non-abelian, soluble, monomial
Aliases: D10⋊2S4, A4⋊2D20, A4⋊C4⋊D5, (C5×A4)⋊3D4, C2.15(D5×S4), C5⋊1(A4⋊D4), C10.15(C2×S4), (C2×A4).7D10, C22⋊(C3⋊D20), (C23×D5)⋊2S3, C23.7(S3×D5), (C22×C10).7D6, (C10×A4).7C22, (C2×C5⋊S4)⋊4C2, (C2×D5×A4)⋊2C2, (C5×A4⋊C4)⋊3C2, (C2×C10)⋊2(C3⋊D4), SmallGroup(480,981)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for A4⋊D20
G = < a,b,c,d,e | a2=b2=c3=d20=e2=1, cac-1=dad-1=eae=ab=ba, cbc-1=a, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 1196 in 124 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, A4, D6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C3⋊D4, S4, C2×A4, C2×A4, C3×D5, D15, C30, C22≀C2, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, A4⋊C4, C2×S4, C22×A4, C5×Dic3, C5×A4, C6×D5, D30, D10⋊C4, C5×C22⋊C4, C2×D20, C2×C5⋊D4, C23×D5, A4⋊D4, C3⋊D20, C5⋊S4, D5×A4, C10×A4, C22⋊D20, C5×A4⋊C4, C2×C5⋊S4, C2×D5×A4, A4⋊D20
Quotients: C1, C2, C22, S3, D4, D5, D6, D10, C3⋊D4, S4, D20, C2×S4, S3×D5, A4⋊D4, C3⋊D20, D5×S4, A4⋊D20
►On 60 points
(1 11)(3 13)(5 15)(7 17)(9 19)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(43 53)(45 55)(47 57)(49 59)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(1 54 21)(2 22 55)(3 56 23)(4 24 57)(5 58 25)(6 26 59)(7 60 27)(8 28 41)(9 42 29)(10 30 43)(11 44 31)(12 32 45)(13 46 33)(14 34 47)(15 48 35)(16 36 49)(17 50 37)(18 38 51)(19 52 39)(20 40 53)
(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 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 46)(42 45)(43 44)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)
G:=sub<Sym(60)| (1,11)(3,13)(5,15)(7,17)(9,19)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(43,53)(45,55)(47,57)(49,59), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,54,21)(2,22,55)(3,56,23)(4,24,57)(5,58,25)(6,26,59)(7,60,27)(8,28,41)(9,42,29)(10,30,43)(11,44,31)(12,32,45)(13,46,33)(14,34,47)(15,48,35)(16,36,49)(17,50,37)(18,38,51)(19,52,39)(20,40,53), (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)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)>;
G:=Group( (1,11)(3,13)(5,15)(7,17)(9,19)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(43,53)(45,55)(47,57)(49,59), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60), (1,54,21)(2,22,55)(3,56,23)(4,24,57)(5,58,25)(6,26,59)(7,60,27)(8,28,41)(9,42,29)(10,30,43)(11,44,31)(12,32,45)(13,46,33)(14,34,47)(15,48,35)(16,36,49)(17,50,37)(18,38,51)(19,52,39)(20,40,53), (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)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54) );
G=PermutationGroup([[(1,11),(3,13),(5,15),(7,17),(9,19),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(43,53),(45,55),(47,57),(49,59)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60)], [(1,54,21),(2,22,55),(3,56,23),(4,24,57),(5,58,25),(6,26,59),(7,60,27),(8,28,41),(9,42,29),(10,30,43),(11,44,31),(12,32,45),(13,46,33),(14,34,47),(15,48,35),(16,36,49),(17,50,37),(18,38,51),(19,52,39),(20,40,53)], [(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),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(41,46),(42,45),(43,44),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54)]])
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 10C | 10D | 10E | 10F | 15A | 15B | 20A | ··· | 20H | 30A | 30B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 15 | 15 | 20 | ··· | 20 | 30 | 30 |
| size | 1 | 1 | 3 | 3 | 10 | 30 | 60 | 8 | 12 | 12 | 60 | 2 | 2 | 8 | 40 | 40 | 2 | 2 | 6 | 6 | 6 | 6 | 16 | 16 | 12 | ··· | 12 | 16 | 16 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D10 | C3⋊D4 | D20 | S4 | C2×S4 | S3×D5 | C3⋊D20 | A4⋊D4 | D5×S4 | A4⋊D20 |
| kernel | A4⋊D20 | C5×A4⋊C4 | C2×C5⋊S4 | C2×D5×A4 | C23×D5 | C5×A4 | A4⋊C4 | C22×C10 | C2×A4 | C2×C10 | A4 | D10 | C10 | C23 | C22 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 1 | 4 | 4 |
Matrix representation of A4⋊D20 ►in 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 | 0 | 1 | 60 |
| 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 46 | 52 | 0 | 0 | 0 |
| 9 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 0 | 60 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 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],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[46,9,0,0,0,52,46,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,60],[0,60,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;
A4⋊D20 in GAP, Magma, Sage, TeX
A_4\rtimes D_{20} % in TeX
G:=Group("A4:D20"); // GroupNames label
G:=SmallGroup(480,981);
// by ID
G=gap.SmallGroup(480,981);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,85,36,234,3364,5052,1286,2953,2232]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^20=e^2=1,c*a*c^-1=d*a*d^-1=e*a*e=a*b=b*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations