metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊4C4, C42⋊3D5, C4.17D20, C20.33D4, Dic10⋊4C4, C5⋊3C4≀C2, (C4×C20)⋊6C2, C4.6(C4×D5), C20.37(C2×C4), C4○D20.1C2, (C2×C10).26D4, (C2×C4).66D10, C4.Dic5⋊1C2, (C2×C20).96C22, C22.7(C5⋊D4), C2.3(D10⋊C4), C10.12(C22⋊C4), SmallGroup(160,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊4C4
G = < a,b,c | a20=b2=c4=1, bab=a-1, ac=ca, cbc-1=a15b >
Smallest permutation representation of D20⋊4C4
►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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 40)(17 39)(18 38)(19 37)(20 36)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)
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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,40)(17,39)(18,38)(19,37)(20,36), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)>;
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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,40)(17,39)(18,38)(19,37)(20,36), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40) );
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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,40),(17,39),(18,38),(19,37),(20,36)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,26,31,36),(22,27,32,37),(23,28,33,38),(24,29,34,39),(25,30,35,40)]])
D20⋊4C4 is a maximal subgroup of
D40⋊17C4 D40⋊10C4 D5×C4≀C2 C42⋊D10 D4⋊4D20 M4(2).22D10 C42.196D10 M4(2)⋊D10 D4.9D20 D4.10D20 C42⋊4D10 C42⋊5D10 D20.14D4 D20⋊5D4 D20.15D4 C60.97D4 D60⋊13C4 D60⋊7C4 C42⋊D15
D20⋊4C4 is a maximal quotient of
(C2×D20)⋊C4 C4⋊Dic5⋊C4 Dic10⋊3C8 D20⋊3C8 C42.D10 C42.2D10 C42⋊6Dic5 C60.97D4 D60⋊13C4 D60⋊7C4
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | ··· | 4G | 4H | 5A | 5B | 8A | 8B | 10A | ··· | 10F | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 20 | 1 | 1 | 2 | ··· | 2 | 20 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | D10 | C4≀C2 | C4×D5 | D20 | C5⋊D4 | D20⋊4C4 |
| kernel | D20⋊4C4 | C4.Dic5 | C4×C20 | C4○D20 | Dic10 | D20 | C20 | C2×C10 | C42 | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 16 |
Matrix representation of D20⋊4C4 ►in GL2(𝔽41) generated by
| 5 | 0 |
| 0 | 33 |
| 0 | 33 |
| 5 | 0 |
| 40 | 0 |
| 0 | 32 |
G:=sub<GL(2,GF(41))| [5,0,0,33],[0,5,33,0],[40,0,0,32] >;
D20⋊4C4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_4C_4 % in TeX
G:=Group("D20:4C4"); // GroupNames label
G:=SmallGroup(160,12);
// by ID
G=gap.SmallGroup(160,12);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,362,579,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations
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