metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊4D20, Q8⋊4D20, D20⋊15D4, C42⋊2D10, Dic10⋊15D4, M4(2)⋊3D10, C4≀C2⋊1D5, (C5×D4)⋊3D4, (C5×Q8)⋊3D4, C4.9(C2×D20), C8⋊D10⋊8C2, C20⋊4D4⋊6C2, C5⋊2(D4⋊4D4), C4○D4.1D10, C4.125(D4×D5), D20⋊4C4⋊5C2, D4⋊D10⋊1C2, D4⋊8D10⋊1C2, (C4×C20)⋊11C22, C20.337(C2×D4), (C22×D5).2D4, C22.29(D4×D5), C10.27C22≀C2, C20.46D4⋊1C2, (C2×D20)⋊13C22, C4.Dic5⋊4C22, (C2×C20).262C23, C4○D20.11C22, C2.30(C22⋊D20), (C5×M4(2))⋊10C22, (C5×C4≀C2)⋊1C2, (C2×C10).26(C2×D4), (C5×C4○D4).3C22, (C2×C4).109(C22×D5), SmallGroup(320,449)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊4D20
G = < a,b,c,d | a4=b2=c20=d2=1, bab=dad=a-1, ac=ca, cbc-1=a-1b, dbd=ab, dcd=c-1 >
Subgroups: 974 in 168 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×4], C22, C22 [×11], C5, C8 [×2], C2×C4, C2×C4 [×5], D4, D4 [×15], Q8, Q8, C23 [×5], D5 [×4], C10, C10 [×2], C42, M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4 [×8], C4○D4, C4○D4 [×3], Dic5, C20 [×2], C20 [×3], D10 [×10], C2×C10, C2×C10, C4.D4, C4≀C2, C4≀C2, C4⋊1D4, C8⋊C22 [×2], 2+ 1+4, C5⋊2C8, C40, Dic10, C4×D5 [×3], D20, D20 [×10], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C22×D5 [×2], C22×D5 [×3], D4⋊4D4, C40⋊C2, D40, C4.Dic5, D4⋊D5, Q8⋊D5, C4×C20, C5×M4(2), C2×D20 [×2], C2×D20 [×3], C4○D20, C4○D20, D4×D5 [×3], Q8⋊2D5, C5×C4○D4, D20⋊4C4, C20.46D4, C5×C4≀C2, C20⋊4D4, C8⋊D10, D4⋊D10, D4⋊8D10, D4⋊4D20
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, D4⋊4D20
►On 40 points
(1 20 8 12)(2 16 9 13)(3 17 10 14)(4 18 6 15)(5 19 7 11)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)
(1 22)(2 38)(3 34)(4 30)(5 26)(6 40)(7 36)(8 32)(9 28)(10 24)(11 21)(12 37)(13 33)(14 29)(15 25)(16 23)(17 39)(18 35)(19 31)(20 27)
(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 2)(3 5)(7 10)(8 9)(11 17)(12 16)(13 20)(14 19)(15 18)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)
G:=sub<Sym(40)| (1,20,8,12)(2,16,9,13)(3,17,10,14)(4,18,6,15)(5,19,7,11)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30), (1,22)(2,38)(3,34)(4,30)(5,26)(6,40)(7,36)(8,32)(9,28)(10,24)(11,21)(12,37)(13,33)(14,29)(15,25)(16,23)(17,39)(18,35)(19,31)(20,27), (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), (1,2)(3,5)(7,10)(8,9)(11,17)(12,16)(13,20)(14,19)(15,18)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)>;
G:=Group( (1,20,8,12)(2,16,9,13)(3,17,10,14)(4,18,6,15)(5,19,7,11)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30), (1,22)(2,38)(3,34)(4,30)(5,26)(6,40)(7,36)(8,32)(9,28)(10,24)(11,21)(12,37)(13,33)(14,29)(15,25)(16,23)(17,39)(18,35)(19,31)(20,27), (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), (1,2)(3,5)(7,10)(8,9)(11,17)(12,16)(13,20)(14,19)(15,18)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33) );
G=PermutationGroup([(1,20,8,12),(2,16,9,13),(3,17,10,14),(4,18,6,15),(5,19,7,11),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30)], [(1,22),(2,38),(3,34),(4,30),(5,26),(6,40),(7,36),(8,32),(9,28),(10,24),(11,21),(12,37),(13,33),(14,29),(15,25),(16,23),(17,39),(18,35),(19,31),(20,27)], [(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)], [(1,2),(3,5),(7,10),(8,9),(11,17),(12,16),(13,20),(14,19),(15,18),(21,24),(22,23),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33)])
44 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 | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 40A | 40B | 40C | 40D |
| 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 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 4 | 20 | 2 | 2 | 8 | 40 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D20 | D4⋊4D4 | D4×D5 | D4×D5 | D4⋊4D20 |
| kernel | D4⋊4D20 | D20⋊4C4 | C20.46D4 | C5×C4≀C2 | C20⋊4D4 | C8⋊D10 | D4⋊D10 | D4⋊8D10 | Dic10 | D20 | C5×D4 | C5×Q8 | C22×D5 | C4≀C2 | C42 | M4(2) | C4○D4 | D4 | Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of D4⋊4D20 ►in GL4(𝔽41) generated by
| 30 | 9 | 0 | 0 |
| 32 | 11 | 0 | 0 |
| 0 | 0 | 11 | 32 |
| 0 | 0 | 9 | 30 |
| 0 | 0 | 11 | 32 |
| 0 | 0 | 9 | 30 |
| 30 | 9 | 0 | 0 |
| 32 | 11 | 0 | 0 |
| 34 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 27 | 30 |
| 0 | 0 | 11 | 32 |
| 1 | 34 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 11 | 14 |
| 0 | 0 | 9 | 30 |
G:=sub<GL(4,GF(41))| [30,32,0,0,9,11,0,0,0,0,11,9,0,0,32,30],[0,0,30,32,0,0,9,11,11,9,0,0,32,30,0,0],[34,40,0,0,1,0,0,0,0,0,27,11,0,0,30,32],[1,0,0,0,34,40,0,0,0,0,11,9,0,0,14,30] >;
D4⋊4D20 in GAP, Magma, Sage, TeX
D_4\rtimes_4D_{20} % in TeX
G:=Group("D4:4D20"); // GroupNames label
G:=SmallGroup(320,449);
// by ID
G=gap.SmallGroup(320,449);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,570,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations