metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊2Q8, C20.11D4, C4.13D20, C4⋊C4⋊5D5, C2.7(Q8×D5), C4⋊Dic5⋊6C2, C10.8(C2×D4), C5⋊3(C22⋊Q8), (C2×C4).14D10, C2.10(C2×D20), C10.14(C2×Q8), (C2×Dic10)⋊7C2, (C2×C20).6C22, D10⋊C4.3C2, C10.27(C4○D4), (C2×C10).38C23, C2.13(D4⋊2D5), C22.52(C22×D5), (C2×Dic5).13C22, (C22×D5).25C22, (C5×C4⋊C4)⋊8C2, (C2×C4×D5).3C2, SmallGroup(160,118)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊2Q8
G = < a,b,c,d | a10=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 240 in 74 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, C22⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, D10⋊2Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, D20, C22×D5, C2×D20, D4⋊2D5, Q8×D5, D10⋊2Q8
►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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 80)(19 79)(20 78)(31 41)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 70)(58 69)(59 68)(60 67)
(1 33 30 45)(2 32 21 44)(3 31 22 43)(4 40 23 42)(5 39 24 41)(6 38 25 50)(7 37 26 49)(8 36 27 48)(9 35 28 47)(10 34 29 46)(11 70 73 58)(12 69 74 57)(13 68 75 56)(14 67 76 55)(15 66 77 54)(16 65 78 53)(17 64 79 52)(18 63 80 51)(19 62 71 60)(20 61 72 59)
(1 70 30 58)(2 61 21 59)(3 62 22 60)(4 63 23 51)(5 64 24 52)(6 65 25 53)(7 66 26 54)(8 67 27 55)(9 68 28 56)(10 69 29 57)(11 45 73 33)(12 46 74 34)(13 47 75 35)(14 48 76 36)(15 49 77 37)(16 50 78 38)(17 41 79 39)(18 42 80 40)(19 43 71 31)(20 44 72 32)
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,80)(19,79)(20,78)(31,41)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67), (1,33,30,45)(2,32,21,44)(3,31,22,43)(4,40,23,42)(5,39,24,41)(6,38,25,50)(7,37,26,49)(8,36,27,48)(9,35,28,47)(10,34,29,46)(11,70,73,58)(12,69,74,57)(13,68,75,56)(14,67,76,55)(15,66,77,54)(16,65,78,53)(17,64,79,52)(18,63,80,51)(19,62,71,60)(20,61,72,59), (1,70,30,58)(2,61,21,59)(3,62,22,60)(4,63,23,51)(5,64,24,52)(6,65,25,53)(7,66,26,54)(8,67,27,55)(9,68,28,56)(10,69,29,57)(11,45,73,33)(12,46,74,34)(13,47,75,35)(14,48,76,36)(15,49,77,37)(16,50,78,38)(17,41,79,39)(18,42,80,40)(19,43,71,31)(20,44,72,32)>;
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,80)(19,79)(20,78)(31,41)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67), (1,33,30,45)(2,32,21,44)(3,31,22,43)(4,40,23,42)(5,39,24,41)(6,38,25,50)(7,37,26,49)(8,36,27,48)(9,35,28,47)(10,34,29,46)(11,70,73,58)(12,69,74,57)(13,68,75,56)(14,67,76,55)(15,66,77,54)(16,65,78,53)(17,64,79,52)(18,63,80,51)(19,62,71,60)(20,61,72,59), (1,70,30,58)(2,61,21,59)(3,62,22,60)(4,63,23,51)(5,64,24,52)(6,65,25,53)(7,66,26,54)(8,67,27,55)(9,68,28,56)(10,69,29,57)(11,45,73,33)(12,46,74,34)(13,47,75,35)(14,48,76,36)(15,49,77,37)(16,50,78,38)(17,41,79,39)(18,42,80,40)(19,43,71,31)(20,44,72,32) );
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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,30),(11,77),(12,76),(13,75),(14,74),(15,73),(16,72),(17,71),(18,80),(19,79),(20,78),(31,41),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,70),(58,69),(59,68),(60,67)], [(1,33,30,45),(2,32,21,44),(3,31,22,43),(4,40,23,42),(5,39,24,41),(6,38,25,50),(7,37,26,49),(8,36,27,48),(9,35,28,47),(10,34,29,46),(11,70,73,58),(12,69,74,57),(13,68,75,56),(14,67,76,55),(15,66,77,54),(16,65,78,53),(17,64,79,52),(18,63,80,51),(19,62,71,60),(20,61,72,59)], [(1,70,30,58),(2,61,21,59),(3,62,22,60),(4,63,23,51),(5,64,24,52),(6,65,25,53),(7,66,26,54),(8,67,27,55),(9,68,28,56),(10,69,29,57),(11,45,73,33),(12,46,74,34),(13,47,75,35),(14,48,76,36),(15,49,77,37),(16,50,78,38),(17,41,79,39),(18,42,80,40),(19,43,71,31),(20,44,72,32)]])
D10⋊2Q8 is a maximal subgroup of
D10.1Q16 D20.8D4 D10⋊SD16 C5⋊2C8⋊D4 D4.D20 D10⋊4Q16 Q8.D20 D10⋊Q16 C5⋊2C8.D4 D10.12SD16 C8⋊8D20 C20.(C4○D4) C8.2D20 D10.8Q16 C8⋊3D20 D10⋊2Q16 C2.D8⋊7D5 C10.2+ 1+4 C10.102+ 1+4 C10.52- 1+4 C42⋊8D10 C42.92D10 C42.94D10 C42.98D10 D4⋊5D20 D4⋊6D20 C42.229D10 C42.115D10 Q8×D20 Q8⋊5D20 C42.232D10 C42.134D10 Dic10⋊19D4 C4⋊C4⋊21D10 C10.392+ 1+4 D20⋊20D4 D5×C22⋊Q8 C10.162- 1+4 D20⋊22D4 Dic10⋊21D4 C10.512+ 1+4 C10.1182+ 1+4 C10.242- 1+4 C10.262- 1+4 C10.612+ 1+4 C10.1222+ 1+4 C10.852- 1+4 C10.692+ 1+4 C42.148D10 D20⋊7Q8 C42.237D10 C42.150D10 C42.152D10 C42.155D10 C42.157D10 C42⋊24D10 C42.161D10 C42.164D10 C42.165D10 C42.241D10 D20⋊9Q8 C42.177D10 C42.178D10 C60.68D4 D30⋊10Q8 D10⋊2Dic6 D30⋊4Q8 D30⋊6Q8
D10⋊2Q8 is a maximal quotient of
C2.(C4×D20) (C2×Dic5)⋊Q8 (C2×C20).28D4 D10⋊3(C4⋊C4) (C2×C4).20D20 (C2×C20).33D4 Dic10.3Q8 D20⋊3Q8 D20⋊4Q8 D20.3Q8 C20.7Q16 Dic10⋊4Q8 (C2×Dic5)⋊6Q8 (C2×C20).53D4 C20⋊6(C4⋊C4) D10⋊4(C4⋊C4) (C2×C20).56D4 C60.68D4 D30⋊10Q8 D10⋊2Dic6 D30⋊4Q8 D30⋊6Q8
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 | D20 | D4⋊2D5 | Q8×D5 |
| kernel | D10⋊2Q8 | C4⋊Dic5 | D10⋊C4 | C5×C4⋊C4 | C2×Dic10 | C2×C4×D5 | C20 | D10 | C4⋊C4 | C10 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of D10⋊2Q8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 35 | 6 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 32 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 7 |
| 0 | 0 | 0 | 0 | 36 | 6 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 9 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 39 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 0 | 0 | 6 | 35 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,35,0,0,0,0,7,6],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,32,0,0,0,0,0,40,0,0,0,0,0,0,35,36,0,0,0,0,7,6],[9,9,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,39,9,0,0,0,0,0,0,6,6,0,0,0,0,1,35],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
D10⋊2Q8 in GAP, Magma, Sage, TeX
D_{10}\rtimes_2Q_8 % in TeX
G:=Group("D10:2Q8"); // GroupNames label
G:=SmallGroup(160,118);
// by ID
G=gap.SmallGroup(160,118);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,122,4613]);
// 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=a^-1,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations