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, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, D4.9D4, C40⋊C2, Dic20, C4×Dic5, C23.D5, C5×M4(2), C2×Dic10, C4○D20, D4×D5, D4⋊2D5, C2×C5⋊D4, D4×C10, D20⋊7C4, C5×C4.D4, C8.D10, C20.17D4, D4⋊6D10, D20.1D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, D4.9D4, C2×D20, D4×D5, C22⋊D20, D20.1D4
►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 22)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)
(1 79 53 22 11 69 43 32)(2 70 54 33 12 80 44 23)(3 61 55 24 13 71 45 34)(4 72 56 35 14 62 46 25)(5 63 57 26 15 73 47 36)(6 74 58 37 16 64 48 27)(7 65 59 28 17 75 49 38)(8 76 60 39 18 66 50 29)(9 67 41 30 19 77 51 40)(10 78 42 21 20 68 52 31)
(1 22 6 27 11 32 16 37)(2 23 7 28 12 33 17 38)(3 24 8 29 13 34 18 39)(4 25 9 30 14 35 19 40)(5 26 10 31 15 36 20 21)(41 67 46 72 51 77 56 62)(42 68 47 73 52 78 57 63)(43 69 48 74 53 79 58 64)(44 70 49 75 54 80 59 65)(45 71 50 76 55 61 60 66)
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)(21,22)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79), (1,79,53,22,11,69,43,32)(2,70,54,33,12,80,44,23)(3,61,55,24,13,71,45,34)(4,72,56,35,14,62,46,25)(5,63,57,26,15,73,47,36)(6,74,58,37,16,64,48,27)(7,65,59,28,17,75,49,38)(8,76,60,39,18,66,50,29)(9,67,41,30,19,77,51,40)(10,78,42,21,20,68,52,31), (1,22,6,27,11,32,16,37)(2,23,7,28,12,33,17,38)(3,24,8,29,13,34,18,39)(4,25,9,30,14,35,19,40)(5,26,10,31,15,36,20,21)(41,67,46,72,51,77,56,62)(42,68,47,73,52,78,57,63)(43,69,48,74,53,79,58,64)(44,70,49,75,54,80,59,65)(45,71,50,76,55,61,60,66)>;
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)(21,22)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,80)(78,79), (1,79,53,22,11,69,43,32)(2,70,54,33,12,80,44,23)(3,61,55,24,13,71,45,34)(4,72,56,35,14,62,46,25)(5,63,57,26,15,73,47,36)(6,74,58,37,16,64,48,27)(7,65,59,28,17,75,49,38)(8,76,60,39,18,66,50,29)(9,67,41,30,19,77,51,40)(10,78,42,21,20,68,52,31), (1,22,6,27,11,32,16,37)(2,23,7,28,12,33,17,38)(3,24,8,29,13,34,18,39)(4,25,9,30,14,35,19,40)(5,26,10,31,15,36,20,21)(41,67,46,72,51,77,56,62)(42,68,47,73,52,78,57,63)(43,69,48,74,53,79,58,64)(44,70,49,75,54,80,59,65)(45,71,50,76,55,61,60,66) );
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),(21,22),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(31,32),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,80),(78,79)], [(1,79,53,22,11,69,43,32),(2,70,54,33,12,80,44,23),(3,61,55,24,13,71,45,34),(4,72,56,35,14,62,46,25),(5,63,57,26,15,73,47,36),(6,74,58,37,16,64,48,27),(7,65,59,28,17,75,49,38),(8,76,60,39,18,66,50,29),(9,67,41,30,19,77,51,40),(10,78,42,21,20,68,52,31)], [(1,22,6,27,11,32,16,37),(2,23,7,28,12,33,17,38),(3,24,8,29,13,34,18,39),(4,25,9,30,14,35,19,40),(5,26,10,31,15,36,20,21),(41,67,46,72,51,77,56,62),(42,68,47,73,52,78,57,63),(43,69,48,74,53,79,58,64),(44,70,49,75,54,80,59,65),(45,71,50,76,55,61,60,66)]])
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