metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.1D4, C23.6D20, M4(2)⋊1D10, Dic10.1D4, C4.78(D4×D5), C4.D4⋊1D5, C20.91(C2×D4), D20⋊7C4⋊1C2, (C2×D4).13D10, C8.D10⋊5C2, C5⋊1(D4.9D4), (C2×C20).3C23, C10.15C22≀C2, D4⋊6D10.2C2, C20.17D4⋊1C2, (C4×Dic5)⋊1C22, C4○D20.1C22, C22.10(C2×D20), (C22×C10).19D4, (D4×C10).13C22, (C5×M4(2))⋊8C22, C2.18(C22⋊D20), (C2×Dic10)⋊12C22, (C5×C4.D4)⋊3C2, (C2×C10).20(C2×D4), (C2×C4).3(C22×D5), SmallGroup(320,373)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.1D4
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=a5b, dbd-1=a15b, dcd-1=a15c3 >
Subgroups: 750 in 152 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×6], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D5 [×2], C10, C10 [×3], C42, C22⋊C4 [×2], M4(2) [×2], SD16 [×2], Q16 [×2], C2×D4, C2×D4 [×4], C2×Q8, C4○D4 [×4], Dic5 [×4], C20 [×2], D10 [×4], C2×C10, C2×C10 [×4], C4.D4, C4≀C2 [×2], C4.4D4, C8.C22 [×2], 2+ 1+4, C40 [×2], Dic10 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×4], C5⋊D4 [×6], C2×C20, C5×D4 [×2], C22×D5 [×2], C22×C10 [×2], D4.9D4, C40⋊C2 [×2], Dic20 [×2], C4×Dic5, C23.D5 [×2], C5×M4(2) [×2], C2×Dic10, C4○D20 [×2], D4×D5 [×2], D4⋊2D5 [×2], C2×C5⋊D4 [×2], D4×C10, D20⋊7C4 [×2], C5×C4.D4, C8.D10 [×2], C20.17D4, D4⋊6D10, D20.1D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, D4.9D4, C2×D20, D4×D5 [×2], C22⋊D20, D20.1D4
►On 80 points
(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 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)
(1 28 65 51 11 38 75 41)(2 39 66 42 12 29 76 52)(3 30 67 53 13 40 77 43)(4 21 68 44 14 31 78 54)(5 32 69 55 15 22 79 45)(6 23 70 46 16 33 80 56)(7 34 71 57 17 24 61 47)(8 25 72 48 18 35 62 58)(9 36 73 59 19 26 63 49)(10 27 74 50 20 37 64 60)
(1 51 6 56 11 41 16 46)(2 52 7 57 12 42 17 47)(3 53 8 58 13 43 18 48)(4 54 9 59 14 44 19 49)(5 55 10 60 15 45 20 50)(21 63 26 68 31 73 36 78)(22 64 27 69 32 74 37 79)(23 65 28 70 33 75 38 80)(24 66 29 71 34 76 39 61)(25 67 30 72 35 77 40 62)
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)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73), (1,28,65,51,11,38,75,41)(2,39,66,42,12,29,76,52)(3,30,67,53,13,40,77,43)(4,21,68,44,14,31,78,54)(5,32,69,55,15,22,79,45)(6,23,70,46,16,33,80,56)(7,34,71,57,17,24,61,47)(8,25,72,48,18,35,62,58)(9,36,73,59,19,26,63,49)(10,27,74,50,20,37,64,60), (1,51,6,56,11,41,16,46)(2,52,7,57,12,42,17,47)(3,53,8,58,13,43,18,48)(4,54,9,59,14,44,19,49)(5,55,10,60,15,45,20,50)(21,63,26,68,31,73,36,78)(22,64,27,69,32,74,37,79)(23,65,28,70,33,75,38,80)(24,66,29,71,34,76,39,61)(25,67,30,72,35,77,40,62)>;
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)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73), (1,28,65,51,11,38,75,41)(2,39,66,42,12,29,76,52)(3,30,67,53,13,40,77,43)(4,21,68,44,14,31,78,54)(5,32,69,55,15,22,79,45)(6,23,70,46,16,33,80,56)(7,34,71,57,17,24,61,47)(8,25,72,48,18,35,62,58)(9,36,73,59,19,26,63,49)(10,27,74,50,20,37,64,60), (1,51,6,56,11,41,16,46)(2,52,7,57,12,42,17,47)(3,53,8,58,13,43,18,48)(4,54,9,59,14,44,19,49)(5,55,10,60,15,45,20,50)(21,63,26,68,31,73,36,78)(22,64,27,69,32,74,37,79)(23,65,28,70,33,75,38,80)(24,66,29,71,34,76,39,61)(25,67,30,72,35,77,40,62) );
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),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,63),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73)], [(1,28,65,51,11,38,75,41),(2,39,66,42,12,29,76,52),(3,30,67,53,13,40,77,43),(4,21,68,44,14,31,78,54),(5,32,69,55,15,22,79,45),(6,23,70,46,16,33,80,56),(7,34,71,57,17,24,61,47),(8,25,72,48,18,35,62,58),(9,36,73,59,19,26,63,49),(10,27,74,50,20,37,64,60)], [(1,51,6,56,11,41,16,46),(2,52,7,57,12,42,17,47),(3,53,8,58,13,43,18,48),(4,54,9,59,14,44,19,49),(5,55,10,60,15,45,20,50),(21,63,26,68,31,73,36,78),(22,64,27,69,32,74,37,79),(23,65,28,70,33,75,38,80),(24,66,29,71,34,76,39,61),(25,67,30,72,35,77,40,62)])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 40 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D20 | D4.9D4 | D4×D5 | D20.1D4 |
| kernel | D20.1D4 | D20⋊7C4 | C5×C4.D4 | C8.D10 | C20.17D4 | D4⋊6D10 | Dic10 | D20 | C22×C10 | C4.D4 | M4(2) | C2×D4 | C23 | C5 | C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 8 | 2 | 4 | 2 |
Matrix representation of D20.1D4 ►in GL8(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 36 | 17 | 8 |
| 0 | 0 | 0 | 0 | 26 | 7 | 25 | 38 |
| 0 | 0 | 0 | 0 | 7 | 5 | 12 | 37 |
| 0 | 0 | 0 | 0 | 31 | 15 | 5 | 29 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 31 | 3 | 40 | 40 | 0 | 0 | 0 | 0 |
| 13 | 28 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 35 | 1 |
| 29 | 25 | 38 | 38 | 0 | 0 | 0 | 0 |
| 16 | 18 | 3 | 0 | 0 | 0 | 0 | 0 |
| 9 | 18 | 39 | 16 | 0 | 0 | 0 | 0 |
| 36 | 32 | 39 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 24 | 36 |
| 0 | 0 | 0 | 0 | 24 | 32 | 4 | 6 |
| 0 | 0 | 0 | 0 | 27 | 2 | 26 | 5 |
| 0 | 0 | 0 | 0 | 23 | 26 | 29 | 15 |
| 29 | 25 | 38 | 38 | 0 | 0 | 0 | 0 |
| 16 | 18 | 3 | 0 | 0 | 0 | 0 | 0 |
| 9 | 5 | 39 | 16 | 0 | 0 | 0 | 0 |
| 8 | 4 | 39 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 35 | 5 |
| 0 | 0 | 0 | 0 | 24 | 32 | 26 | 33 |
| 0 | 0 | 0 | 0 | 27 | 2 | 32 | 0 |
| 0 | 0 | 0 | 0 | 23 | 26 | 28 | 9 |
G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,35,5,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,34,26,7,31,0,0,0,0,36,7,5,15,0,0,0,0,17,25,12,5,0,0,0,0,8,38,37,29],[0,1,31,13,0,0,0,0,1,0,3,28,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,0,40,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,35,0,0,0,0,0,0,0,1],[29,16,9,36,0,0,0,0,25,18,18,32,0,0,0,0,38,3,39,39,0,0,0,0,38,0,16,37,0,0,0,0,0,0,0,0,9,24,27,23,0,0,0,0,0,32,2,26,0,0,0,0,24,4,26,29,0,0,0,0,36,6,5,15],[29,16,9,8,0,0,0,0,25,18,5,4,0,0,0,0,38,3,39,39,0,0,0,0,38,0,16,37,0,0,0,0,0,0,0,0,9,24,27,23,0,0,0,0,0,32,2,26,0,0,0,0,35,26,32,28,0,0,0,0,5,33,0,9] >;
D20.1D4 in GAP, Magma, Sage, TeX
D_{20}._1D_4 % in TeX
G:=Group("D20.1D4"); // GroupNames label
G:=SmallGroup(320,373);
// by ID
G=gap.SmallGroup(320,373);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,1123,570,136,1684,438,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^5*b,d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^3>;
// generators/relations