metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊4D4, C22⋊2D20, C23.15D10, C5⋊1C22≀C2, (C2×C10)⋊1D4, (C2×C4)⋊1D10, C2.7(D4×D5), (C2×D20)⋊2C2, C22⋊C4⋊2D5, C2.7(C2×D20), C10.5(C2×D4), (C2×C20)⋊1C22, (C23×D5)⋊1C2, D10⋊C4⋊4C2, (C2×C10).23C23, (C2×Dic5)⋊1C22, (C22×D5)⋊1C22, C22.41(C22×D5), (C22×C10).12C22, (C2×C5⋊D4)⋊1C2, (C5×C22⋊C4)⋊3C2, SmallGroup(160,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊D20
G = < a,b,c,d | a2=b2=c20=d2=1, cac-1=dad=ab=ba, bc=cb, bd=db, dcd=c-1 >
Subgroups: 544 in 130 conjugacy classes, 37 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22≀C2, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×D5, C22×C10, D10⋊C4, C5×C22⋊C4, C2×D20, C2×C5⋊D4, C23×D5, C22⋊D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, C2×D20, D4×D5, C22⋊D20
►On 40 points
(2 36)(4 38)(6 40)(8 22)(10 24)(12 26)(14 28)(16 30)(18 32)(20 34)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)
(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 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)
G:=sub<Sym(40)| (2,36)(4,38)(6,40)(8,22)(10,24)(12,26)(14,28)(16,30)(18,32)(20,34), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34), (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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)>;
G:=Group( (2,36)(4,38)(6,40)(8,22)(10,24)(12,26)(14,28)(16,30)(18,32)(20,34), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34), (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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35) );
G=PermutationGroup([[(2,36),(4,38),(6,40),(8,22),(10,24),(12,26),(14,28),(16,30),(18,32),(20,34)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34)], [(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)], [(1,34),(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,40),(16,39),(17,38),(18,37),(19,36),(20,35)]])
C22⋊D20 is a maximal subgroup of
C5⋊C2≀C4 C23⋊D20 C24.27D10 C23⋊3D20 C42⋊8D10 C42⋊9D10 C42⋊10D10 C42⋊12D10 D4×D20 D4⋊5D20 C42⋊17D10 D5×C22≀C2 C24⋊4D10 C24.34D10 C10.372+ 1+4 C10.382+ 1+4 D20⋊19D4 C10.482+ 1+4 C4⋊C4⋊26D10 D20⋊21D4 C10.532+ 1+4 C10.562+ 1+4 C10.1202+ 1+4 C10.1212+ 1+4 C4⋊C4⋊28D10 C10.612+ 1+4 C10.682+ 1+4 C42⋊18D10 D20⋊10D4 C42⋊20D10 C42⋊22D10 C42⋊23D10 C42⋊24D10 C42⋊25D10 D6⋊4D20 D30⋊5D4 (C2×C6)⋊8D20 D30⋊18D4 D30⋊16D4 A4⋊D20
C22⋊D20 is a maximal quotient of
(C2×Dic5)⋊Q8 (C2×C4)⋊9D20 D10⋊3(C4⋊C4) (C2×C20)⋊5D4 (C2×Dic5)⋊3D4 (C2×C4).20D20 D20.31D4 D20⋊13D4 D20.32D4 D20⋊14D4 Dic10⋊14D4 C22⋊Dic20 C23⋊D20 C23.5D20 D20.1D4 D20⋊1D4 D20.4D4 D20.5D4 D4⋊D20 D20.8D4 D4⋊3D20 D4.D20 Q8⋊2D20 D10⋊4Q16 Q8.D20 D20⋊4D4 D4⋊4D20 M4(2)⋊D10 D4.9D20 D4.10D20 C24.47D10 C24.48D10 C23.45D20 C23⋊2D20 C24.16D10 D6⋊4D20 D30⋊5D4 (C2×C6)⋊8D20 D30⋊18D4 D30⋊16D4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 4 | 4 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D20 | D4×D5 |
| kernel | C22⋊D20 | D10⋊C4 | C5×C22⋊C4 | C2×D20 | C2×C5⋊D4 | C23×D5 | D10 | C2×C10 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 8 | 4 |
Matrix representation of C22⋊D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 21 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 9 | 11 | 0 | 0 |
| 30 | 14 | 0 | 0 |
| 0 | 0 | 1 | 37 |
| 0 | 0 | 0 | 40 |
| 32 | 30 | 0 | 0 |
| 11 | 9 | 0 | 0 |
| 0 | 0 | 40 | 4 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,21,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,30,0,0,11,14,0,0,0,0,1,0,0,0,37,40],[32,11,0,0,30,9,0,0,0,0,40,0,0,0,4,1] >;
C22⋊D20 in GAP, Magma, Sage, TeX
C_2^2\rtimes D_{20} % in TeX
G:=Group("C2^2:D20"); // GroupNames label
G:=SmallGroup(160,103);
// by ID
G=gap.SmallGroup(160,103);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,188,50,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^20=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations