metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊10D10, Q16⋊9D10, SD16⋊10D10, D40⋊22C22, C20.15C24, C40.37C23, D20.10C23, Dic20⋊19C22, Dic10.10C23, C4○D8⋊3D5, C4○D4⋊8D10, (C2×C8)⋊12D10, D8⋊D5⋊7C2, D40⋊C2⋊7C2, (C2×C40)⋊5C22, C4.222(D4×D5), (D4×D5)⋊7C22, D40⋊7C2⋊7C2, (Q8×D5)⋊8C22, C22.5(D4×D5), D4⋊D5⋊13C22, Q16⋊D5⋊7C2, (C4×D5).107D4, D10.87(C2×D4), C20.381(C2×D4), SD16⋊D5⋊7C2, C4○D20⋊6C22, (C5×D8)⋊15C22, Q8⋊D5⋊12C22, C5⋊2C8.6C23, D4.9(C22×D5), (C5×D4).9C23, (C4×D5).8C23, C8.15(C22×D5), C4.15(C23×D5), D4.8D10⋊2C2, (C5×Q8).9C23, Q8.9(C22×D5), C8⋊D5⋊15C22, C40⋊C2⋊16C22, C5⋊2(D8⋊C22), D4.D5⋊12C22, Dic5.98(C2×D4), (C2×Dic5).88D4, (C5×Q16)⋊13C22, C5⋊Q16⋊11C22, (C22×D5).50D4, (C2×C20).532C23, (C5×SD16)⋊10C22, D4⋊2D5.9C22, C10.116(C22×D4), Q8⋊2D5.9C22, C2.89(C2×D4×D5), (C5×C4○D8)⋊3C2, (D5×C4○D4)⋊2C2, (C2×C8⋊D5)⋊1C2, (C2×C10).12(C2×D4), (C5×C4○D4)⋊2C22, (C2×C5⋊2C8)⋊16C22, (C2×C4×D5).168C22, (C2×C4).619(C22×D5), SmallGroup(320,1440)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1022 in 262 conjugacy classes, 99 normal (33 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×11], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23 [×3], D5 [×4], C10, C10 [×3], C2×C8, C2×C8, M4(2) [×4], D8, D8 [×3], SD16 [×2], SD16 [×6], Q16, Q16 [×3], C22×C4 [×3], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×2], C4○D4 [×10], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×7], C2×C10, C2×C10 [×2], C2×M4(2), C4○D8, C4○D8 [×3], C8⋊C22 [×4], C8.C22 [×4], C2×C4○D4 [×2], C5⋊2C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×2], C4×D5 [×4], C4×D5 [×6], D20 [×2], D20 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×6], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], D8⋊C22, C8⋊D5 [×4], C40⋊C2 [×2], D40, Dic20, C2×C5⋊2C8, D4⋊D5 [×2], D4.D5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C2×C4×D5, C2×C4×D5 [×2], C4○D20 [×2], C4○D20 [×2], D4×D5 [×2], D4×D5 [×2], D4⋊2D5 [×2], D4⋊2D5 [×2], Q8×D5 [×2], Q8⋊2D5 [×2], C5×C4○D4 [×2], C2×C8⋊D5, D40⋊7C2, D8⋊D5 [×2], D40⋊C2 [×2], SD16⋊D5 [×2], Q16⋊D5 [×2], D4.8D10 [×2], C5×C4○D8, D5×C4○D4 [×2], Q16⋊D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D8⋊C22, D4×D5 [×2], C23×D5, C2×D4×D5, Q16⋊D10
Generators and relations
G = < a,b,c,d | a8=c10=d2=1, b2=a4, bab-1=cac-1=a-1, dad=a3, cbc-1=a6b, dbd=a2b, dcd=c-1 >
►On 80 points
(1 50 12 38 33 17 45 6)(2 7 46 18 34 39 13 41)(3 42 14 40 35 19 47 8)(4 9 48 20 36 31 15 43)(5 44 16 32 37 11 49 10)(21 26 78 68 55 60 63 73)(22 74 64 51 56 69 79 27)(23 28 80 70 57 52 65 75)(24 76 66 53 58 61 71 29)(25 30 72 62 59 54 67 77)
(1 68 33 73)(2 51 34 27)(3 70 35 75)(4 53 36 29)(5 62 37 77)(6 55 38 21)(7 64 39 79)(8 57 40 23)(9 66 31 71)(10 59 32 25)(11 67 44 72)(12 26 45 60)(13 69 46 74)(14 28 47 52)(15 61 48 76)(16 30 49 54)(17 63 50 78)(18 22 41 56)(19 65 42 80)(20 24 43 58)
(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 72)(2 71)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 30)(18 29)(19 28)(20 27)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)(37 63)(38 62)(39 61)(40 70)(41 53)(42 52)(43 51)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)
G:=sub<Sym(80)| (1,50,12,38,33,17,45,6)(2,7,46,18,34,39,13,41)(3,42,14,40,35,19,47,8)(4,9,48,20,36,31,15,43)(5,44,16,32,37,11,49,10)(21,26,78,68,55,60,63,73)(22,74,64,51,56,69,79,27)(23,28,80,70,57,52,65,75)(24,76,66,53,58,61,71,29)(25,30,72,62,59,54,67,77), (1,68,33,73)(2,51,34,27)(3,70,35,75)(4,53,36,29)(5,62,37,77)(6,55,38,21)(7,64,39,79)(8,57,40,23)(9,66,31,71)(10,59,32,25)(11,67,44,72)(12,26,45,60)(13,69,46,74)(14,28,47,52)(15,61,48,76)(16,30,49,54)(17,63,50,78)(18,22,41,56)(19,65,42,80)(20,24,43,58), (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), (1,72)(2,71)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,30)(18,29)(19,28)(20,27)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,70)(41,53)(42,52)(43,51)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)>;
G:=Group( (1,50,12,38,33,17,45,6)(2,7,46,18,34,39,13,41)(3,42,14,40,35,19,47,8)(4,9,48,20,36,31,15,43)(5,44,16,32,37,11,49,10)(21,26,78,68,55,60,63,73)(22,74,64,51,56,69,79,27)(23,28,80,70,57,52,65,75)(24,76,66,53,58,61,71,29)(25,30,72,62,59,54,67,77), (1,68,33,73)(2,51,34,27)(3,70,35,75)(4,53,36,29)(5,62,37,77)(6,55,38,21)(7,64,39,79)(8,57,40,23)(9,66,31,71)(10,59,32,25)(11,67,44,72)(12,26,45,60)(13,69,46,74)(14,28,47,52)(15,61,48,76)(16,30,49,54)(17,63,50,78)(18,22,41,56)(19,65,42,80)(20,24,43,58), (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), (1,72)(2,71)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,30)(18,29)(19,28)(20,27)(31,69)(32,68)(33,67)(34,66)(35,65)(36,64)(37,63)(38,62)(39,61)(40,70)(41,53)(42,52)(43,51)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54) );
G=PermutationGroup([(1,50,12,38,33,17,45,6),(2,7,46,18,34,39,13,41),(3,42,14,40,35,19,47,8),(4,9,48,20,36,31,15,43),(5,44,16,32,37,11,49,10),(21,26,78,68,55,60,63,73),(22,74,64,51,56,69,79,27),(23,28,80,70,57,52,65,75),(24,76,66,53,58,61,71,29),(25,30,72,62,59,54,67,77)], [(1,68,33,73),(2,51,34,27),(3,70,35,75),(4,53,36,29),(5,62,37,77),(6,55,38,21),(7,64,39,79),(8,57,40,23),(9,66,31,71),(10,59,32,25),(11,67,44,72),(12,26,45,60),(13,69,46,74),(14,28,47,52),(15,61,48,76),(16,30,49,54),(17,63,50,78),(18,22,41,56),(19,65,42,80),(20,24,43,58)], [(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)], [(1,72),(2,71),(3,80),(4,79),(5,78),(6,77),(7,76),(8,75),(9,74),(10,73),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,30),(18,29),(19,28),(20,27),(31,69),(32,68),(33,67),(34,66),(35,65),(36,64),(37,63),(38,62),(39,61),(40,70),(41,53),(42,52),(43,51),(44,60),(45,59),(46,58),(47,57),(48,56),(49,55),(50,54)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 40 | 35 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[40,6,0,0,0,0,35,35,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,9,0,0,0,0,32,0,0,0] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | D8⋊C22 | D4×D5 | D4×D5 | Q16⋊D10 |
| kernel | Q16⋊D10 | C2×C8⋊D5 | D40⋊7C2 | D8⋊D5 | D40⋊C2 | SD16⋊D5 | Q16⋊D5 | D4.8D10 | C5×C4○D8 | D5×C4○D4 | C4×D5 | C2×Dic5 | C22×D5 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
Q_{16}\rtimes D_{10} % in TeX
G:=Group("Q16:D10"); // GroupNames label
G:=SmallGroup(320,1440);
// by ID
G=gap.SmallGroup(320,1440);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,570,185,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^10=d^2=1,b^2=a^4,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^3,c*b*c^-1=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations