Aliases: C20⋊4D4⋊C3, (C4×C20)⋊3C6, C42⋊C3⋊3D5, C5⋊(C23.A4), C42⋊2(C3×D5), C22.1(D5×A4), (C22×D5).1A4, (C5×C42⋊C3)⋊3C2, (C2×C10).1(C2×A4), SmallGroup(480,262)
Series: Derived ►Chief ►Lower central ►Upper central
| C4×C20 — C20⋊4D4⋊C3 |
Generators and relations for C20⋊4D4⋊C3
G = < a,b,c,d | a4=b20=c2=d3=1, ab=ba, cac=a-1, dad-1=a-1b15, cbc=b-1, dbd-1=ab16, dcd-1=ab15c >
Subgroups: 648 in 56 conjugacy classes, 11 normal (all characteristic)
C1, C2, C3, C4, C22, C22, C5, C6, C2×C4, D4, C23, D5, C10, A4, C15, C42, C2×D4, C20, D10, C2×C10, C2×A4, C3×D5, C4⋊1D4, D20, C2×C20, C22×D5, C22×D5, C42⋊C3, C5×A4, C4×C20, C2×D20, C23.A4, D5×A4, C20⋊4D4, C5×C42⋊C3, C20⋊4D4⋊C3
Quotients: C1, C2, C3, C6, D5, A4, C2×A4, C3×D5, C23.A4, D5×A4, C20⋊4D4⋊C3
Character table of C20⋊4D4⋊C3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 10A | 10B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 3 | 20 | 60 | 16 | 16 | 6 | 6 | 2 | 2 | 80 | 80 | 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 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ65 | ζ6 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ6 | ζ65 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | -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 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | -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 |
| ρ9 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | 2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ3ζ54+ζ3ζ5 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ53+ζ3ζ52 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | complex lifted from C3×D5 |
| ρ10 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | 2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ32ζ53+ζ32ζ52 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ54+ζ32ζ5 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | complex lifted from C3×D5 |
| ρ11 | 2 | 2 | 0 | 0 | -1+√-3 | -1-√-3 | 2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ3ζ53+ζ3ζ52 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ54+ζ3ζ5 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | complex lifted from C3×D5 |
| ρ12 | 2 | 2 | 0 | 0 | -1-√-3 | -1+√-3 | 2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ32ζ54+ζ32ζ5 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ53+ζ32ζ52 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | complex lifted from C3×D5 |
| ρ13 | 3 | 3 | 3 | -1 | 0 | 0 | -1 | -1 | 3 | 3 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ14 | 3 | 3 | -3 | 1 | 0 | 0 | -1 | -1 | 3 | 3 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from C2×A4 |
| ρ15 | 6 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 6 | 6 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | orthogonal lifted from C23.A4 |
| ρ16 | 6 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 6 | 6 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | orthogonal lifted from C23.A4 |
| ρ17 | 6 | 6 | 0 | 0 | 0 | 0 | -2 | -2 | -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 D5×A4 |
| ρ18 | 6 | 6 | 0 | 0 | 0 | 0 | -2 | -2 | -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 D5×A4 |
| ρ19 | 6 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -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 | -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | -1+√5/2 | -1-√5/2 | 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 | -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | -1-√5/2 | orthogonal faithful |
| ρ20 | 6 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -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 | -1-√5/2 | -1-√5/2 | -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | orthogonal faithful |
| ρ21 | 6 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -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 | -1-√5/2 | -1-√5/2 | 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 | orthogonal faithful |
| ρ22 | 6 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -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 | 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 | -1-√5/2 | -1+√5/2 | 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 | -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | -1+√5/2 | orthogonal faithful |
| ρ23 | 6 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -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 | -1+√5/2 | -1+√5/2 | -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | orthogonal faithful |
| ρ24 | 6 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -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 | -1+√5/2 | -1+√5/2 | 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 | orthogonal faithful |
| ρ25 | 6 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -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 | 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 | -1+√5/2 | -1-√5/2 | -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | 2ζ4ζ53-2ζ4ζ52-ζ53-ζ52 | -1-√5/2 | orthogonal faithful |
| ρ26 | 6 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -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 | -2ζ4ζ53+2ζ4ζ52-ζ53-ζ52 | -1-√5/2 | -1+√5/2 | -2ζ43ζ54+2ζ43ζ5-ζ54-ζ5 | 2ζ43ζ54-2ζ43ζ5-ζ54-ζ5 | -1+√5/2 | orthogonal faithful |
►On 60 points
Generators in S60
(1 9 16 15)(2 10 17 11)(3 6 18 12)(4 7 19 13)(5 8 20 14)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)
(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 19)(2 18)(3 17)(4 16)(5 20)(6 10)(7 9)(11 12)(13 15)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 60)(56 59)(57 58)
(1 41 21)(2 57 37)(3 53 33)(4 49 29)(5 45 25)(6 58 28)(7 54 24)(8 50 40)(9 46 36)(10 42 32)(11 52 22)(12 48 38)(13 44 34)(14 60 30)(15 56 26)(16 51 31)(17 47 27)(18 43 23)(19 59 39)(20 55 35)
G:=sub<Sym(60)| (1,9,16,15)(2,10,17,11)(3,6,18,12)(4,7,19,13)(5,8,20,14)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50), (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), (1,19)(2,18)(3,17)(4,16)(5,20)(6,10)(7,9)(11,12)(13,15)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58), (1,41,21)(2,57,37)(3,53,33)(4,49,29)(5,45,25)(6,58,28)(7,54,24)(8,50,40)(9,46,36)(10,42,32)(11,52,22)(12,48,38)(13,44,34)(14,60,30)(15,56,26)(16,51,31)(17,47,27)(18,43,23)(19,59,39)(20,55,35)>;
G:=Group( (1,9,16,15)(2,10,17,11)(3,6,18,12)(4,7,19,13)(5,8,20,14)(41,56,51,46)(42,57,52,47)(43,58,53,48)(44,59,54,49)(45,60,55,50), (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), (1,19)(2,18)(3,17)(4,16)(5,20)(6,10)(7,9)(11,12)(13,15)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,60)(56,59)(57,58), (1,41,21)(2,57,37)(3,53,33)(4,49,29)(5,45,25)(6,58,28)(7,54,24)(8,50,40)(9,46,36)(10,42,32)(11,52,22)(12,48,38)(13,44,34)(14,60,30)(15,56,26)(16,51,31)(17,47,27)(18,43,23)(19,59,39)(20,55,35) );
G=PermutationGroup([[(1,9,16,15),(2,10,17,11),(3,6,18,12),(4,7,19,13),(5,8,20,14),(41,56,51,46),(42,57,52,47),(43,58,53,48),(44,59,54,49),(45,60,55,50)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,20),(6,10),(7,9),(11,12),(13,15),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(55,60),(56,59),(57,58)], [(1,41,21),(2,57,37),(3,53,33),(4,49,29),(5,45,25),(6,58,28),(7,54,24),(8,50,40),(9,46,36),(10,42,32),(11,52,22),(12,48,38),(13,44,34),(14,60,30),(15,56,26),(16,51,31),(17,47,27),(18,43,23),(19,59,39),(20,55,35)]])
Matrix representation of C20⋊4D4⋊C3 ►in GL6(𝔽61)
| 36 | 4 | 0 | 0 | 0 | 0 |
| 57 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 57 |
| 0 | 0 | 0 | 0 | 4 | 36 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 36 | 0 | 0 |
| 0 | 0 | 25 | 57 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 25 |
| 0 | 0 | 0 | 0 | 36 | 4 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 18 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 36 | 0 | 0 |
| 0 | 0 | 34 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 25 |
| 0 | 0 | 0 | 0 | 27 | 34 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(61))| [36,57,0,0,0,0,4,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,4,0,0,0,0,57,36],[43,1,0,0,0,0,60,0,0,0,0,0,0,0,34,25,0,0,0,0,36,57,0,0,0,0,0,0,27,36,0,0,0,0,25,4],[43,18,0,0,0,0,60,18,0,0,0,0,0,0,34,34,0,0,0,0,36,27,0,0,0,0,0,0,27,27,0,0,0,0,25,34],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C20⋊4D4⋊C3 in GAP, Magma, Sage, TeX
C_{20}\rtimes_4D_4\rtimes C_3 % in TeX
G:=Group("C20:4D4:C3"); // GroupNames label
G:=SmallGroup(480,262);
// by ID
G=gap.SmallGroup(480,262);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-5,-2,2,7688,198,1276,7059,3454,584,3364,5052,8833]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^20=c^2=d^3=1,a*b=b*a,c*a*c=a^-1,d*a*d^-1=a^-1*b^15,c*b*c=b^-1,d*b*d^-1=a*b^16,d*c*d^-1=a*b^15*c>;
// generators/relations
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