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.A4)⋊4C2, (C5×C4○D4)⋊2C6, (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 |
Subgroups: 590 in 92 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, C6 [×3], C2×C4 [×5], D4 [×4], Q8, Q8 [×3], D5 [×2], C10, C10, C12, C2×C6 [×2], C15, C2×Q8 [×3], C4○D4, C4○D4 [×3], Dic5 [×2], C20, C20, D10 [×2], C2×C10, SL2(𝔽3), C3×D4, C3×D5 [×2], C30, 2- (1+4), Dic10 [×3], C4×D5 [×2], D20, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C5×D4, C5×Q8, C2×SL2(𝔽3) [×2], C4.A4, C60, C6×D5 [×2], C2×Dic10, C4○D20, D4⋊2D5 [×2], Q8×D5 [×2], C5×C4○D4, D4.A4, C5×SL2(𝔽3), C3×D20, D4.10D10, D5×SL2(𝔽3) [×2], C5×C4.A4, D20.A4
Quotients:
C1, C2 [×3], C3, C22, C6 [×3], D5, A4, C2×C6, D10, C2×A4 [×3], C3×D5, C22×A4, C6×D5, D4.A4, D5×A4, C2×D5×A4, D20.A4
Generators and relations
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 >
►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 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 65)(62 64)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)
(1 36 11 26)(2 37 12 27)(3 38 13 28)(4 39 14 29)(5 40 15 30)(6 21 16 31)(7 22 17 32)(8 23 18 33)(9 24 19 34)(10 25 20 35)(41 80 51 70)(42 61 52 71)(43 62 53 72)(44 63 54 73)(45 64 55 74)(46 65 56 75)(47 66 57 76)(48 67 58 77)(49 68 59 78)(50 69 60 79)
(1 47 11 57)(2 48 12 58)(3 49 13 59)(4 50 14 60)(5 51 15 41)(6 52 16 42)(7 53 17 43)(8 54 18 44)(9 55 19 45)(10 56 20 46)(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 52 71)(22 53 72)(23 54 73)(24 55 74)(25 56 75)(26 57 76)(27 58 77)(28 59 78)(29 60 79)(30 41 80)(31 42 61)(32 43 62)(33 44 63)(34 45 64)(35 46 65)(36 47 66)(37 48 67)(38 49 68)(39 50 69)(40 51 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)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74), (1,36,11,26)(2,37,12,27)(3,38,13,28)(4,39,14,29)(5,40,15,30)(6,21,16,31)(7,22,17,32)(8,23,18,33)(9,24,19,34)(10,25,20,35)(41,80,51,70)(42,61,52,71)(43,62,53,72)(44,63,54,73)(45,64,55,74)(46,65,56,75)(47,66,57,76)(48,67,58,77)(49,68,59,78)(50,69,60,79), (1,47,11,57)(2,48,12,58)(3,49,13,59)(4,50,14,60)(5,51,15,41)(6,52,16,42)(7,53,17,43)(8,54,18,44)(9,55,19,45)(10,56,20,46)(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), (21,52,71)(22,53,72)(23,54,73)(24,55,74)(25,56,75)(26,57,76)(27,58,77)(28,59,78)(29,60,79)(30,41,80)(31,42,61)(32,43,62)(33,44,63)(34,45,64)(35,46,65)(36,47,66)(37,48,67)(38,49,68)(39,50,69)(40,51,70)>;
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)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,65)(62,64)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74), (1,36,11,26)(2,37,12,27)(3,38,13,28)(4,39,14,29)(5,40,15,30)(6,21,16,31)(7,22,17,32)(8,23,18,33)(9,24,19,34)(10,25,20,35)(41,80,51,70)(42,61,52,71)(43,62,53,72)(44,63,54,73)(45,64,55,74)(46,65,56,75)(47,66,57,76)(48,67,58,77)(49,68,59,78)(50,69,60,79), (1,47,11,57)(2,48,12,58)(3,49,13,59)(4,50,14,60)(5,51,15,41)(6,52,16,42)(7,53,17,43)(8,54,18,44)(9,55,19,45)(10,56,20,46)(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), (21,52,71)(22,53,72)(23,54,73)(24,55,74)(25,56,75)(26,57,76)(27,58,77)(28,59,78)(29,60,79)(30,41,80)(31,42,61)(32,43,62)(33,44,63)(34,45,64)(35,46,65)(36,47,66)(37,48,67)(38,49,68)(39,50,69)(40,51,70) );
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),(21,25),(22,24),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,47),(42,46),(43,45),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(61,65),(62,64),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74)], [(1,36,11,26),(2,37,12,27),(3,38,13,28),(4,39,14,29),(5,40,15,30),(6,21,16,31),(7,22,17,32),(8,23,18,33),(9,24,19,34),(10,25,20,35),(41,80,51,70),(42,61,52,71),(43,62,53,72),(44,63,54,73),(45,64,55,74),(46,65,56,75),(47,66,57,76),(48,67,58,77),(49,68,59,78),(50,69,60,79)], [(1,47,11,57),(2,48,12,58),(3,49,13,59),(4,50,14,60),(5,51,15,41),(6,52,16,42),(7,53,17,43),(8,54,18,44),(9,55,19,45),(10,56,20,46),(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)], [(21,52,71),(22,53,72),(23,54,73),(24,55,74),(25,56,75),(26,57,76),(27,58,77),(28,59,78),(29,60,79),(30,41,80),(31,42,61),(32,43,62),(33,44,63),(34,45,64),(35,46,65),(36,47,66),(37,48,67),(38,49,68),(39,50,69),(40,51,70)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