metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊6D10, SD16⋊4D10, D20.42D4, C40.3C23, C20.22C24, M4(2)⋊10D10, Dic10.42D4, Dic20⋊2C22, D20.15C23, Dic10.15C23, C4○D4⋊4D10, C8⋊C22⋊5D5, C5⋊D4.5D4, D8⋊3D5⋊2C2, D8⋊D5⋊4C2, (D5×SD16)⋊2C2, C4.116(D4×D5), C5⋊4(D4○SD16), (C5×D8)⋊4C22, (C8×D5)⋊4C22, D4⋊6D10⋊8C2, (Q8×D5)⋊3C22, C8.3(C22×D5), D4⋊D5⋊15C22, D10.56(C2×D4), C8.D10⋊2C2, SD16⋊D5⋊2C2, C20.243(C2×D4), C40⋊C2⋊4C22, C8⋊D5⋊4C22, Q8⋊D5⋊14C22, (D4×D5).3C22, C22.15(D4×D5), C4.22(C23×D5), D4.8D10⋊4C2, D20.2C4⋊2C2, (C2×D4).117D10, D4⋊2D5⋊4C22, C5⋊2C8.26C23, D4.D5⋊14C22, Dic5.62(C2×D4), (C5×SD16)⋊4C22, C5⋊Q16⋊13C22, D4.15(C22×D5), (C5×D4).15C23, (C4×D5).14C23, D4.10D10⋊7C2, (C5×Q8).15C23, Q8.15(C22×D5), (C2×C20).113C23, C4○D20.29C22, C10.123(C22×D4), (C5×M4(2))⋊4C22, (C2×Dic10)⋊40C22, (D4×C10).168C22, C2.96(C2×D4×D5), (C5×C8⋊C22)⋊4C2, (C2×D4.D5)⋊29C2, (C2×C10).68(C2×D4), (C5×C4○D4)⋊7C22, (C2×C5⋊2C8)⋊18C22, (C2×C4).97(C22×D5), SmallGroup(320,1447)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 998 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4, D4 [×2], D4 [×13], Q8, Q8 [×7], C23 [×3], D5 [×3], C10, C10 [×4], C2×C8 [×3], M4(2), M4(2) [×2], D8 [×2], D8, SD16 [×2], SD16 [×8], Q16 [×3], C2×D4, C2×D4 [×5], C2×Q8 [×4], C4○D4, C4○D4 [×10], Dic5 [×2], Dic5 [×3], C20 [×2], C20, D10 [×2], D10 [×3], C2×C10, C2×C10 [×4], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22, C8⋊C22 [×2], C8.C22 [×3], 2+ (1+4), 2- (1+4), C5⋊2C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×2], Dic10 [×3], C4×D5 [×2], C4×D5 [×3], D20 [×2], C2×Dic5 [×5], C5⋊D4 [×2], C5⋊D4 [×7], C2×C20, C2×C20, C5×D4, C5×D4 [×2], C5×D4 [×2], C5×Q8, C22×D5 [×2], C22×C10, D4○SD16, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], Dic20 [×2], C2×C5⋊2C8, D4⋊D5, D4.D5, D4.D5 [×4], Q8⋊D5, C5⋊Q16, C5×M4(2), C5×D8 [×2], C5×SD16 [×2], C2×Dic10, C2×Dic10, C4○D20 [×2], C4○D20, D4×D5 [×2], D4×D5, D4⋊2D5 [×4], D4⋊2D5 [×3], Q8×D5 [×2], C2×C5⋊D4 [×2], D4×C10, C5×C4○D4, D20.2C4, C8.D10, D8⋊D5 [×2], D8⋊3D5 [×2], D5×SD16 [×2], SD16⋊D5 [×2], C2×D4.D5, D4.8D10, C5×C8⋊C22, D4⋊6D10, D4.10D10, D8⋊6D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○SD16, D4×D5 [×2], C23×D5, C2×D4×D5, D8⋊6D10
Generators and relations
G = < a,b,c,d | a8=b2=c10=d2=1, bab=a-1, cac-1=dad=a3, bc=cb, dbd=a4b, dcd=c-1 >
►On 80 points
(1 41 40 78 63 17 53 25)(2 79 54 42 64 26 31 18)(3 43 32 80 65 19 55 27)(4 71 56 44 66 28 33 20)(5 45 34 72 67 11 57 29)(6 73 58 46 68 30 35 12)(7 47 36 74 69 13 59 21)(8 75 60 48 70 22 37 14)(9 49 38 76 61 15 51 23)(10 77 52 50 62 24 39 16)
(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 71)(21 47)(22 48)(23 49)(24 50)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 51)(39 52)(40 53)
(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 57)(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 60)(9 59)(10 58)(11 17)(12 16)(13 15)(18 20)(21 76)(22 75)(23 74)(24 73)(25 72)(26 71)(27 80)(28 79)(29 78)(30 77)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 70)(38 69)(39 68)(40 67)(41 45)(42 44)(46 50)(47 49)
G:=sub<Sym(80)| (1,41,40,78,63,17,53,25)(2,79,54,42,64,26,31,18)(3,43,32,80,65,19,55,27)(4,71,56,44,66,28,33,20)(5,45,34,72,67,11,57,29)(6,73,58,46,68,30,35,12)(7,47,36,74,69,13,59,21)(8,75,60,48,70,22,37,14)(9,49,38,76,61,15,51,23)(10,77,52,50,62,24,39,16), (11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,71)(21,47)(22,48)(23,49)(24,50)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,51)(39,52)(40,53), (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), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,60)(9,59)(10,58)(11,17)(12,16)(13,15)(18,20)(21,76)(22,75)(23,74)(24,73)(25,72)(26,71)(27,80)(28,79)(29,78)(30,77)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,70)(38,69)(39,68)(40,67)(41,45)(42,44)(46,50)(47,49)>;
G:=Group( (1,41,40,78,63,17,53,25)(2,79,54,42,64,26,31,18)(3,43,32,80,65,19,55,27)(4,71,56,44,66,28,33,20)(5,45,34,72,67,11,57,29)(6,73,58,46,68,30,35,12)(7,47,36,74,69,13,59,21)(8,75,60,48,70,22,37,14)(9,49,38,76,61,15,51,23)(10,77,52,50,62,24,39,16), (11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,71)(21,47)(22,48)(23,49)(24,50)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,51)(39,52)(40,53), (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), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,60)(9,59)(10,58)(11,17)(12,16)(13,15)(18,20)(21,76)(22,75)(23,74)(24,73)(25,72)(26,71)(27,80)(28,79)(29,78)(30,77)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,70)(38,69)(39,68)(40,67)(41,45)(42,44)(46,50)(47,49) );
G=PermutationGroup([(1,41,40,78,63,17,53,25),(2,79,54,42,64,26,31,18),(3,43,32,80,65,19,55,27),(4,71,56,44,66,28,33,20),(5,45,34,72,67,11,57,29),(6,73,58,46,68,30,35,12),(7,47,36,74,69,13,59,21),(8,75,60,48,70,22,37,14),(9,49,38,76,61,15,51,23),(10,77,52,50,62,24,39,16)], [(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,71),(21,47),(22,48),(23,49),(24,50),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,51),(39,52),(40,53)], [(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)], [(1,57),(2,56),(3,55),(4,54),(5,53),(6,52),(7,51),(8,60),(9,59),(10,58),(11,17),(12,16),(13,15),(18,20),(21,76),(22,75),(23,74),(24,73),(25,72),(26,71),(27,80),(28,79),(29,78),(30,77),(31,66),(32,65),(33,64),(34,63),(35,62),(36,61),(37,70),(38,69),(39,68),(40,67),(41,45),(42,44),(46,50),(47,49)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 1 | 0 | 13 | 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 40 | 0 | 0 | 0 | 0 | 0 |
| 38 | 3 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 26 | 26 | 0 |
| 0 | 0 | 0 | 0 | 15 | 26 | 15 | 26 |
| 0 | 0 | 0 | 0 | 0 | 30 | 0 | 15 |
| 0 | 0 | 0 | 0 | 11 | 30 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 40 | 0 | 0 | 0 | 0 | 0 |
| 38 | 3 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 23 | 18 | 0 | 35 | 0 | 0 | 0 | 0 |
| 20 | 0 | 7 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 39 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 23 | 38 | 7 | 35 | 0 | 0 | 0 | 0 |
| 20 | 38 | 8 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 39 | 1 |
G:=sub<GL(8,GF(41))| [1,0,0,38,0,0,0,0,0,1,3,3,0,0,0,0,13,13,40,0,0,0,0,0,28,0,0,40,0,0,0,0,0,0,0,0,15,15,0,11,0,0,0,0,26,26,30,30,0,0,0,0,26,15,0,0,0,0,0,0,0,26,15,0],[1,0,0,38,0,0,0,0,0,1,3,3,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,40,40],[1,35,23,20,0,0,0,0,6,6,18,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,1,40,39,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40],[1,35,23,20,0,0,0,0,0,40,38,38,0,0,0,0,0,0,7,8,0,0,0,0,0,0,35,34,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,40,39,0,0,0,0,0,0,0,1] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 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 | 5 | 5 | 8 | 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 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 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 | D4○SD16 | D4×D5 | D4×D5 | D8⋊6D10 |
| kernel | D8⋊6D10 | D20.2C4 | C8.D10 | D8⋊D5 | D8⋊3D5 | D5×SD16 | SD16⋊D5 | C2×D4.D5 | D4.8D10 | C5×C8⋊C22 | D4⋊6D10 | D4.10D10 | Dic10 | D20 | C5⋊D4 | 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 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_8\rtimes_6D_{10} % in TeX
G:=Group("D8:6D10"); // GroupNames label
G:=SmallGroup(320,1447);
// by ID
G=gap.SmallGroup(320,1447);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,570,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations