direct product, metabelian, soluble, monomial
Aliases: A4×D20, C4⋊(D5×A4), C5⋊1(D4×A4), C20⋊1(C2×A4), (C5×A4)⋊5D4, (C4×A4)⋊3D5, (C22×D20)⋊C3, C22⋊(C3×D20), (A4×C20)⋊3C2, D10⋊1(C2×A4), (C22×C20)⋊1C6, (C23×D5)⋊1C6, (C2×A4).15D10, C10.3(C22×A4), C23.12(C6×D5), (C10×A4).15C22, (C2×D5×A4)⋊4C2, C2.4(C2×D5×A4), (C22×C4)⋊(C3×D5), (C2×C10)⋊1(C3×D4), (C22×C10).3(C2×C6), SmallGroup(480,1037)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1080 in 132 conjugacy classes, 27 normal (21 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22, C22 [×14], C5, C6 [×3], C2×C4 [×2], D4 [×6], C23, C23 [×8], D5 [×4], C10, C10 [×2], C12, A4, C2×C6 [×2], C15, C22×C4, C2×D4 [×4], C24 [×2], C20, C20, D10 [×2], D10 [×10], C2×C10, C2×C10 [×2], C3×D4, C2×A4, C2×A4 [×2], C3×D5 [×2], C30, C22×D4, D20, D20 [×5], C2×C20 [×2], C22×D5 [×8], C22×C10, C4×A4, C22×A4 [×2], C60, C5×A4, C6×D5 [×2], C2×D20 [×4], C22×C20, C23×D5 [×2], D4×A4, C3×D20, D5×A4 [×2], C10×A4, C22×D20, A4×C20, C2×D5×A4 [×2], A4×D20
Quotients:
C1, C2 [×3], C3, C22, C6 [×3], D4, D5, A4, C2×C6, D10, C3×D4, C2×A4 [×3], C3×D5, D20, C22×A4, C6×D5, D4×A4, C3×D20, D5×A4, C2×D5×A4, A4×D20
Generators and relations
G = < a,b,c,d,e | a2=b2=c3=d20=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
►On 60 points
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(1 37 60)(2 38 41)(3 39 42)(4 40 43)(5 21 44)(6 22 45)(7 23 46)(8 24 47)(9 25 48)(10 26 49)(11 27 50)(12 28 51)(13 29 52)(14 30 53)(15 31 54)(16 32 55)(17 33 56)(18 34 57)(19 35 58)(20 36 59)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(33 40)(34 39)(35 38)(36 37)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)
G:=sub<Sym(60)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,37,60)(2,38,41)(3,39,42)(4,40,43)(5,21,44)(6,22,45)(7,23,46)(8,24,47)(9,25,48)(10,26,49)(11,27,50)(12,28,51)(13,29,52)(14,30,53)(15,31,54)(16,32,55)(17,33,56)(18,34,57)(19,35,58)(20,36,59), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,37,60)(2,38,41)(3,39,42)(4,40,43)(5,21,44)(6,22,45)(7,23,46)(8,24,47)(9,25,48)(10,26,49)(11,27,50)(12,28,51)(13,29,52)(14,30,53)(15,31,54)(16,32,55)(17,33,56)(18,34,57)(19,35,58)(20,36,59), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60) );
G=PermutationGroup([(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(1,37,60),(2,38,41),(3,39,42),(4,40,43),(5,21,44),(6,22,45),(7,23,46),(8,24,47),(9,25,48),(10,26,49),(11,27,50),(12,28,51),(13,29,52),(14,30,53),(15,31,54),(16,32,55),(17,33,56),(18,34,57),(19,35,58),(20,36,59)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(33,40),(34,39),(35,38),(36,37),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60)])
Matrix representation ►G ⊆ GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 1 | 60 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 | 1 |
| 0 | 0 | 60 | 1 | 0 |
| 13 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 26 | 22 | 0 | 0 | 0 |
| 39 | 26 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 39 | 26 | 0 | 0 | 0 |
| 26 | 22 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,1,0],[13,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[26,39,0,0,0,22,26,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[39,26,0,0,0,26,22,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 3 | 3 | 10 | 10 | 30 | 30 | 4 | 4 | 2 | 6 | 2 | 2 | 4 | 4 | 40 | 40 | 40 | 40 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D5 | D10 | C3×D4 | C3×D5 | D20 | C6×D5 | C3×D20 | A4 | C2×A4 | C2×A4 | D4×A4 | D5×A4 | C2×D5×A4 | A4×D20 |
| kernel | A4×D20 | A4×C20 | C2×D5×A4 | C22×D20 | C22×C20 | C23×D5 | C5×A4 | C4×A4 | C2×A4 | C2×C10 | C22×C4 | A4 | C23 | C22 | D20 | C20 | D10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 1 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
A_4\times D_{20} % in TeX
G:=Group("A4xD20"); // GroupNames label
G:=SmallGroup(480,1037);
// by ID
G=gap.SmallGroup(480,1037);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,197,92,648,271,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^20=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations