metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.35D4, D4.10D20, Q8.10D20, C42.26D10, Dic10.35D4, M4(2).8D10, C4≀C2⋊4D5, (C5×D4).5D4, C20.6(C2×D4), (C5×Q8).5D4, C8⋊D10⋊9C2, C4○D4.4D10, C4.12(C2×D20), C4.128(D4×D5), D4⋊D10⋊2C2, C5⋊2(D4.8D4), (C2×Dic5).3D4, D20⋊4C4⋊10C2, C22.32(D4×D5), C10.30C22≀C2, C20.47D4⋊2C2, C4.D20⋊11C2, (C4×C20).53C22, D4.10D10⋊1C2, (C2×C20).267C23, C4○D20.16C22, (C2×D20).75C22, C2.33(C22⋊D20), (C5×M4(2)).5C22, C4.Dic5.11C22, (C2×Dic10).81C22, (C5×C4≀C2)⋊4C2, (C2×C10).29(C2×D4), (C5×C4○D4).8C22, (C2×C4).112(C22×D5), SmallGroup(320,454)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.35D4
G = < a,b,c,d | a20=b2=c4=1, d2=a10, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=a5c-1 >
Subgroups: 686 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×5], C22, C22 [×5], C5, C8 [×2], C2×C4, C2×C4 [×9], D4, D4 [×7], Q8, Q8 [×5], C23, D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×2], M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4, C2×Q8 [×3], C4○D4, C4○D4 [×5], Dic5 [×3], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10, C4.10D4, C4≀C2, C4≀C2, C4.4D4, C8⋊C22 [×2], 2- 1+4, C5⋊2C8, C40, Dic10, Dic10 [×4], C4×D5 [×3], D20, D20 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C22×D5, D4.8D4, C40⋊C2, D40, C4.Dic5, D10⋊C4 [×2], D4⋊D5, Q8⋊D5, C4×C20, C5×M4(2), C2×Dic10, C2×Dic10, C2×D20, C4○D20, C4○D20, D4⋊2D5 [×3], Q8×D5, C5×C4○D4, D20⋊4C4, C20.47D4, C5×C4≀C2, C4.D20, C8⋊D10, D4⋊D10, D4.10D10, D20.35D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, D4.8D4, C2×D20, D4×D5 [×2], C22⋊D20, D20.35D4
►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 21)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)
(1 79)(2 80)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 44 31 54)(22 45 32 55)(23 46 33 56)(24 47 34 57)(25 48 35 58)(26 49 36 59)(27 50 37 60)(28 51 38 41)(29 52 39 42)(30 53 40 43)
(1 60 11 50)(2 41 12 51)(3 42 13 52)(4 43 14 53)(5 44 15 54)(6 45 16 55)(7 46 17 56)(8 47 18 57)(9 48 19 58)(10 49 20 59)(21 68 31 78)(22 69 32 79)(23 70 33 80)(24 71 34 61)(25 72 35 62)(26 73 36 63)(27 74 37 64)(28 75 38 65)(29 76 39 66)(30 77 40 67)
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), (1,21)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68), (1,79)(2,80)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,44,31,54)(22,45,32,55)(23,46,33,56)(24,47,34,57)(25,48,35,58)(26,49,36,59)(27,50,37,60)(28,51,38,41)(29,52,39,42)(30,53,40,43), (1,60,11,50)(2,41,12,51)(3,42,13,52)(4,43,14,53)(5,44,15,54)(6,45,16,55)(7,46,17,56)(8,47,18,57)(9,48,19,58)(10,49,20,59)(21,68,31,78)(22,69,32,79)(23,70,33,80)(24,71,34,61)(25,72,35,62)(26,73,36,63)(27,74,37,64)(28,75,38,65)(29,76,39,66)(30,77,40,67)>;
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), (1,21)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(41,67)(42,66)(43,65)(44,64)(45,63)(46,62)(47,61)(48,80)(49,79)(50,78)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68), (1,79)(2,80)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,44,31,54)(22,45,32,55)(23,46,33,56)(24,47,34,57)(25,48,35,58)(26,49,36,59)(27,50,37,60)(28,51,38,41)(29,52,39,42)(30,53,40,43), (1,60,11,50)(2,41,12,51)(3,42,13,52)(4,43,14,53)(5,44,15,54)(6,45,16,55)(7,46,17,56)(8,47,18,57)(9,48,19,58)(10,49,20,59)(21,68,31,78)(22,69,32,79)(23,70,33,80)(24,71,34,61)(25,72,35,62)(26,73,36,63)(27,74,37,64)(28,75,38,65)(29,76,39,66)(30,77,40,67) );
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)], [(1,21),(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(41,67),(42,66),(43,65),(44,64),(45,63),(46,62),(47,61),(48,80),(49,79),(50,78),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72),(57,71),(58,70),(59,69),(60,68)], [(1,79),(2,80),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,44,31,54),(22,45,32,55),(23,46,33,56),(24,47,34,57),(25,48,35,58),(26,49,36,59),(27,50,37,60),(28,51,38,41),(29,52,39,42),(30,53,40,43)], [(1,60,11,50),(2,41,12,51),(3,42,13,52),(4,43,14,53),(5,44,15,54),(6,45,16,55),(7,46,17,56),(8,47,18,57),(9,48,19,58),(10,49,20,59),(21,68,31,78),(22,69,32,79),(23,70,33,80),(24,71,34,61),(25,72,35,62),(26,73,36,63),(27,74,37,64),(28,75,38,65),(29,76,39,66),(30,77,40,67)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 20 | 40 | 2 | 2 | 4 | 4 | 4 | 20 | 20 | 20 | 2 | 2 | 8 | 40 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D20 | D4.8D4 | D4×D5 | D4×D5 | D20.35D4 |
| kernel | D20.35D4 | D20⋊4C4 | C20.47D4 | C5×C4≀C2 | C4.D20 | C8⋊D10 | D4⋊D10 | D4.10D10 | Dic10 | D20 | C2×Dic5 | C5×D4 | C5×Q8 | C4≀C2 | C42 | M4(2) | C4○D4 | D4 | Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of D20.35D4 ►in GL4(𝔽41) generated by
| 2 | 14 | 0 | 0 |
| 11 | 16 | 0 | 0 |
| 0 | 0 | 2 | 14 |
| 0 | 0 | 11 | 16 |
| 0 | 0 | 39 | 4 |
| 0 | 0 | 30 | 2 |
| 39 | 4 | 0 | 0 |
| 30 | 2 | 0 | 0 |
| 18 | 5 | 0 | 0 |
| 1 | 23 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 23 | 36 |
| 0 | 0 | 40 | 18 |
| 18 | 5 | 0 | 0 |
| 1 | 23 | 0 | 0 |
G:=sub<GL(4,GF(41))| [2,11,0,0,14,16,0,0,0,0,2,11,0,0,14,16],[0,0,39,30,0,0,4,2,39,30,0,0,4,2,0,0],[18,1,0,0,5,23,0,0,0,0,32,0,0,0,0,32],[0,0,18,1,0,0,5,23,23,40,0,0,36,18,0,0] >;
D20.35D4 in GAP, Magma, Sage, TeX
D_{20}._{35}D_4 % in TeX
G:=Group("D20.35D4"); // GroupNames label
G:=SmallGroup(320,454);
// by ID
G=gap.SmallGroup(320,454);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,226,1123,136,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=a^5*c^-1>;
// generators/relations