metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊13D10, D20.28D4, C20.4C24, D40⋊16C22, C40.40C23, D20.2C23, Dic10.28D4, Dic20⋊14C22, Dic10.2C23, (D5×D8)⋊6C2, (C2×C8)⋊9D10, C5⋊2(D4○D8), (C10×D8)⋊3C2, (C2×D8)⋊12D5, C4.75(D4×D5), C5⋊D4.8D4, (C2×D4)⋊14D10, D8⋊3D5⋊6C2, D8⋊D5⋊5C2, (C2×C40)⋊3C22, D4⋊D5⋊1C22, C20.79(C2×D4), (C8×D5)⋊7C22, (D4×D5)⋊1C22, D4⋊6D10⋊5C2, D40⋊7C2⋊3C2, C4.4(C23×D5), D10.49(C2×D4), C4○D20⋊3C22, (C5×D8)⋊11C22, C5⋊2C8.1C23, D4.D5⋊1C22, D4.2(C22×D5), (C5×D4).2C23, (C4×D5).2C23, C22.20(D4×D5), C8.10(C22×D5), D20.3C4⋊2C2, D4.D10⋊7C2, D4⋊2D5⋊1C22, (D4×C10)⋊20C22, C40⋊C2⋊14C22, C8⋊D5⋊13C22, Dic5.55(C2×D4), (C2×C20).521C23, C10.105(C22×D4), C4.Dic5⋊28C22, C2.78(C2×D4×D5), (C2×C10).394(C2×D4), (C2×C4).229(C22×D5), SmallGroup(320,1429)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1142 in 268 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×4], C22, C22 [×14], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4 [×4], D4 [×17], Q8 [×3], C23 [×6], D5 [×4], C10, C10 [×5], C2×C8, C2×C8 [×2], M4(2) [×3], D8 [×4], D8 [×5], SD16 [×6], Q16, C2×D4 [×2], C2×D4 [×10], C4○D4 [×9], Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10, C2×C10 [×6], C8○D4, C2×D8, C2×D8 [×2], C4○D8 [×3], C8⋊C22 [×6], 2+ (1+4) [×2], C5⋊2C8 [×2], C40 [×2], Dic10, Dic10 [×2], C4×D5 [×2], C4×D5 [×2], D20, D20 [×2], C2×Dic5 [×4], C5⋊D4 [×2], C5⋊D4 [×10], C2×C20, C5×D4 [×4], C5×D4 [×2], C22×D5 [×4], C22×C10 [×2], D4○D8, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40, Dic20, C4.Dic5, D4⋊D5 [×4], D4.D5 [×4], C2×C40, C5×D8 [×4], C4○D20, C4○D20 [×2], D4×D5 [×4], D4×D5 [×2], D4⋊2D5 [×4], D4⋊2D5 [×2], C2×C5⋊D4 [×4], D4×C10 [×2], D20.3C4, D40⋊7C2, D5×D8 [×2], D8⋊D5 [×4], D8⋊3D5 [×2], D4.D10 [×2], C10×D8, D4⋊6D10 [×2], D8⋊13D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D8⋊13D10
Generators and relations
G = < a,b,c,d | a8=b2=c10=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a6b, dbd=a4b, dcd=c-1 >
►On 80 points
(1 45 22 20 73 53 66 31)(2 32 67 54 74 11 23 46)(3 47 24 12 75 55 68 33)(4 34 69 56 76 13 25 48)(5 49 26 14 77 57 70 35)(6 36 61 58 78 15 27 50)(7 41 28 16 79 59 62 37)(8 38 63 60 80 17 29 42)(9 43 30 18 71 51 64 39)(10 40 65 52 72 19 21 44)
(2 23)(4 25)(6 27)(8 29)(10 21)(11 32)(12 55)(13 34)(14 57)(15 36)(16 59)(17 38)(18 51)(19 40)(20 53)(22 66)(24 68)(26 70)(28 62)(30 64)(31 45)(33 47)(35 49)(37 41)(39 43)(61 78)(63 80)(65 72)(67 74)(69 76)
(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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 71)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 53)(22 52)(23 51)(24 60)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 70)
G:=sub<Sym(80)| (1,45,22,20,73,53,66,31)(2,32,67,54,74,11,23,46)(3,47,24,12,75,55,68,33)(4,34,69,56,76,13,25,48)(5,49,26,14,77,57,70,35)(6,36,61,58,78,15,27,50)(7,41,28,16,79,59,62,37)(8,38,63,60,80,17,29,42)(9,43,30,18,71,51,64,39)(10,40,65,52,72,19,21,44), (2,23)(4,25)(6,27)(8,29)(10,21)(11,32)(12,55)(13,34)(14,57)(15,36)(16,59)(17,38)(18,51)(19,40)(20,53)(22,66)(24,68)(26,70)(28,62)(30,64)(31,45)(33,47)(35,49)(37,41)(39,43)(61,78)(63,80)(65,72)(67,74)(69,76), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,71)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,53)(22,52)(23,51)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,70)>;
