metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊3D4, C23.9D10, (C2×D4)⋊5D5, (C2×C10)⋊3D4, (D4×C10)⋊9C2, C5⋊5(C4⋊D4), C2.27(D4×D5), C10.51(C2×D4), (C2×C4).19D10, C22⋊1(C5⋊D4), C23.D5⋊12C2, D10⋊C4⋊15C2, C10.32(C4○D4), C10.D4⋊15C2, (C2×C10).54C23, (C2×C20).62C22, (C22×Dic5)⋊6C2, C2.18(D4⋊2D5), C22.61(C22×D5), (C22×C10).21C22, (C2×Dic5).41C22, (C22×D5).11C22, (C2×C5⋊D4)⋊6C2, C2.15(C2×C5⋊D4), SmallGroup(160,160)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊D4
G = < a,b,c,d | a10=c4=d2=1, b2=a5, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c-1 >
Subgroups: 304 in 94 conjugacy classes, 35 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C4⋊D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C10.D4, D10⋊C4, C23.D5, C22×Dic5, C2×C5⋊D4, D4×C10, Dic5⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4×D5, D4⋊2D5, C2×C5⋊D4, Dic5⋊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 48 6 43)(2 47 7 42)(3 46 8 41)(4 45 9 50)(5 44 10 49)(11 54 16 59)(12 53 17 58)(13 52 18 57)(14 51 19 56)(15 60 20 55)(21 33 26 38)(22 32 27 37)(23 31 28 36)(24 40 29 35)(25 39 30 34)(61 73 66 78)(62 72 67 77)(63 71 68 76)(64 80 69 75)(65 79 70 74)
(1 14 28 76)(2 15 29 77)(3 16 30 78)(4 17 21 79)(5 18 22 80)(6 19 23 71)(7 20 24 72)(8 11 25 73)(9 12 26 74)(10 13 27 75)(31 63 43 51)(32 64 44 52)(33 65 45 53)(34 66 46 54)(35 67 47 55)(36 68 48 56)(37 69 49 57)(38 70 50 58)(39 61 41 59)(40 62 42 60)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 51)(7 52)(8 53)(9 54)(10 55)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 41)(18 42)(19 43)(20 44)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
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), (1,48,6,43)(2,47,7,42)(3,46,8,41)(4,45,9,50)(5,44,10,49)(11,54,16,59)(12,53,17,58)(13,52,18,57)(14,51,19,56)(15,60,20,55)(21,33,26,38)(22,32,27,37)(23,31,28,36)(24,40,29,35)(25,39,30,34)(61,73,66,78)(62,72,67,77)(63,71,68,76)(64,80,69,75)(65,79,70,74), (1,14,28,76)(2,15,29,77)(3,16,30,78)(4,17,21,79)(5,18,22,80)(6,19,23,71)(7,20,24,72)(8,11,25,73)(9,12,26,74)(10,13,27,75)(31,63,43,51)(32,64,44,52)(33,65,45,53)(34,66,46,54)(35,67,47,55)(36,68,48,56)(37,69,49,57)(38,70,50,58)(39,61,41,59)(40,62,42,60), (1,56)(2,57)(3,58)(4,59)(5,60)(6,51)(7,52)(8,53)(9,54)(10,55)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,41)(18,42)(19,43)(20,44)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)>;
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), (1,48,6,43)(2,47,7,42)(3,46,8,41)(4,45,9,50)(5,44,10,49)(11,54,16,59)(12,53,17,58)(13,52,18,57)(14,51,19,56)(15,60,20,55)(21,33,26,38)(22,32,27,37)(23,31,28,36)(24,40,29,35)(25,39,30,34)(61,73,66,78)(62,72,67,77)(63,71,68,76)(64,80,69,75)(65,79,70,74), (1,14,28,76)(2,15,29,77)(3,16,30,78)(4,17,21,79)(5,18,22,80)(6,19,23,71)(7,20,24,72)(8,11,25,73)(9,12,26,74)(10,13,27,75)(31,63,43,51)(32,64,44,52)(33,65,45,53)(34,66,46,54)(35,67,47,55)(36,68,48,56)(37,69,49,57)(38,70,50,58)(39,61,41,59)(40,62,42,60), (1,56)(2,57)(3,58)(4,59)(5,60)(6,51)(7,52)(8,53)(9,54)(10,55)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,41)(18,42)(19,43)(20,44)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80) );
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)], [(1,48,6,43),(2,47,7,42),(3,46,8,41),(4,45,9,50),(5,44,10,49),(11,54,16,59),(12,53,17,58),(13,52,18,57),(14,51,19,56),(15,60,20,55),(21,33,26,38),(22,32,27,37),(23,31,28,36),(24,40,29,35),(25,39,30,34),(61,73,66,78),(62,72,67,77),(63,71,68,76),(64,80,69,75),(65,79,70,74)], [(1,14,28,76),(2,15,29,77),(3,16,30,78),(4,17,21,79),(5,18,22,80),(6,19,23,71),(7,20,24,72),(8,11,25,73),(9,12,26,74),(10,13,27,75),(31,63,43,51),(32,64,44,52),(33,65,45,53),(34,66,46,54),(35,67,47,55),(36,68,48,56),(37,69,49,57),(38,70,50,58),(39,61,41,59),(40,62,42,60)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,51),(7,52),(8,53),(9,54),(10,55),(11,45),(12,46),(13,47),(14,48),(15,49),(16,50),(17,41),(18,42),(19,43),(20,44),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80)]])
Dic5⋊D4 is a maximal subgroup of
C42.102D10 C42.104D10 C42⋊12D10 Dic10⋊23D4 C42⋊17D10 C42.118D10 C42.119D10 C24.56D10 C24⋊4D10 C24.33D10 C24.34D10 C24.35D10 C24.36D10 C20⋊(C4○D4) C10.682- 1+4 Dic10⋊19D4 Dic10⋊20D4 C10.342+ 1+4 D5×C4⋊D4 C10.372+ 1+4 C10.392+ 1+4 C10.402+ 1+4 C10.422+ 1+4 C10.442+ 1+4 C10.452+ 1+4 C10.462+ 1+4 C10.1152+ 1+4 C10.472+ 1+4 C10.482+ 1+4 C10.742- 1+4 C4⋊C4.197D10 C10.1212+ 1+4 C10.822- 1+4 C10.642+ 1+4 C10.842- 1+4 C10.662+ 1+4 C10.852- 1+4 C10.692+ 1+4 C42.137D10 C42.138D10 C42⋊20D10 C42.145D10 C42⋊26D10 Dic10⋊11D4 C42.168D10 C42⋊28D10 D4×C5⋊D4 C24⋊8D10 C24.42D10 C10.1042- 1+4 C10.1452+ 1+4 (C2×C20)⋊17D4 C10.1472+ 1+4 Dic5⋊D12 Dic15⋊2D4 Dic15⋊4D4 (S3×C10)⋊D4 (C2×C10)⋊4D12 Dic15⋊18D4 Dic15⋊12D4 Dic5⋊S4
Dic5⋊D4 is a maximal quotient of
C24.44D10 C24.4D10 C24.6D10 C24.9D10 C24.13D10 C23.45D20 C24.14D10 C24.16D10 C10.96(C4×D4) (C2×C20).54D4 C10.90(C4×D4) (C2×C20).56D4 (C2×C10)⋊D8 C4⋊D4⋊D5 C5⋊2C8⋊23D4 C4.(D4×D5) C5⋊2C8⋊24D4 C22⋊Q8⋊D5 (C2×C10)⋊Q16 C5⋊(C8.D4) Dic5⋊D8 (C2×D8).D5 Dic5⋊3SD16 Dic5⋊5SD16 (C5×D4).D4 (C5×Q8).D4 Dic5⋊3Q16 (C2×Q16)⋊D5 M4(2).D10 M4(2).13D10 M4(2).15D10 M4(2).16D10 C24.18D10 C24.20D10 C24.21D10 Dic5⋊D12 Dic15⋊2D4 Dic15⋊4D4 (S3×C10)⋊D4 (C2×C10)⋊4D12 Dic15⋊18D4 Dic15⋊12D4
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 | 2 | 2 | 4 | 20 | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D4×D5 | D4⋊2D5 |
| kernel | Dic5⋊D4 | C10.D4 | D10⋊C4 | C23.D5 | C22×Dic5 | C2×C5⋊D4 | D4×C10 | Dic5 | C2×C10 | C2×D4 | C10 | C2×C4 | C23 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 |
Matrix representation of Dic5⋊D4 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 40 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 40 | 0 | 0 |
| 1 | 24 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,1,0,0,0,0,1],[17,3,0,0,40,24,0,0,0,0,1,0,0,0,0,1],[17,1,0,0,40,24,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;
Dic5⋊D4 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes D_4 % in TeX
G:=Group("Dic5:D4"); // GroupNames label
G:=SmallGroup(160,160);
// by ID
G=gap.SmallGroup(160,160);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,218,188,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^4=d^2=1,b^2=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations