metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊3D4, Dic5⋊1D4, C23.10D10, (C2×D4)⋊6D5, (D4×C10)⋊4C2, (C2×D20)⋊9C2, C4⋊1(C5⋊D4), C5⋊2(C4⋊1D4), C2.28(D4×D5), (C4×Dic5)⋊6C2, C10.52(C2×D4), (C2×C4).52D10, (C2×C10).55C23, (C2×C20).35C22, C22.62(C22×D5), (C22×C10).22C22, (C2×Dic5).42C22, (C22×D5).12C22, (C2×C5⋊D4)⋊7C2, C2.16(C2×C5⋊D4), SmallGroup(160,161)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊D4
G = < a,b,c | a20=b4=c2=1, bab-1=a9, cac=a-1, cbc=b-1 >
Subgroups: 400 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C42, C2×D4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4⋊1D4, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C4×Dic5, C2×D20, C2×C5⋊D4, D4×C10, C20⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4⋊1D4, C5⋊D4, C22×D5, D4×D5, C2×C5⋊D4, C20⋊D4
►On 80 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)(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 22 78 50)(2 31 79 59)(3 40 80 48)(4 29 61 57)(5 38 62 46)(6 27 63 55)(7 36 64 44)(8 25 65 53)(9 34 66 42)(10 23 67 51)(11 32 68 60)(12 21 69 49)(13 30 70 58)(14 39 71 47)(15 28 72 56)(16 37 73 45)(17 26 74 54)(18 35 75 43)(19 24 76 52)(20 33 77 41)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 80)(77 79)
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), (1,22,78,50)(2,31,79,59)(3,40,80,48)(4,29,61,57)(5,38,62,46)(6,27,63,55)(7,36,64,44)(8,25,65,53)(9,34,66,42)(10,23,67,51)(11,32,68,60)(12,21,69,49)(13,30,70,58)(14,39,71,47)(15,28,72,56)(16,37,73,45)(17,26,74,54)(18,35,75,43)(19,24,76,52)(20,33,77,41), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,80)(77,79)>;
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), (1,22,78,50)(2,31,79,59)(3,40,80,48)(4,29,61,57)(5,38,62,46)(6,27,63,55)(7,36,64,44)(8,25,65,53)(9,34,66,42)(10,23,67,51)(11,32,68,60)(12,21,69,49)(13,30,70,58)(14,39,71,47)(15,28,72,56)(16,37,73,45)(17,26,74,54)(18,35,75,43)(19,24,76,52)(20,33,77,41), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,80)(77,79) );
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)], [(1,22,78,50),(2,31,79,59),(3,40,80,48),(4,29,61,57),(5,38,62,46),(6,27,63,55),(7,36,64,44),(8,25,65,53),(9,34,66,42),(10,23,67,51),(11,32,68,60),(12,21,69,49),(13,30,70,58),(14,39,71,47),(15,28,72,56),(16,37,73,45),(17,26,74,54),(18,35,75,43),(19,24,76,52),(20,33,77,41)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,51),(22,50),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(76,80),(77,79)]])
C20⋊D4 is a maximal subgroup of
C23.2D20 (C2×D4)⋊F5 Dic5.SD16 D20⋊1D4 Dic5.5D8 Dic10⋊2D4 C4⋊C4.D10 D20⋊3D4 Dic5⋊D8 C40⋊5D4 C40⋊11D4 Dic5⋊5SD16 C40⋊15D4 C40⋊9D4 D20⋊18D4 2+ 1+4⋊D5 C42.228D10 Dic10⋊24D4 C42.114D10 C42.116D10 C24⋊4D10 C24.34D10 C24.36D10 C20⋊(C4○D4) Dic10⋊20D4 C10.382+ 1+4 D20⋊19D4 C10.442+ 1+4 C10.472+ 1+4 C10.482+ 1+4 C10.662+ 1+4 C10.672+ 1+4 C10.682+ 1+4 C42.233D10 C42⋊18D10 C42.143D10 D5×C4⋊1D4 C42⋊26D10 Dic10⋊11D4 D4×C5⋊D4 C24.41D10 C10.1462+ 1+4 (C2×C20)⋊17D4 C10.1482+ 1+4 C20⋊D12 C60⋊10D4 Dic15⋊5D4 C60⋊3D4
C20⋊D4 is a maximal quotient of
C24.7D10 C24.13D10 C23⋊2D20 C20⋊5(C4⋊C4) (C2×D20)⋊22C4 (C2×C20).289D4 C42.64D10 C42.214D10 C42.65D10 C20⋊D8 C42.74D10 C20⋊4SD16 C20⋊6SD16 C42.80D10 C20⋊3Q16 C40⋊5D4 C40⋊11D4 C40.22D4 C40.31D4 C40.43D4 C40⋊15D4 C40⋊9D4 C40.26D4 C40.37D4 C40.28D4 C24.19D10 C24.21D10 C20⋊D12 C60⋊10D4 Dic15⋊5D4 C60⋊3D4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | C5⋊D4 | D4×D5 |
| kernel | C20⋊D4 | C4×Dic5 | C2×D20 | C2×C5⋊D4 | D4×C10 | Dic5 | C20 | C2×D4 | C2×C4 | C23 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 4 |
Matrix representation of C20⋊D4 ►in GL4(𝔽41) generated by
| 7 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 17 | 40 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [7,40,0,0,1,0,0,0,0,0,0,40,0,0,1,0],[17,3,0,0,40,24,0,0,0,0,0,1,0,0,40,0],[1,34,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;
C20⋊D4 in GAP, Magma, Sage, TeX
C_{20}\rtimes D_4 % in TeX
G:=Group("C20:D4"); // GroupNames label
G:=SmallGroup(160,161);
// by ID
G=gap.SmallGroup(160,161);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,218,188,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations