-√5/2 | -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 | -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | -1-√5/2 | -1+√5/2 | orthogonal faithful |
| ρ23 | 6 | -2 | 0 | 0 | -2+4i | -2-4i | 2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | 2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | 2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 2ζ4ζ54+2ζ4ζ5-ζ54-ζ5 | 2ζ43ζ53+2ζ43ζ52-ζ53-ζ52 | complex faithful |
| ρ24 | 6 | -2 | 0 | 0 | -2-4i | -2+4i | 2 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | 2ζ43ζ53+2ζ43ζ52-ζ53-ζ52 | 2ζ4ζ54+2ζ4ζ5-ζ54-ζ5 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | 2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | complex faithful |
| ρ25 | 6 | -2 | 0 | 0 | -2-4i | -2+4i | 2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | 2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | 2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 2ζ43ζ53+2ζ43ζ52-ζ53-ζ52 | 2ζ4ζ54+2ζ4ζ5-ζ54-ζ5 | complex faithful |
| ρ26 | 6 | -2 | 0 | 0 | -2+4i | -2-4i | 2 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | 2ζ4ζ54+2ζ4ζ5-ζ54-ζ5 | 2ζ43ζ53+2ζ43ζ52-ζ53-ζ52 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | 2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | complex faithful |
►On 60 points
Generators in S60
(1 22 60 42)(2 46)(3 24 47 44)(4 25 48 45)(5 49)(6 27 50 32)(7 28 51 33)(8 52)(9 30 53 35)(10 16 54 36)(11 55)(12 18 56 38)(13 19 57 39)(14 58)(15 21 59 41)(17 37)(20 40)(23 43)(26 31)(29 34)
(1 22 60 42)(2 23 46 43)(3 47)(4 25 48 45)(5 26 49 31)(6 50)(7 28 51 33)(8 29 52 34)(9 53)(10 16 54 36)(11 17 55 37)(12 56)(13 19 57 39)(14 20 58 40)(15 59)(18 38)(21 41)(24 44)(27 32)(30 35)
(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)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 60)(16 32)(17 31)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)
G:=sub<Sym(60)| (1,22,60,42)(2,46)(3,24,47,44)(4,25,48,45)(5,49)(6,27,50,32)(7,28,51,33)(8,52)(9,30,53,35)(10,16,54,36)(11,55)(12,18,56,38)(13,19,57,39)(14,58)(15,21,59,41)(17,37)(20,40)(23,43)(26,31)(29,34), (1,22,60,42)(2,23,46,43)(3,47)(4,25,48,45)(5,26,49,31)(6,50)(7,28,51,33)(8,29,52,34)(9,53)(10,16,54,36)(11,17,55,37)(12,56)(13,19,57,39)(14,20,58,40)(15,59)(18,38)(21,41)(24,44)(27,32)(30,35), (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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,60)(16,32)(17,31)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)>;
G:=Group( (1,22,60,42)(2,46)(3,24,47,44)(4,25,48,45)(5,49)(6,27,50,32)(7,28,51,33)(8,52)(9,30,53,35)(10,16,54,36)(11,55)(12,18,56,38)(13,19,57,39)(14,58)(15,21,59,41)(17,37)(20,40)(23,43)(26,31)(29,34), (1,22,60,42)(2,23,46,43)(3,47)(4,25,48,45)(5,26,49,31)(6,50)(7,28,51,33)(8,29,52,34)(9,53)(10,16,54,36)(11,17,55,37)(12,56)(13,19,57,39)(14,20,58,40)(15,59)(18,38)(21,41)(24,44)(27,32)(30,35), (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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,60)(16,32)(17,31)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33) );
G=PermutationGroup([[(1,22,60,42),(2,46),(3,24,47,44),(4,25,48,45),(5,49),(6,27,50,32),(7,28,51,33),(8,52),(9,30,53,35),(10,16,54,36),(11,55),(12,18,56,38),(13,19,57,39),(14,58),(15,21,59,41),(17,37),(20,40),(23,43),(26,31),(29,34)], [(1,22,60,42),(2,23,46,43),(3,47),(4,25,48,45),(5,26,49,31),(6,50),(7,28,51,33),(8,29,52,34),(9,53),(10,16,54,36),(11,17,55,37),(12,56),(13,19,57,39),(14,20,58,40),(15,59),(18,38),(21,41),(24,44),(27,32),(30,35)], [(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)], [(1,59),(2,58),(3,57),(4,56),(5,55),(6,54),(7,53),(8,52),(9,51),(10,50),(11,49),(12,48),(13,47),(14,46),(15,60),(16,32),(17,31),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33)]])
Matrix representation of C42⋊D15 ►in GL5(𝔽241)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 45 |
| 0 | 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 177 | 107 | 0 |
| 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 177 |
| 189 | 51 | 0 | 0 | 0 |
| 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 41 | 200 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 239 | 34 | 208 |
| 1 | 240 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 34 | 25 | 216 |
| 0 | 0 | 2 | 207 | 33 |
| 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,1,0,0,0,0,0,177,0,0,0,0,0,177,0,0,0,45,0,240],[1,0,0,0,0,0,1,0,0,0,0,0,177,0,0,0,0,107,240,0,0,0,0,0,177],[189,189,0,0,0,51,0,0,0,0,0,0,33,0,239,0,0,41,0,34,0,0,200,1,208],[1,0,0,0,0,240,240,0,0,0,0,0,34,2,0,0,0,25,207,0,0,0,216,33,240] >;
C42⋊D15 in GAP, Magma, Sage, TeX
C_4^2\rtimes D_{15} % in TeX
G:=Group("C4^2:D15"); // GroupNames label
G:=SmallGroup(480,258);
// by ID
G=gap.SmallGroup(480,258);
# by ID
G:=PCGroup([7,-2,-3,-5,-2,2,-2,2,57,506,1683,850,360,1054,5786,102,15125,5052,8833]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^15=d^2=1,a*b=b*a,c*a*c^-1=d*b*d=a^-1*b^-1,a*d=d*a,c*b*c^-1=a,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C42⋊D15 in TeX
Character table of C42⋊D15 in TeX