G:=Group( (1,45,22,20,73,53,66,31)(2,32,67,54,74,11,23,46)(3,47,24,12,75,55,68,33)(4,34,69,56,76,13,25,48)(5,49,26,14,77,57,70,35)(6,36,61,58,78,15,27,50)(7,41,28,16,79,59,62,37)(8,38,63,60,80,17,29,42)(9,43,30,18,71,51,64,39)(10,40,65,52,72,19,21,44), (2,23)(4,25)(6,27)(8,29)(10,21)(11,32)(12,55)(13,34)(14,57)(15,36)(16,59)(17,38)(18,51)(19,40)(20,53)(22,66)(24,68)(26,70)(28,62)(30,64)(31,45)(33,47)(35,49)(37,41)(39,43)(61,78)(63,80)(65,72)(67,74)(69,76), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,71)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,53)(22,52)(23,51)(24,60)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,70) );
G=PermutationGroup([(1,45,22,20,73,53,66,31),(2,32,67,54,74,11,23,46),(3,47,24,12,75,55,68,33),(4,34,69,56,76,13,25,48),(5,49,26,14,77,57,70,35),(6,36,61,58,78,15,27,50),(7,41,28,16,79,59,62,37),(8,38,63,60,80,17,29,42),(9,43,30,18,71,51,64,39),(10,40,65,52,72,19,21,44)], [(2,23),(4,25),(6,27),(8,29),(10,21),(11,32),(12,55),(13,34),(14,57),(15,36),(16,59),(17,38),(18,51),(19,40),(20,53),(22,66),(24,68),(26,70),(28,62),(30,64),(31,45),(33,47),(35,49),(37,41),(39,43),(61,78),(63,80),(65,72),(67,74),(69,76)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,71),(12,80),(13,79),(14,78),(15,77),(16,76),(17,75),(18,74),(19,73),(20,72),(21,53),(22,52),(23,51),(24,60),(25,59),(26,58),(27,57),(28,56),(29,55),(30,54),(41,69),(42,68),(43,67),(44,66),(45,65),(46,64),(47,63),(48,62),(49,61),(50,70)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 24 | 0 | 0 |
| 0 | 0 | 29 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 12 | 29 |
| 0 | 0 | 29 | 29 | 12 | 12 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 0 | 0 | 40 |
| 35 | 35 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 17 |
| 0 | 0 | 29 | 0 | 24 | 0 |
| 0 | 0 | 0 | 12 | 29 | 29 |
| 0 | 0 | 29 | 29 | 12 | 12 |
| 35 | 35 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 39 |
| 0 | 0 | 1 | 0 | 40 | 1 |
| 0 | 0 | 40 | 40 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,24,29,0,29,0,0,24,0,12,29,0,0,0,0,12,12,0,0,0,0,29,12],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,40,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[35,6,0,0,0,0,35,40,0,0,0,0,0,0,0,29,0,29,0,0,0,0,12,29,0,0,17,24,29,12,0,0,17,0,29,12],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,1,40,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,39,1,40,1] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D4○D8 | D4×D5 | D4×D5 | D8⋊13D10 |
| kernel | D8⋊13D10 | D20.3C4 | D40⋊7C2 | D5×D8 | D8⋊D5 | D8⋊3D5 | D4.D10 | C10×D8 | D4⋊6D10 | Dic10 | D20 | C5⋊D4 | C2×D8 | C2×C8 | D8 | C2×D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 8 | 4 | 2 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
D_8\rtimes_{13}D_{10} % in TeX
G:=Group("D8:13D10"); // GroupNames label
G:=SmallGroup(320,1429);
// by ID
G=gap.SmallGroup(320,1429);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,185,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^6*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations