metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊6SD16, D20.16D4, (C2×C8)⋊17D10, (C2×Q8)⋊3D10, (C5×D4).9D4, C4.62(D4×D5), C20.47(C2×D4), (C2×C40)⋊33C22, D10⋊3Q8⋊3C2, (C2×SD16)⋊10D5, D4.8(C5⋊D4), C5⋊4(C22⋊SD16), D20⋊5C4⋊35C2, D10⋊1C8⋊32C2, C2.28(D5×SD16), (Q8×C10)⋊3C22, C10.57C22≀C2, (C10×SD16)⋊20C2, (C2×D4).146D10, D4⋊Dic5⋊33C2, C4⋊Dic5⋊20C22, (C2×Dic5).79D4, C10.45(C2×SD16), C22.266(D4×D5), C2.28(D40⋊C2), C10.78(C8⋊C22), (C2×C20).446C23, (C22×D5).128D4, (D4×C10).95C22, C2.25(C23⋊D10), (C2×D20).124C22, (C2×D4×D5).6C2, (C2×Q8⋊D5)⋊17C2, C4.42(C2×C5⋊D4), (C2×C5⋊2C8)⋊8C22, (C2×C4×D5).51C22, (C2×C10).358(C2×D4), (C2×C4).535(C22×D5), SmallGroup(320,796)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊6SD16
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd=c3 >
Subgroups: 990 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×3], C22, C22 [×20], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], D4 [×8], Q8 [×2], C23 [×11], D5 [×4], C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×4], C22×C4, C2×D4, C2×D4 [×6], C2×Q8, C24, Dic5 [×2], C20 [×2], C20, D10 [×2], D10 [×14], C2×C10, C2×C10 [×4], C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16, C2×SD16, C22×D4, C5⋊2C8, C40, C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C22×D5, C22×D5 [×9], C22×C10, C22⋊SD16, C2×C5⋊2C8, C10.D4, C4⋊Dic5, D10⋊C4, Q8⋊D5 [×2], C2×C40, C5×SD16 [×2], C2×C4×D5, C2×D20, D4×D5 [×4], C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, D10⋊1C8, D20⋊5C4, D4⋊Dic5, C2×Q8⋊D5, D10⋊3Q8, C10×SD16, C2×D4×D5, D10⋊6SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8⋊C22, C5⋊D4 [×2], C22×D5, C22⋊SD16, D4×D5 [×2], C2×C5⋊D4, D5×SD16, D40⋊C2, C23⋊D10, D10⋊6SD16
►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 10)(2 9)(3 8)(4 7)(5 6)(11 14)(12 13)(15 20)(16 19)(17 18)(21 25)(22 24)(26 30)(27 29)(31 34)(32 33)(35 40)(36 39)(37 38)(41 48)(42 47)(43 46)(44 45)(49 50)(52 60)(53 59)(54 58)(55 57)(61 67)(62 66)(63 65)(68 70)(71 79)(72 78)(73 77)(74 76)
(1 26 18 73 33 62 50 54)(2 27 19 74 34 63 41 55)(3 28 20 75 35 64 42 56)(4 29 11 76 36 65 43 57)(5 30 12 77 37 66 44 58)(6 21 13 78 38 67 45 59)(7 22 14 79 39 68 46 60)(8 23 15 80 40 69 47 51)(9 24 16 71 31 70 48 52)(10 25 17 72 32 61 49 53)
(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 41)(20 42)(21 78)(22 79)(23 80)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(51 69)(52 70)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(15,20)(16,19)(17,18)(21,25)(22,24)(26,30)(27,29)(31,34)(32,33)(35,40)(36,39)(37,38)(41,48)(42,47)(43,46)(44,45)(49,50)(52,60)(53,59)(54,58)(55,57)(61,67)(62,66)(63,65)(68,70)(71,79)(72,78)(73,77)(74,76), (1,26,18,73,33,62,50,54)(2,27,19,74,34,63,41,55)(3,28,20,75,35,64,42,56)(4,29,11,76,36,65,43,57)(5,30,12,77,37,66,44,58)(6,21,13,78,38,67,45,59)(7,22,14,79,39,68,46,60)(8,23,15,80,40,69,47,51)(9,24,16,71,31,70,48,52)(10,25,17,72,32,61,49,53), (11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,41)(20,42)(21,78)(22,79)(23,80)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)>;
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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,14)(12,13)(15,20)(16,19)(17,18)(21,25)(22,24)(26,30)(27,29)(31,34)(32,33)(35,40)(36,39)(37,38)(41,48)(42,47)(43,46)(44,45)(49,50)(52,60)(53,59)(54,58)(55,57)(61,67)(62,66)(63,65)(68,70)(71,79)(72,78)(73,77)(74,76), (1,26,18,73,33,62,50,54)(2,27,19,74,34,63,41,55)(3,28,20,75,35,64,42,56)(4,29,11,76,36,65,43,57)(5,30,12,77,37,66,44,58)(6,21,13,78,38,67,45,59)(7,22,14,79,39,68,46,60)(8,23,15,80,40,69,47,51)(9,24,16,71,31,70,48,52)(10,25,17,72,32,61,49,53), (11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,41)(20,42)(21,78)(22,79)(23,80)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(51,69)(52,70)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68) );
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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13),(15,20),(16,19),(17,18),(21,25),(22,24),(26,30),(27,29),(31,34),(32,33),(35,40),(36,39),(37,38),(41,48),(42,47),(43,46),(44,45),(49,50),(52,60),(53,59),(54,58),(55,57),(61,67),(62,66),(63,65),(68,70),(71,79),(72,78),(73,77),(74,76)], [(1,26,18,73,33,62,50,54),(2,27,19,74,34,63,41,55),(3,28,20,75,35,64,42,56),(4,29,11,76,36,65,43,57),(5,30,12,77,37,66,44,58),(6,21,13,78,38,67,45,59),(7,22,14,79,39,68,46,60),(8,23,15,80,40,69,47,51),(9,24,16,71,31,70,48,52),(10,25,17,72,32,61,49,53)], [(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,41),(20,42),(21,78),(22,79),(23,80),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(51,69),(52,70),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68)])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 8 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | SD16 | D10 | D10 | D10 | C5⋊D4 | C8⋊C22 | D4×D5 | D4×D5 | D5×SD16 | D40⋊C2 |
| kernel | D10⋊6SD16 | D10⋊1C8 | D20⋊5C4 | D4⋊Dic5 | C2×Q8⋊D5 | D10⋊3Q8 | C10×SD16 | C2×D4×D5 | D20 | C2×Dic5 | C5×D4 | C22×D5 | C2×SD16 | D10 | C2×C8 | C2×D4 | C2×Q8 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D10⋊6SD16 ►in GL6(𝔽41)
| 40 | 7 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 19 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 15 |
| 0 | 0 | 0 | 0 | 26 | 15 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,40,24,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,12,19,0,0,0,0,1,29,0,0,0,0,0,0,15,26,0,0,0,0,15,15],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
D10⋊6SD16 in GAP, Magma, Sage, TeX
D_{10}\rtimes_6{\rm SD}_{16} % in TeX
G:=Group("D10:6SD16"); // GroupNames label
G:=SmallGroup(320,796);
// by ID
G=gap.SmallGroup(320,796);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,184,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^8=d^2=1,b*a*b=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^3>;
// generators/relations