metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊4D10, SD16⋊2D10, C40.1C23, M4(2)⋊8D10, C20.20C24, Dic20⋊1C22, D20.13C23, Dic10.13C23, C8⋊C22⋊6D5, D8⋊D5⋊2C2, (C2×D4)⋊30D10, D8⋊3D5⋊1C2, D4⋊D5⋊6C22, (C4×D5).99D4, C4.190(D4×D5), (C8×D5)⋊2C22, (C5×D8)⋊2C22, (D4×D5)⋊9C22, C8.1(C22×D5), C4○D4.28D10, D10.88(C2×D4), C20.241(C2×D4), C8.D10⋊1C2, SD16⋊D5⋊1C2, C8⋊D5⋊2C22, C40⋊C2⋊2C22, (D5×M4(2))⋊2C2, D4.D5⋊5C22, (Q8×D5)⋊10C22, C5⋊Q16⋊3C22, C22.47(D4×D5), C4.20(C23×D5), SD16⋊3D5⋊1C2, D4.9D10⋊9C2, (D4×C10)⋊22C22, C5⋊3(D8⋊C22), C5⋊2C8.10C23, (C5×SD16)⋊2C22, (C4×D5).66C23, (C5×D4).13C23, (C22×D5).51D4, D4.13(C22×D5), D4.D10⋊10C2, Q8.13(C22×D5), (C5×Q8).13C23, D4⋊2D5⋊10C22, (C2×C20).111C23, Dic5.100(C2×D4), (C2×Dic5).251D4, Q8⋊2D5⋊10C22, C4○D20.28C22, C10.121(C22×D4), (C5×M4(2))⋊2C22, C4.Dic5⋊13C22, (C2×Dic10)⋊39C22, C2.94(C2×D4×D5), (D5×C4○D4)⋊4C2, (C5×C8⋊C22)⋊2C2, (C2×C10).66(C2×D4), (C2×D4⋊2D5)⋊26C2, (C2×C4×D5).170C22, (C2×C4).95(C22×D5), (C5×C4○D4).24C22, SmallGroup(320,1445)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 974 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×11], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4, D4 [×2], D4 [×11], Q8, Q8 [×5], C23 [×3], D5 [×3], C10, C10 [×4], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×2], D8 [×2], SD16 [×2], SD16 [×6], Q16 [×4], C22×C4 [×3], C2×D4, C2×D4 [×3], C2×Q8 [×2], C4○D4, C4○D4 [×11], Dic5 [×2], Dic5 [×3], C20 [×2], C20, D10 [×2], D10 [×4], C2×C10, C2×C10 [×5], C2×M4(2), C4○D8 [×4], C8⋊C22, C8⋊C22 [×3], C8.C22 [×4], C2×C4○D4 [×2], C5⋊2C8 [×2], C40 [×2], Dic10, Dic10 [×2], Dic10 [×2], C4×D5 [×4], C4×D5 [×3], D20, D20, C2×Dic5, C2×Dic5 [×6], C5⋊D4 [×7], C2×C20, C2×C20, C5×D4, C5×D4 [×2], C5×D4 [×2], C5×Q8, C22×D5, C22×D5, C22×C10, D8⋊C22, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], Dic20 [×2], C4.Dic5, D4⋊D5 [×2], D4.D5 [×4], C5⋊Q16 [×2], C5×M4(2), C5×D8 [×2], C5×SD16 [×2], C2×Dic10, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5 [×4], D4⋊2D5 [×3], Q8×D5, Q8⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C5×C4○D4, D5×M4(2), C8.D10, D8⋊D5 [×2], D8⋊3D5 [×2], SD16⋊D5 [×2], SD16⋊3D5 [×2], D4.D10, D4.9D10, C5×C8⋊C22, C2×D4⋊2D5, D5×C4○D4, SD16⋊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, SD16⋊D10
Generators and relations
G = < a,b,c,d | a8=b2=c10=d2=1, bab=dad=a3, cac-1=a-1, cbc-1=a6b, dbd=a2b, dcd=c-1 >
►On 80 points
(1 62 50 76 19 71 45 67)(2 68 46 72 20 77 41 63)(3 64 42 78 16 73 47 69)(4 70 48 74 17 79 43 65)(5 66 44 80 18 75 49 61)(6 58 39 24 15 29 34 53)(7 54 35 30 11 25 40 59)(8 60 31 26 12 21 36 55)(9 56 37 22 13 27 32 51)(10 52 33 28 14 23 38 57)
(1 28)(2 24)(3 30)(4 26)(5 22)(6 63)(7 69)(8 65)(9 61)(10 67)(11 78)(12 74)(13 80)(14 76)(15 72)(16 59)(17 55)(18 51)(19 57)(20 53)(21 43)(23 45)(25 47)(27 49)(29 41)(31 79)(32 66)(33 71)(34 68)(35 73)(36 70)(37 75)(38 62)(39 77)(40 64)(42 54)(44 56)(46 58)(48 60)(50 52)
(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)
(2 5)(3 4)(6 9)(7 8)(11 12)(13 15)(16 17)(18 20)(21 59)(22 58)(23 57)(24 56)(25 55)(26 54)(27 53)(28 52)(29 51)(30 60)(31 40)(32 39)(33 38)(34 37)(35 36)(41 44)(42 43)(45 50)(46 49)(47 48)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 80)(69 79)(70 78)
G:=sub<Sym(80)| (1,62,50,76,19,71,45,67)(2,68,46,72,20,77,41,63)(3,64,42,78,16,73,47,69)(4,70,48,74,17,79,43,65)(5,66,44,80,18,75,49,61)(6,58,39,24,15,29,34,53)(7,54,35,30,11,25,40,59)(8,60,31,26,12,21,36,55)(9,56,37,22,13,27,32,51)(10,52,33,28,14,23,38,57), (1,28)(2,24)(3,30)(4,26)(5,22)(6,63)(7,69)(8,65)(9,61)(10,67)(11,78)(12,74)(13,80)(14,76)(15,72)(16,59)(17,55)(18,51)(19,57)(20,53)(21,43)(23,45)(25,47)(27,49)(29,41)(31,79)(32,66)(33,71)(34,68)(35,73)(36,70)(37,75)(38,62)(39,77)(40,64)(42,54)(44,56)(46,58)(48,60)(50,52), (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), (2,5)(3,4)(6,9)(7,8)(11,12)(13,15)(16,17)(18,20)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,60)(31,40)(32,39)(33,38)(34,37)(35,36)(41,44)(42,43)(45,50)(46,49)(47,48)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,80)(69,79)(70,78)>;
G:=Group( (1,62,50,76,19,71,45,67)(2,68,46,72,20,77,41,63)(3,64,42,78,16,73,47,69)(4,70,48,74,17,79,43,65)(5,66,44,80,18,75,49,61)(6,58,39,24,15,29,34,53)(7,54,35,30,11,25,40,59)(8,60,31,26,12,21,36,55)(9,56,37,22,13,27,32,51)(10,52,33,28,14,23,38,57), (1,28)(2,24)(3,30)(4,26)(5,22)(6,63)(7,69)(8,65)(9,61)(10,67)(11,78)(12,74)(13,80)(14,76)(15,72)(16,59)(17,55)(18,51)(19,57)(20,53)(21,43)(23,45)(25,47)(27,49)(29,41)(31,79)(32,66)(33,71)(34,68)(35,73)(36,70)(37,75)(38,62)(39,77)(40,64)(42,54)(44,56)(46,58)(48,60)(50,52), (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), (2,5)(3,4)(6,9)(7,8)(11,12)(13,15)(16,17)(18,20)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,60)(31,40)(32,39)(33,38)(34,37)(35,36)(41,44)(42,43)(45,50)(46,49)(47,48)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,80)(69,79)(70,78) );
G=PermutationGroup([(1,62,50,76,19,71,45,67),(2,68,46,72,20,77,41,63),(3,64,42,78,16,73,47,69),(4,70,48,74,17,79,43,65),(5,66,44,80,18,75,49,61),(6,58,39,24,15,29,34,53),(7,54,35,30,11,25,40,59),(8,60,31,26,12,21,36,55),(9,56,37,22,13,27,32,51),(10,52,33,28,14,23,38,57)], [(1,28),(2,24),(3,30),(4,26),(5,22),(6,63),(7,69),(8,65),(9,61),(10,67),(11,78),(12,74),(13,80),(14,76),(15,72),(16,59),(17,55),(18,51),(19,57),(20,53),(21,43),(23,45),(25,47),(27,49),(29,41),(31,79),(32,66),(33,71),(34,68),(35,73),(36,70),(37,75),(38,62),(39,77),(40,64),(42,54),(44,56),(46,58),(48,60),(50,52)], [(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)], [(2,5),(3,4),(6,9),(7,8),(11,12),(13,15),(16,17),(18,20),(21,59),(22,58),(23,57),(24,56),(25,55),(26,54),(27,53),(28,52),(29,51),(30,60),(31,40),(32,39),(33,38),(34,37),(35,36),(41,44),(42,43),(45,50),(46,49),(47,48),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,80),(69,79),(70,78)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 0 | 0 | 1 | 40 | 0 | 0 | 0 | 0 |
| 7 | 1 | 2 | 7 | 0 | 0 | 0 | 0 |
| 3 | 18 | 40 | 0 | 0 | 0 | 0 | 0 |
| 4 | 18 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 | 0 | 0 |
| 7 | 1 | 2 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 6 | 34 | 34 | 0 | 0 | 0 | 0 |
| 16 | 13 | 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 35 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 36 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 6 | 34 | 34 | 0 | 0 | 0 | 0 |
| 16 | 13 | 1 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(8,GF(41))| [0,7,3,4,0,0,0,0,0,1,18,18,0,0,0,0,1,2,40,40,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,7,0,40,0,0,0,0,0,1,0,0,0,0,0,0,1,2,40,40,0,0,0,0,40,7,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0],[0,35,8,16,0,0,0,0,7,6,6,13,0,0,0,0,0,0,34,7,0,0,0,0,0,0,34,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[35,36,8,16,0,0,0,0,7,6,6,13,0,0,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0] >;
44 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 | ··· | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 40A | 40B | 40C | 40D |
| 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 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 2 | 2 | 4 | 5 | 5 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | D8⋊C22 | D4×D5 | D4×D5 | SD16⋊D10 |
| kernel | SD16⋊D10 | D5×M4(2) | C8.D10 | D8⋊D5 | D8⋊3D5 | SD16⋊D5 | SD16⋊3D5 | D4.D10 | D4.9D10 | C5×C8⋊C22 | C2×D4⋊2D5 | D5×C4○D4 | C4×D5 | C2×Dic5 | C22×D5 | C8⋊C22 | M4(2) | D8 | SD16 | C2×D4 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
SD_{16}\rtimes D_{10} % in TeX
G:=Group("SD16:D10"); // GroupNames label
G:=SmallGroup(320,1445);
// by ID
G=gap.SmallGroup(320,1445);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,185,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=d*a*d=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations