metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊1D4, Dic10⋊1D4, C23.7D20, M4(2)⋊2D10, C4.79(D4×D5), C20⋊D4⋊1C2, C8⋊D10⋊5C2, C5⋊1(D4⋊4D4), C20.92(C2×D4), C4.D4⋊2D5, D20⋊7C4⋊2C2, D4⋊6D10⋊2C2, (C2×D4).14D10, (C2×C20).4C23, C10.16C22≀C2, (C2×D20)⋊11C22, (C4×Dic5)⋊2C22, C4○D20.2C22, (C22×C10).20D4, C22.11(C2×D20), (D4×C10).14C22, (C5×M4(2))⋊9C22, C2.19(C22⋊D20), (C5×C4.D4)⋊4C2, (C2×C10).21(C2×D4), (C2×C4).4(C22×D5), SmallGroup(320,374)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊1D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a13b, dbd=a3b, dcd=c-1 >
Subgroups: 942 in 168 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×4], C22, C22 [×11], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×16], Q8 [×2], C23 [×2], C23 [×3], D5 [×3], C10, C10 [×3], C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×D4 [×7], C4○D4 [×4], Dic5 [×4], C20 [×2], D10 [×7], C2×C10, C2×C10 [×4], C4.D4, C4≀C2 [×2], C4⋊1D4, C8⋊C22 [×2], 2+ 1+4, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], D20 [×2], C2×Dic5 [×3], C5⋊D4 [×10], C2×C20, C5×D4 [×2], C22×D5 [×3], C22×C10 [×2], D4⋊4D4, C40⋊C2 [×2], D40 [×2], C4×Dic5, C5×M4(2) [×2], C2×D20, C4○D20 [×2], D4×D5 [×2], D4⋊2D5 [×2], C2×C5⋊D4 [×4], D4×C10, D20⋊7C4 [×2], C5×C4.D4, C8⋊D10 [×2], C20⋊D4, 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⋊4D4, C2×D20, D4×D5 [×2], C22⋊D20, D20⋊1D4
►On 40 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)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)
(1 6 11 16)(2 15 12 5)(3 4 13 14)(7 20 17 10)(8 9 18 19)(22 30)(23 39)(24 28)(25 37)(27 35)(29 33)(32 40)(34 38)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(37 40)(38 39)
G:=sub<Sym(40)| (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31), (1,6,11,16)(2,15,12,5)(3,4,13,14)(7,20,17,10)(8,9,18,19)(22,30)(23,39)(24,28)(25,37)(27,35)(29,33)(32,40)(34,38), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31), (1,6,11,16)(2,15,12,5)(3,4,13,14)(7,20,17,10)(8,9,18,19)(22,30)(23,39)(24,28)(25,37)(27,35)(29,33)(32,40)(34,38), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(37,40)(38,39) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31)], [(1,6,11,16),(2,15,12,5),(3,4,13,14),(7,20,17,10),(8,9,18,19),(22,30),(23,39),(24,28),(25,37),(27,35),(29,33),(32,40),(34,38)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(37,40),(38,39)])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 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 | 2 | 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 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 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⋊4D4 | D4×D5 | D20⋊1D4 |
| kernel | D20⋊1D4 | D20⋊7C4 | C5×C4.D4 | C8⋊D10 | C20⋊D4 | 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 GL6(𝔽41)
| 6 | 40 | 0 | 0 | 0 | 0 |
| 36 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 28 | 40 | 5 |
| 0 | 0 | 0 | 3 | 16 | 1 |
| 2 | 11 | 0 | 0 | 0 | 0 |
| 37 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 28 | 28 | 40 | 5 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 32 | 32 | 0 | 13 |
| 35 | 34 | 0 | 0 | 0 | 0 |
| 5 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 38 | 0 | 40 |
| 35 | 34 | 0 | 0 | 0 | 0 |
| 5 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 28 | 40 | 5 |
| 0 | 0 | 3 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [6,36,0,0,0,0,40,1,0,0,0,0,0,0,0,1,28,0,0,0,40,0,28,3,0,0,0,0,40,16,0,0,0,0,5,1],[2,37,0,0,0,0,11,39,0,0,0,0,0,0,0,28,1,32,0,0,0,28,0,32,0,0,1,40,0,0,0,0,0,5,0,13],[35,5,0,0,0,0,34,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,38,0,0,0,0,40,0,0,0,0,0,0,40],[35,5,0,0,0,0,34,6,0,0,0,0,0,0,40,0,28,3,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,5,1] >;
D20⋊1D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_1D_4 % in TeX
G:=Group("D20:1D4"); // GroupNames label
G:=SmallGroup(320,374);
// by ID
G=gap.SmallGroup(320,374);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,226,1123,570,136,1684,438,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^13*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations