metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5⋊4D4, C23.14D10, C5⋊3(C4×D4), C5⋊D4⋊2C4, C2.2(D4×D5), D10⋊3(C2×C4), C22⋊C4⋊7D5, C22⋊1(C4×D5), Dic5⋊2(C2×C4), (C2×C4).28D10, C10.18(C2×D4), (C4×Dic5)⋊11C2, C10.D4⋊9C2, D10⋊C4⋊10C2, C10.22(C4○D4), C2.2(D4⋊2D5), (C2×C10).22C23, (C2×C20).51C22, C10.20(C22×C4), (C22×Dic5)⋊1C2, C22.14(C22×D5), (C22×C10).11C22, (C2×Dic5).61C22, (C22×D5).19C22, (C2×C4×D5)⋊9C2, C2.9(C2×C4×D5), (C2×C10)⋊5(C2×C4), (C5×C22⋊C4)⋊9C2, (C2×C5⋊D4).2C2, SmallGroup(160,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5⋊4D4
G = < a,b,c,d | a10=c4=d2=1, b2=a5, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 280 in 94 conjugacy classes, 43 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C4×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, Dic5⋊4D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C4×D5, C22×D5, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4
►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 38 6 33)(2 37 7 32)(3 36 8 31)(4 35 9 40)(5 34 10 39)(11 66 16 61)(12 65 17 70)(13 64 18 69)(14 63 19 68)(15 62 20 67)(21 45 26 50)(22 44 27 49)(23 43 28 48)(24 42 29 47)(25 41 30 46)(51 75 56 80)(52 74 57 79)(53 73 58 78)(54 72 59 77)(55 71 60 76)
(1 63 28 53)(2 62 29 52)(3 61 30 51)(4 70 21 60)(5 69 22 59)(6 68 23 58)(7 67 24 57)(8 66 25 56)(9 65 26 55)(10 64 27 54)(11 46 75 36)(12 45 76 35)(13 44 77 34)(14 43 78 33)(15 42 79 32)(16 41 80 31)(17 50 71 40)(18 49 72 39)(19 48 73 38)(20 47 74 37)
(1 58)(2 59)(3 60)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(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), (1,38,6,33)(2,37,7,32)(3,36,8,31)(4,35,9,40)(5,34,10,39)(11,66,16,61)(12,65,17,70)(13,64,18,69)(14,63,19,68)(15,62,20,67)(21,45,26,50)(22,44,27,49)(23,43,28,48)(24,42,29,47)(25,41,30,46)(51,75,56,80)(52,74,57,79)(53,73,58,78)(54,72,59,77)(55,71,60,76), (1,63,28,53)(2,62,29,52)(3,61,30,51)(4,70,21,60)(5,69,22,59)(6,68,23,58)(7,67,24,57)(8,66,25,56)(9,65,26,55)(10,64,27,54)(11,46,75,36)(12,45,76,35)(13,44,77,34)(14,43,78,33)(15,42,79,32)(16,41,80,31)(17,50,71,40)(18,49,72,39)(19,48,73,38)(20,47,74,37), (1,58)(2,59)(3,60)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(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), (1,38,6,33)(2,37,7,32)(3,36,8,31)(4,35,9,40)(5,34,10,39)(11,66,16,61)(12,65,17,70)(13,64,18,69)(14,63,19,68)(15,62,20,67)(21,45,26,50)(22,44,27,49)(23,43,28,48)(24,42,29,47)(25,41,30,46)(51,75,56,80)(52,74,57,79)(53,73,58,78)(54,72,59,77)(55,71,60,76), (1,63,28,53)(2,62,29,52)(3,61,30,51)(4,70,21,60)(5,69,22,59)(6,68,23,58)(7,67,24,57)(8,66,25,56)(9,65,26,55)(10,64,27,54)(11,46,75,36)(12,45,76,35)(13,44,77,34)(14,43,78,33)(15,42,79,32)(16,41,80,31)(17,50,71,40)(18,49,72,39)(19,48,73,38)(20,47,74,37), (1,58)(2,59)(3,60)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(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)], [(1,38,6,33),(2,37,7,32),(3,36,8,31),(4,35,9,40),(5,34,10,39),(11,66,16,61),(12,65,17,70),(13,64,18,69),(14,63,19,68),(15,62,20,67),(21,45,26,50),(22,44,27,49),(23,43,28,48),(24,42,29,47),(25,41,30,46),(51,75,56,80),(52,74,57,79),(53,73,58,78),(54,72,59,77),(55,71,60,76)], [(1,63,28,53),(2,62,29,52),(3,61,30,51),(4,70,21,60),(5,69,22,59),(6,68,23,58),(7,67,24,57),(8,66,25,56),(9,65,26,55),(10,64,27,54),(11,46,75,36),(12,45,76,35),(13,44,77,34),(14,43,78,33),(15,42,79,32),(16,41,80,31),(17,50,71,40),(18,49,72,39),(19,48,73,38),(20,47,74,37)], [(1,58),(2,59),(3,60),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(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⋊4D4 is a maximal subgroup of
C5⋊C8⋊8D4 C5⋊C8⋊D4 D10⋊M4(2) Dic5⋊M4(2) C24.24D10 C24.27D10 C24.31D10 C42.188D10 C42.91D10 C42⋊10D10 C42.96D10 C4×D4⋊2D5 C42.104D10 C4×D4×D5 C42⋊11D10 C42.108D10 Dic10⋊23D4 C42.119D10 C24.56D10 C24⋊4D10 C24.33D10 C24.35D10 C20⋊(C4○D4) Dic10⋊20D4 C10.342+ 1+4 C4⋊C4⋊21D10 C10.402+ 1+4 C10.422+ 1+4 C10.432+ 1+4 C10.442+ 1+4 C22⋊Q8⋊25D5 C4⋊C4⋊26D10 Dic10⋊21D4 C10.1182+ 1+4 C10.522+ 1+4 C10.562+ 1+4 C10.572+ 1+4 C4⋊C4.197D10 C10.1212+ 1+4 C10.822- 1+4 C4⋊C4⋊28D10 C10.1222+ 1+4 C10.632+ 1+4 C10.642+ 1+4 C10.662+ 1+4 C10.672+ 1+4 C10.852- 1+4 C42.233D10 C42.137D10 C42.138D10 Dic10⋊10D4 C42⋊20D10 C42⋊21D10 C42.234D10 C42.143D10 C42.160D10 C42.189D10 C42.161D10 C42.162D10 C42.163D10 C42.164D10 Dic5⋊4D12 Dic15⋊14D4 D6⋊(C4×D5) Dic15⋊9D4 C15⋊26(C4×D4) Dic15⋊16D4 Dic15⋊19D4 Dic5⋊2S4
Dic5⋊4D4 is a maximal quotient of
C10.49(C4×D4) Dic5⋊2C42 C10.51(C4×D4) C10.52(C4×D4) D10⋊2C42 D10⋊3(C4⋊C4) C10.54(C4×D4) C10.55(C4×D4) C5⋊5(C8×D4) D10⋊4M4(2) Dic5⋊2M4(2) C5⋊2C8⋊26D4 Dic5⋊4D8 D4.D5⋊5C4 Dic5⋊6SD16 D4⋊D5⋊6C4 Dic5⋊7SD16 C5⋊Q16⋊5C4 Dic5⋊4Q16 Q8⋊D5⋊6C4 M4(2).22D10 C42.196D10 C22⋊C4×Dic5 C24.3D10 C24.4D10 C24.46D10 C24.12D10 C24.13D10 C23.45D20 Dic5⋊4D12 Dic15⋊14D4 D6⋊(C4×D5) Dic15⋊9D4 C15⋊26(C4×D4) Dic15⋊16D4 Dic15⋊19D4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D4×D5 | D4⋊2D5 |
| kernel | Dic5⋊4D4 | C4×Dic5 | C10.D4 | D10⋊C4 | C5×C22⋊C4 | C2×C4×D5 | C22×Dic5 | C2×C5⋊D4 | C5⋊D4 | Dic5 | C22⋊C4 | C10 | C2×C4 | C23 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 4 | 2 | 8 | 2 | 2 |
Matrix representation of Dic5⋊4D4 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 |
| 13 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 20 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 37 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,40,0,0,0,0,40],[9,13,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[1,6,0,0,0,40,0,0,0,0,1,20,0,0,4,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,37,1] >;
Dic5⋊4D4 in GAP, Magma, Sage, TeX
{\rm Dic}_5\rtimes_4D_4 % in TeX
G:=Group("Dic5:4D4"); // GroupNames label
G:=SmallGroup(160,102);
// by ID
G=gap.SmallGroup(160,102);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,188,50,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^4=d^2=1,b^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations