Aliases: D20.A4, SL2(𝔽3).12D10, C4.A4⋊4D5, (Q8×D5)⋊2C6, C4.3(D5×A4), C20.3(C2×A4), C5⋊2(D4.A4), Q8.5(C6×D5), D4.10D10⋊C3, D10.2(C2×A4), C10.9(C22×A4), (D5×SL2(𝔽3))⋊5C2, (C5×SL2(𝔽3)).12C22, C4○D4⋊(C3×D5), C2.10(C2×D5×A4), (C5×C4○D4)⋊2C6, (C5×C4.A4)⋊4C2, (C5×Q8).5(C2×C6), SmallGroup(480,1043)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — C10 — C5×Q8 — C5×SL2(𝔽3) — D5×SL2(𝔽3) — D20.A4 |
| C5×Q8 — D20.A4 |
Generators and relations for D20.A4
G = < a,b,c,d,e | a20=b2=e3=1, c2=d2=a10, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a10c, ece-1=a10cd, ede-1=c >
Subgroups: 590 in 92 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C2×C4, D4, Q8, Q8, D5, C10, C10, C12, C2×C6, C15, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, SL2(𝔽3), C3×D4, C3×D5, C30, 2- 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C2×SL2(𝔽3), C4.A4, C60, C6×D5, C2×Dic10, C4○D20, D4⋊2D5, Q8×D5, C5×C4○D4, D4.A4, C5×SL2(𝔽3), C3×D20, D4.10D10, D5×SL2(𝔽3), C5×C4.A4, D20.A4
Quotients: C1, C2, C3, C22, C6, D5, A4, C2×C6, D10, C2×A4, C3×D5, C22×A4, C6×D5, D4.A4, D5×A4, C2×D5×A4, D20.A4
►On 80 points
Generators in S80
(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)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)
(1 23 11 33)(2 24 12 34)(3 25 13 35)(4 26 14 36)(5 27 15 37)(6 28 16 38)(7 29 17 39)(8 30 18 40)(9 31 19 21)(10 32 20 22)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)
(1 43 11 53)(2 44 12 54)(3 45 13 55)(4 46 14 56)(5 47 15 57)(6 48 16 58)(7 49 17 59)(8 50 18 60)(9 51 19 41)(10 52 20 42)(21 61 31 71)(22 62 32 72)(23 63 33 73)(24 64 34 74)(25 65 35 75)(26 66 36 76)(27 67 37 77)(28 68 38 78)(29 69 39 79)(30 70 40 80)
(21 41 71)(22 42 72)(23 43 73)(24 44 74)(25 45 75)(26 46 76)(27 47 77)(28 48 78)(29 49 79)(30 50 80)(31 51 61)(32 52 62)(33 53 63)(34 54 64)(35 55 65)(36 56 66)(37 57 67)(38 58 68)(39 59 69)(40 60 70)
G:=sub<Sym(80)| (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71), (1,23,11,33)(2,24,12,34)(3,25,13,35)(4,26,14,36)(5,27,15,37)(6,28,16,38)(7,29,17,39)(8,30,18,40)(9,31,19,21)(10,32,20,22)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (1,43,11,53)(2,44,12,54)(3,45,13,55)(4,46,14,56)(5,47,15,57)(6,48,16,58)(7,49,17,59)(8,50,18,60)(9,51,19,41)(10,52,20,42)(21,61,31,71)(22,62,32,72)(23,63,33,73)(24,64,34,74)(25,65,35,75)(26,66,36,76)(27,67,37,77)(28,68,38,78)(29,69,39,79)(30,70,40,80), (21,41,71)(22,42,72)(23,43,73)(24,44,74)(25,45,75)(26,46,76)(27,47,77)(28,48,78)(29,49,79)(30,50,80)(31,51,61)(32,52,62)(33,53,63)(34,54,64)(35,55,65)(36,56,66)(37,57,67)(38,58,68)(39,59,69)(40,60,70)>;
G:=Group( (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71), (1,23,11,33)(2,24,12,34)(3,25,13,35)(4,26,14,36)(5,27,15,37)(6,28,16,38)(7,29,17,39)(8,30,18,40)(9,31,19,21)(10,32,20,22)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (1,43,11,53)(2,44,12,54)(3,45,13,55)(4,46,14,56)(5,47,15,57)(6,48,16,58)(7,49,17,59)(8,50,18,60)(9,51,19,41)(10,52,20,42)(21,61,31,71)(22,62,32,72)(23,63,33,73)(24,64,34,74)(25,65,35,75)(26,66,36,76)(27,67,37,77)(28,68,38,78)(29,69,39,79)(30,70,40,80), (21,41,71)(22,42,72)(23,43,73)(24,44,74)(25,45,75)(26,46,76)(27,47,77)(28,48,78)(29,49,79)(30,50,80)(31,51,61)(32,52,62)(33,53,63)(34,54,64)(35,55,65)(36,56,66)(37,57,67)(38,58,68)(39,59,69)(40,60,70) );
G=PermutationGroup([[(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),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71)], [(1,23,11,33),(2,24,12,34),(3,25,13,35),(4,26,14,36),(5,27,15,37),(6,28,16,38),(7,29,17,39),(8,30,18,40),(9,31,19,21),(10,32,20,22),(41,71,51,61),(42,72,52,62),(43,73,53,63),(44,74,54,64),(45,75,55,65),(46,76,56,66),(47,77,57,67),(48,78,58,68),(49,79,59,69),(50,80,60,70)], [(1,43,11,53),(2,44,12,54),(3,45,13,55),(4,46,14,56),(5,47,15,57),(6,48,16,58),(7,49,17,59),(8,50,18,60),(9,51,19,41),(10,52,20,42),(21,61,31,71),(22,62,32,72),(23,63,33,73),(24,64,34,74),(25,65,35,75),(26,66,36,76),(27,67,37,77),(28,68,38,78),(29,69,39,79),(30,70,40,80)], [(21,41,71),(22,42,72),(23,43,73),(24,44,74),(25,45,75),(26,46,76),(27,47,77),(28,48,78),(29,49,79),(30,50,80),(31,51,61),(32,52,62),(33,53,63),(34,54,64),(35,55,65),(36,56,66),(37,57,67),(38,58,68),(39,59,69),(40,60,70)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 30A | 30B | 30C | 30D | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 6 | 10 | 10 | 4 | 4 | 2 | 6 | 30 | 30 | 2 | 2 | 4 | 4 | 40 | 40 | 40 | 40 | 2 | 2 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D5 | D10 | C3×D5 | C6×D5 | A4 | C2×A4 | C2×A4 | D4.A4 | D4.A4 | D20.A4 | D20.A4 | D5×A4 | C2×D5×A4 |
| kernel | D20.A4 | D5×SL2(𝔽3) | C5×C4.A4 | D4.10D10 | Q8×D5 | C5×C4○D4 | C4.A4 | SL2(𝔽3) | C4○D4 | Q8 | D20 | C20 | D10 | C5 | C5 | C1 | C1 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 1 | 2 | 4 | 8 | 2 | 2 |
Matrix representation of D20.A4 ►in GL4(𝔽61) generated by
| 54 | 29 | 0 | 0 |
| 32 | 59 | 0 | 0 |
| 0 | 0 | 54 | 29 |
| 0 | 0 | 32 | 59 |
| 0 | 60 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 48 | 0 | 47 | 0 |
| 0 | 48 | 0 | 47 |
| 47 | 0 | 13 | 0 |
| 0 | 47 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 47 | 0 | 13 | 0 |
| 0 | 47 | 0 | 13 |
G:=sub<GL(4,GF(61))| [54,32,0,0,29,59,0,0,0,0,54,32,0,0,29,59],[0,60,0,0,60,0,0,0,0,0,0,60,0,0,60,0],[0,0,60,0,0,0,0,60,1,0,0,0,0,1,0,0],[48,0,47,0,0,48,0,47,47,0,13,0,0,47,0,13],[1,0,47,0,0,1,0,47,0,0,13,0,0,0,0,13] >;
D20.A4 in GAP, Magma, Sage, TeX
D_{20}.A_4 % in TeX
G:=Group("D20.A4"); // GroupNames label
G:=SmallGroup(480,1043);
// by ID
G=gap.SmallGroup(480,1043);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,2,-5,-2,3389,1688,269,584,123,795,382,8069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=b^2=e^3=1,c^2=d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^10*c,e*c*e^-1=a^10*c*d,e*d*e^-1=c>;
// generators/relations