metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊1Q8, Dic5.15D4, C4⋊C4⋊4D5, C2.6(Q8×D5), C2.14(D4×D5), C5⋊2(C22⋊Q8), C10.26(C2×D4), (C2×C4).13D10, C10.13(C2×Q8), (C2×Dic10)⋊4C2, D10⋊C4.2C2, C10.13(C4○D4), C2.15(C4○D20), C10.D4⋊12C2, (C2×C10).37C23, (C2×C20).58C22, C22.51(C22×D5), (C2×Dic5).12C22, (C22×D5).24C22, (C5×C4⋊C4)⋊7C2, (C2×C4×D5).11C2, SmallGroup(160,117)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊Q8
G = < a,b,c,d | a10=b2=c4=1, d2=c2, bab=cac-1=dad-1=a-1, cbc-1=a3b, dbd-1=a8b, dcd-1=c-1 >
Subgroups: 240 in 74 conjugacy classes, 33 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, Dic5, C20, D10, D10, C2×C10, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×C20, C22×D5, C10.D4, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, D10⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C22×D5, C4○D20, D4×D5, Q8×D5, D10⋊Q8
►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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 30)(10 29)(11 72)(12 71)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 50)(38 49)(39 48)(40 47)(51 61)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)
(1 50 24 38)(2 49 25 37)(3 48 26 36)(4 47 27 35)(5 46 28 34)(6 45 29 33)(7 44 30 32)(8 43 21 31)(9 42 22 40)(10 41 23 39)(11 65 73 53)(12 64 74 52)(13 63 75 51)(14 62 76 60)(15 61 77 59)(16 70 78 58)(17 69 79 57)(18 68 80 56)(19 67 71 55)(20 66 72 54)
(1 70 24 58)(2 69 25 57)(3 68 26 56)(4 67 27 55)(5 66 28 54)(6 65 29 53)(7 64 30 52)(8 63 21 51)(9 62 22 60)(10 61 23 59)(11 33 73 45)(12 32 74 44)(13 31 75 43)(14 40 76 42)(15 39 77 41)(16 38 78 50)(17 37 79 49)(18 36 80 48)(19 35 71 47)(20 34 72 46)
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,30)(10,29)(11,72)(12,71)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,50)(38,49)(39,48)(40,47)(51,61)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62), (1,50,24,38)(2,49,25,37)(3,48,26,36)(4,47,27,35)(5,46,28,34)(6,45,29,33)(7,44,30,32)(8,43,21,31)(9,42,22,40)(10,41,23,39)(11,65,73,53)(12,64,74,52)(13,63,75,51)(14,62,76,60)(15,61,77,59)(16,70,78,58)(17,69,79,57)(18,68,80,56)(19,67,71,55)(20,66,72,54), (1,70,24,58)(2,69,25,57)(3,68,26,56)(4,67,27,55)(5,66,28,54)(6,65,29,53)(7,64,30,52)(8,63,21,51)(9,62,22,60)(10,61,23,59)(11,33,73,45)(12,32,74,44)(13,31,75,43)(14,40,76,42)(15,39,77,41)(16,38,78,50)(17,37,79,49)(18,36,80,48)(19,35,71,47)(20,34,72,46)>;
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,30)(10,29)(11,72)(12,71)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,50)(38,49)(39,48)(40,47)(51,61)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62), (1,50,24,38)(2,49,25,37)(3,48,26,36)(4,47,27,35)(5,46,28,34)(6,45,29,33)(7,44,30,32)(8,43,21,31)(9,42,22,40)(10,41,23,39)(11,65,73,53)(12,64,74,52)(13,63,75,51)(14,62,76,60)(15,61,77,59)(16,70,78,58)(17,69,79,57)(18,68,80,56)(19,67,71,55)(20,66,72,54), (1,70,24,58)(2,69,25,57)(3,68,26,56)(4,67,27,55)(5,66,28,54)(6,65,29,53)(7,64,30,52)(8,63,21,51)(9,62,22,60)(10,61,23,59)(11,33,73,45)(12,32,74,44)(13,31,75,43)(14,40,76,42)(15,39,77,41)(16,38,78,50)(17,37,79,49)(18,36,80,48)(19,35,71,47)(20,34,72,46) );
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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,30),(10,29),(11,72),(12,71),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,50),(38,49),(39,48),(40,47),(51,61),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62)], [(1,50,24,38),(2,49,25,37),(3,48,26,36),(4,47,27,35),(5,46,28,34),(6,45,29,33),(7,44,30,32),(8,43,21,31),(9,42,22,40),(10,41,23,39),(11,65,73,53),(12,64,74,52),(13,63,75,51),(14,62,76,60),(15,61,77,59),(16,70,78,58),(17,69,79,57),(18,68,80,56),(19,67,71,55),(20,66,72,54)], [(1,70,24,58),(2,69,25,57),(3,68,26,56),(4,67,27,55),(5,66,28,54),(6,65,29,53),(7,64,30,52),(8,63,21,51),(9,62,22,60),(10,61,23,59),(11,33,73,45),(12,32,74,44),(13,31,75,43),(14,40,76,42),(15,39,77,41),(16,38,78,50),(17,37,79,49),(18,36,80,48),(19,35,71,47),(20,34,72,46)]])
D10⋊Q8 is a maximal subgroup of
C10.2- 1+4 C10.102+ 1+4 C10.62- 1+4 C42⋊10D10 C42.93D10 C42.96D10 C42.99D10 C42⋊12D10 Dic10⋊23D4 C42⋊16D10 C42.118D10 C42.122D10 C42.232D10 D20⋊10Q8 C42.133D10 C10.682- 1+4 C10.402+ 1+4 C10.422+ 1+4 C10.742- 1+4 D5×C22⋊Q8 C10.172- 1+4 Dic10⋊21D4 Dic10⋊22D4 C10.512+ 1+4 C10.522+ 1+4 C10.222- 1+4 C10.582+ 1+4 C10.262- 1+4 C10.792- 1+4 C10.1212+ 1+4 C10.822- 1+4 C4⋊C4⋊28D10 C10.622+ 1+4 C10.632+ 1+4 C10.842- 1+4 C10.692+ 1+4 C42.236D10 C42.148D10 D20⋊7Q8 C42.150D10 C42.151D10 C42.154D10 C42.157D10 C42.158D10 C42.160D10 C42⋊23D10 C42⋊24D10 C42.189D10 C42.161D10 C42.162D10 C42.164D10 C42.165D10 C42.171D10 D20⋊8Q8 C42.174D10 C42.180D10 D10⋊Dic6 D30⋊8Q8 D30⋊Q8 D30⋊2Q8 Dic15.D4 D10⋊4Dic6 D30⋊5Q8
D10⋊Q8 is a maximal quotient of
(C2×C20)⋊Q8 C10.51(C4×D4) C2.(C20⋊Q8) (C2×Dic5).Q8 D10⋊2(C4⋊C4) D10⋊3(C4⋊C4) (C2×C4).20D20 (C22×D5).Q8 Dic10⋊Q8 Dic10.Q8 D20⋊Q8 D20.Q8 Dic10⋊2Q8 Dic10.2Q8 D20⋊2Q8 D20.2Q8 C10.96(C4×D4) (C2×C4)⋊Dic10 (C2×C20).287D4 C4⋊C4⋊5Dic5 D10⋊5(C4⋊C4) (C2×C20).289D4 (C2×C20).56D4 D10⋊Dic6 D30⋊8Q8 D30⋊Q8 D30⋊2Q8 Dic15.D4 D10⋊4Dic6 D30⋊5Q8
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D5 | C4○D4 | D10 | C4○D20 | D4×D5 | Q8×D5 |
| kernel | D10⋊Q8 | C10.D4 | D10⋊C4 | C5×C4⋊C4 | C2×Dic10 | C2×C4×D5 | Dic5 | D10 | C4⋊C4 | C10 | C2×C4 | C2 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of D10⋊Q8 ►in GL4(𝔽41) generated by
| 1 | 6 | 0 | 0 |
| 35 | 6 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
| 20 | 20 | 0 | 0 |
| 23 | 21 | 0 | 0 |
| 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 1 |
| 13 | 28 | 0 | 0 |
| 32 | 28 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,35,0,0,6,6,0,0,0,0,40,0,0,0,0,40],[1,35,0,0,0,40,0,0,0,0,1,1,0,0,0,40],[20,23,0,0,20,21,0,0,0,0,40,0,0,0,2,1],[13,32,0,0,28,28,0,0,0,0,40,0,0,0,0,40] >;
D10⋊Q8 in GAP, Magma, Sage, TeX
D_{10}\rtimes Q_8 % in TeX
G:=Group("D10:Q8"); // GroupNames label
G:=SmallGroup(160,117);
// by ID
G=gap.SmallGroup(160,117);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,55,506,188,86,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^3*b,d*b*d^-1=a^8*b,d*c*d^-1=c^-1>;
// generators/relations