metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.9D10, C20.50D4, Q8.9D10, C20.19C23, Dic10.12C22, D4.D5⋊6C2, C4○D4.2D5, C5⋊Q16⋊6C2, (C2×C10).10D4, (C2×C4).23D10, C10.61(C2×D4), C5⋊5(C8.C22), C4.25(C5⋊D4), C4.Dic5⋊10C2, C5⋊2C8.4C22, (C5×D4).9C22, C4.19(C22×D5), (C2×Dic10)⋊11C2, (C5×Q8).9C22, (C2×C20).44C22, C22.6(C5⋊D4), (C5×C4○D4).3C2, C2.25(C2×C5⋊D4), SmallGroup(160,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.9D10
G = < a,b,c,d | a4=b2=c10=1, d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >
Subgroups: 160 in 60 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C8.C22, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C4.Dic5, D4.D5, C5⋊Q16, C2×Dic10, C5×C4○D4, D4.9D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8.C22, C5⋊D4, C22×D5, C2×C5⋊D4, D4.9D10
►On 80 points
(1 41 33 15)(2 42 34 16)(3 43 35 17)(4 44 36 18)(5 45 37 19)(6 46 38 20)(7 47 39 11)(8 48 40 12)(9 49 31 13)(10 50 32 14)(21 58 65 78)(22 59 66 79)(23 60 67 80)(24 51 68 71)(25 52 69 72)(26 53 70 73)(27 54 61 74)(28 55 62 75)(29 56 63 76)(30 57 64 77)
(1 15)(2 42)(3 17)(4 44)(5 19)(6 46)(7 11)(8 48)(9 13)(10 50)(12 40)(14 32)(16 34)(18 36)(20 38)(21 65)(23 67)(25 69)(27 61)(29 63)(31 49)(33 41)(35 43)(37 45)(39 47)(51 71)(53 73)(55 75)(57 77)(59 79)
(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 76 33 56)(2 75 34 55)(3 74 35 54)(4 73 36 53)(5 72 37 52)(6 71 38 51)(7 80 39 60)(8 79 40 59)(9 78 31 58)(10 77 32 57)(11 23 47 67)(12 22 48 66)(13 21 49 65)(14 30 50 64)(15 29 41 63)(16 28 42 62)(17 27 43 61)(18 26 44 70)(19 25 45 69)(20 24 46 68)
G:=sub<Sym(80)| (1,41,33,15)(2,42,34,16)(3,43,35,17)(4,44,36,18)(5,45,37,19)(6,46,38,20)(7,47,39,11)(8,48,40,12)(9,49,31,13)(10,50,32,14)(21,58,65,78)(22,59,66,79)(23,60,67,80)(24,51,68,71)(25,52,69,72)(26,53,70,73)(27,54,61,74)(28,55,62,75)(29,56,63,76)(30,57,64,77), (1,15)(2,42)(3,17)(4,44)(5,19)(6,46)(7,11)(8,48)(9,13)(10,50)(12,40)(14,32)(16,34)(18,36)(20,38)(21,65)(23,67)(25,69)(27,61)(29,63)(31,49)(33,41)(35,43)(37,45)(39,47)(51,71)(53,73)(55,75)(57,77)(59,79), (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), (1,76,33,56)(2,75,34,55)(3,74,35,54)(4,73,36,53)(5,72,37,52)(6,71,38,51)(7,80,39,60)(8,79,40,59)(9,78,31,58)(10,77,32,57)(11,23,47,67)(12,22,48,66)(13,21,49,65)(14,30,50,64)(15,29,41,63)(16,28,42,62)(17,27,43,61)(18,26,44,70)(19,25,45,69)(20,24,46,68)>;
G:=Group( (1,41,33,15)(2,42,34,16)(3,43,35,17)(4,44,36,18)(5,45,37,19)(6,46,38,20)(7,47,39,11)(8,48,40,12)(9,49,31,13)(10,50,32,14)(21,58,65,78)(22,59,66,79)(23,60,67,80)(24,51,68,71)(25,52,69,72)(26,53,70,73)(27,54,61,74)(28,55,62,75)(29,56,63,76)(30,57,64,77), (1,15)(2,42)(3,17)(4,44)(5,19)(6,46)(7,11)(8,48)(9,13)(10,50)(12,40)(14,32)(16,34)(18,36)(20,38)(21,65)(23,67)(25,69)(27,61)(29,63)(31,49)(33,41)(35,43)(37,45)(39,47)(51,71)(53,73)(55,75)(57,77)(59,79), (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), (1,76,33,56)(2,75,34,55)(3,74,35,54)(4,73,36,53)(5,72,37,52)(6,71,38,51)(7,80,39,60)(8,79,40,59)(9,78,31,58)(10,77,32,57)(11,23,47,67)(12,22,48,66)(13,21,49,65)(14,30,50,64)(15,29,41,63)(16,28,42,62)(17,27,43,61)(18,26,44,70)(19,25,45,69)(20,24,46,68) );
G=PermutationGroup([[(1,41,33,15),(2,42,34,16),(3,43,35,17),(4,44,36,18),(5,45,37,19),(6,46,38,20),(7,47,39,11),(8,48,40,12),(9,49,31,13),(10,50,32,14),(21,58,65,78),(22,59,66,79),(23,60,67,80),(24,51,68,71),(25,52,69,72),(26,53,70,73),(27,54,61,74),(28,55,62,75),(29,56,63,76),(30,57,64,77)], [(1,15),(2,42),(3,17),(4,44),(5,19),(6,46),(7,11),(8,48),(9,13),(10,50),(12,40),(14,32),(16,34),(18,36),(20,38),(21,65),(23,67),(25,69),(27,61),(29,63),(31,49),(33,41),(35,43),(37,45),(39,47),(51,71),(53,73),(55,75),(57,77),(59,79)], [(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)], [(1,76,33,56),(2,75,34,55),(3,74,35,54),(4,73,36,53),(5,72,37,52),(6,71,38,51),(7,80,39,60),(8,79,40,59),(9,78,31,58),(10,77,32,57),(11,23,47,67),(12,22,48,66),(13,21,49,65),(14,30,50,64),(15,29,41,63),(16,28,42,62),(17,27,43,61),(18,26,44,70),(19,25,45,69),(20,24,46,68)]])
D4.9D10 is a maximal subgroup of
M4(2)⋊D10 D4.9D20 D4.3D20 D4.5D20 M4(2).13D10 M4(2).16D10 2+ 1+4.D5 2- 1+4.2D5 D8⋊11D10 D20.47D4 SD16⋊D10 D5×C8.C22 C20.C24 D20.33C23 D20.35C23 D12.37D10 D12.33D10 C60.10C23 Dic10.26D6 D4.9D30 C20.2S4
D4.9D10 is a maximal quotient of
C20.64(C4⋊C4) C4⋊C4.233D10 C4.(C2×D20) C20.50D8 C42.51D10 D4.2D20 C20.48SD16 C42.59D10 C20⋊7Q16 (C2×C10).D8 Dic10⋊17D4 C4.(D4×D5) C22⋊Q8.D5 Dic10.37D4 C5⋊(C8.D4) C42.61D10 C42.62D10 C42.65D10 Dic10.4Q8 C42.68D10 C42.71D10 C4○D4⋊Dic5 (C5×D4).32D4 D12.37D10 D12.33D10 C60.10C23 Dic10.26D6 D4.9D30
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8.C22 | D4.9D10 |
| kernel | D4.9D10 | C4.Dic5 | D4.D5 | C5⋊Q16 | C2×Dic10 | C5×C4○D4 | C20 | C2×C10 | C4○D4 | C2×C4 | D4 | Q8 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 4 |
Matrix representation of D4.9D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 28 |
| 0 | 1 | 13 | 13 |
| 3 | 3 | 40 | 0 |
| 38 | 0 | 0 | 40 |
| 1 | 0 | 0 | 28 |
| 0 | 1 | 13 | 13 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 25 | 14 | 18 | 21 |
| 29 | 13 | 5 | 38 |
| 39 | 40 | 14 | 27 |
| 7 | 40 | 14 | 30 |
| 25 | 36 | 29 | 10 |
| 27 | 16 | 22 | 31 |
| 37 | 37 | 30 | 0 |
| 24 | 28 | 36 | 11 |
G:=sub<GL(4,GF(41))| [1,0,3,38,0,1,3,0,0,13,40,0,28,13,0,40],[1,0,0,0,0,1,0,0,0,13,40,0,28,13,0,40],[25,29,39,7,14,13,40,40,18,5,14,14,21,38,27,30],[25,27,37,24,36,16,37,28,29,22,30,36,10,31,0,11] >;
D4.9D10 in GAP, Magma, Sage, TeX
D_4._9D_{10} % in TeX
G:=Group("D4.9D10"); // GroupNames label
G:=SmallGroup(160,172);
// by ID
G=gap.SmallGroup(160,172);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,188,579,159,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^10=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations