non-abelian, soluble, monomial
Aliases: A4⋊Dic5, C10.3S4, C23.D15, C22⋊Dic15, (C2×A4).D5, C5⋊2(A4⋊C4), (C5×A4)⋊3C4, C2.1(C5⋊S4), (C10×A4).1C2, (C2×C10)⋊3Dic3, (C22×C10).2S3, SmallGroup(240,107)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×A4 — A4⋊Dic5 |
Generators and relations for A4⋊Dic5
G = < a,b,c,d,e | a2=b2=c3=d10=1, e2=d5, cac-1=eae-1=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Character table of A4⋊Dic5
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 15A | 15B | 15C | 15D | 30A | 30B | 30C | 30D | |
| size | 1 | 1 | 3 | 3 | 8 | 30 | 30 | 30 | 30 | 2 | 2 | 8 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | i | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -i | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 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 | -1-√5/2 | orthogonal lifted from D5 |
| ρ6 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | orthogonal lifted from D15 |
| ρ8 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 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 | -1+√5/2 | orthogonal lifted from D5 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | orthogonal lifted from D15 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | orthogonal lifted from D15 |
| ρ11 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | orthogonal lifted from D15 |
| ρ12 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | 1 | -2 | -2 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -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 | 1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ14 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52+ζ53 | ζ32ζ54-ζ32ζ5+ζ54 | ζ3ζ54-ζ3ζ5+ζ54 | -ζ3ζ53+ζ3ζ52+ζ52 | symplectic lifted from Dic15, Schur index 2 |
| ρ15 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52+ζ52 | ζ3ζ54-ζ3ζ5+ζ54 | ζ32ζ54-ζ32ζ5+ζ54 | ζ3ζ53-ζ3ζ52+ζ53 | symplectic lifted from Dic15, Schur index 2 |
| ρ16 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ54-ζ3ζ5+ζ54 | ζ3ζ53-ζ3ζ52+ζ53 | -ζ3ζ53+ζ3ζ52+ζ52 | ζ32ζ54-ζ32ζ5+ζ54 | symplectic lifted from Dic15, Schur index 2 |
| ρ17 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | ζ32ζ54-ζ32ζ5+ζ54 | -ζ3ζ53+ζ3ζ52+ζ52 | ζ3ζ53-ζ3ζ52+ζ53 | ζ3ζ54-ζ3ζ5+ζ54 | symplectic lifted from Dic15, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -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 | 1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ19 | 3 | 3 | -1 | -1 | 0 | 1 | 1 | -1 | -1 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | -1 | -1 | 0 | -1 | -1 | 1 | 1 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ21 | 3 | -3 | -1 | 1 | 0 | -i | i | i | -i | 3 | 3 | 0 | -3 | -3 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ22 | 3 | -3 | -1 | 1 | 0 | i | -i | -i | i | 3 | 3 | 0 | -3 | -3 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from A4⋊C4 |
| ρ23 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | -3+3√5/2 | -3-3√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C5⋊S4 |
| ρ24 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | -3-3√5/2 | -3+3√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C5⋊S4 |
| ρ25 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 3+3√5/2 | 3-3√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ26 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 3-3√5/2 | 3+3√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 60 points
Generators in S60
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 21)(11 59)(12 60)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 51)(19 52)(20 53)(21 26)(22 27)(23 28)(24 29)(25 30)(31 48)(32 49)(33 50)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
(1 19 44)(2 20 45)(3 11 46)(4 12 47)(5 13 48)(6 14 49)(7 15 50)(8 16 41)(9 17 42)(10 18 43)(21 51 31)(22 52 32)(23 53 33)(24 54 34)(25 55 35)(26 56 36)(27 57 37)(28 58 38)(29 59 39)(30 60 40)
(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 27 6 22)(2 26 7 21)(3 25 8 30)(4 24 9 29)(5 23 10 28)(11 35 16 40)(12 34 17 39)(13 33 18 38)(14 32 19 37)(15 31 20 36)(41 60 46 55)(42 59 47 54)(43 58 48 53)(44 57 49 52)(45 56 50 51)
G:=sub<Sym(60)| (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,21)(11,59)(12,60)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50), (1,6)(2,7)(3,8)(4,9)(5,10)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,51)(19,52)(20,53)(21,26)(22,27)(23,28)(24,29)(25,30)(31,48)(32,49)(33,50)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (1,19,44)(2,20,45)(3,11,46)(4,12,47)(5,13,48)(6,14,49)(7,15,50)(8,16,41)(9,17,42)(10,18,43)(21,51,31)(22,52,32)(23,53,33)(24,54,34)(25,55,35)(26,56,36)(27,57,37)(28,58,38)(29,59,39)(30,60,40), (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), (1,27,6,22)(2,26,7,21)(3,25,8,30)(4,24,9,29)(5,23,10,28)(11,35,16,40)(12,34,17,39)(13,33,18,38)(14,32,19,37)(15,31,20,36)(41,60,46,55)(42,59,47,54)(43,58,48,53)(44,57,49,52)(45,56,50,51)>;
G:=Group( (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,21)(11,59)(12,60)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50), (1,6)(2,7)(3,8)(4,9)(5,10)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,51)(19,52)(20,53)(21,26)(22,27)(23,28)(24,29)(25,30)(31,48)(32,49)(33,50)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (1,19,44)(2,20,45)(3,11,46)(4,12,47)(5,13,48)(6,14,49)(7,15,50)(8,16,41)(9,17,42)(10,18,43)(21,51,31)(22,52,32)(23,53,33)(24,54,34)(25,55,35)(26,56,36)(27,57,37)(28,58,38)(29,59,39)(30,60,40), (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), (1,27,6,22)(2,26,7,21)(3,25,8,30)(4,24,9,29)(5,23,10,28)(11,35,16,40)(12,34,17,39)(13,33,18,38)(14,32,19,37)(15,31,20,36)(41,60,46,55)(42,59,47,54)(43,58,48,53)(44,57,49,52)(45,56,50,51) );
G=PermutationGroup([[(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,21),(11,59),(12,60),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,51),(19,52),(20,53),(21,26),(22,27),(23,28),(24,29),(25,30),(31,48),(32,49),(33,50),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47)], [(1,19,44),(2,20,45),(3,11,46),(4,12,47),(5,13,48),(6,14,49),(7,15,50),(8,16,41),(9,17,42),(10,18,43),(21,51,31),(22,52,32),(23,53,33),(24,54,34),(25,55,35),(26,56,36),(27,57,37),(28,58,38),(29,59,39),(30,60,40)], [(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)], [(1,27,6,22),(2,26,7,21),(3,25,8,30),(4,24,9,29),(5,23,10,28),(11,35,16,40),(12,34,17,39),(13,33,18,38),(14,32,19,37),(15,31,20,36),(41,60,46,55),(42,59,47,54),(43,58,48,53),(44,57,49,52),(45,56,50,51)]])
A4⋊Dic5 is a maximal subgroup of
A4⋊Dic10 Dic5×S4 D5×A4⋊C4 D10⋊S4 C20.1S4 C4×C5⋊S4 C24⋊2D15
A4⋊Dic5 is a maximal quotient of C20.S4 Q8⋊Dic15 C5⋊2U2(𝔽3)
Matrix representation of A4⋊Dic5 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 50 | 11 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 11 | 50 | 60 |
| 0 | 0 | 56 | 55 | 11 |
| 44 | 60 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 4 | 57 | 0 | 0 | 0 |
| 50 | 57 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 6 | 5 | 11 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,50,0,0,0,60,11,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,11,56,0,0,0,50,55,0,0,1,60,11],[44,1,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[4,50,0,0,0,57,57,0,0,0,0,0,0,60,6,0,0,1,0,5,0,0,0,0,11] >;
A4⋊Dic5 in GAP, Magma, Sage, TeX
A_4\rtimes {\rm Dic}_5 % in TeX
G:=Group("A4:Dic5"); // GroupNames label
G:=SmallGroup(240,107);
// by ID
G=gap.SmallGroup(240,107);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,2,12,146,1155,3604,916,2165,1637]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^10=1,e^2=d^5,c*a*c^-1=e*a*e^-1=a*b=b*a,a*d=d*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of A4⋊Dic5 in TeX
Character table of A4⋊Dic5 in TeX