metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.10D20, D20.35D4, 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, C4.128(D4×D5), C4.12(C2×D20), D4⋊D10⋊2C2, C5⋊2(D4.8D4), (C2×Dic5).3D4, D20⋊4C4⋊10C2, C22.32(D4×D5), C10.30C22≀C2, C4.12D20⋊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 D4.10D20
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, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C42, C22⋊C4, M4(2), M4(2), D8, SD16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.10D4, C4≀C2, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, D4.8D4, C40⋊C2, D40, C4.Dic5, D10⋊C4, D4⋊D5, Q8⋊D5, C4×C20, C5×M4(2), C2×Dic10, C2×Dic10, C2×D20, C4○D20, C4○D20, D4⋊2D5, Q8×D5, C5×C4○D4, D20⋊4C4, C4.12D20, C5×C4≀C2, C4.D20, C8⋊D10, D4⋊D10, D4.10D10, D4.10D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, D4.8D4, C2×D20, D4×D5, C22⋊D20, D4.10D20
►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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 40)(17 39)(18 38)(19 37)(20 36)(41 75)(42 74)(43 73)(44 72)(45 71)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)(53 63)(54 62)(55 61)(56 80)(57 79)(58 78)(59 77)(60 76)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 50 31 60)(22 51 32 41)(23 52 33 42)(24 53 34 43)(25 54 35 44)(26 55 36 45)(27 56 37 46)(28 57 38 47)(29 58 39 48)(30 59 40 49)
(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 62 31 72)(22 63 32 73)(23 64 33 74)(24 65 34 75)(25 66 35 76)(26 67 36 77)(27 68 37 78)(28 69 38 79)(29 70 39 80)(30 71 40 61)
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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,40)(17,39)(18,38)(19,37)(20,36)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,80)(57,79)(58,78)(59,77)(60,76), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,50,31,60)(22,51,32,41)(23,52,33,42)(24,53,34,43)(25,54,35,44)(26,55,36,45)(27,56,37,46)(28,57,38,47)(29,58,39,48)(30,59,40,49), (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)(21,62,31,72)(22,63,32,73)(23,64,33,74)(24,65,34,75)(25,66,35,76)(26,67,36,77)(27,68,37,78)(28,69,38,79)(29,70,39,80)(30,71,40,61)>;
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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,40)(17,39)(18,38)(19,37)(20,36)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,80)(57,79)(58,78)(59,77)(60,76), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,50,31,60)(22,51,32,41)(23,52,33,42)(24,53,34,43)(25,54,35,44)(26,55,36,45)(27,56,37,46)(28,57,38,47)(29,58,39,48)(30,59,40,49), (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)(21,62,31,72)(22,63,32,73)(23,64,33,74)(24,65,34,75)(25,66,35,76)(26,67,36,77)(27,68,37,78)(28,69,38,79)(29,70,39,80)(30,71,40,61) );
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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,40),(17,39),(18,38),(19,37),(20,36),(41,75),(42,74),(43,73),(44,72),(45,71),(46,70),(47,69),(48,68),(49,67),(50,66),(51,65),(52,64),(53,63),(54,62),(55,61),(56,80),(57,79),(58,78),(59,77),(60,76)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,50,31,60),(22,51,32,41),(23,52,33,42),(24,53,34,43),(25,54,35,44),(26,55,36,45),(27,56,37,46),(28,57,38,47),(29,58,39,48),(30,59,40,49)], [(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),(21,62,31,72),(22,63,32,73),(23,64,33,74),(24,65,34,75),(25,66,35,76),(26,67,36,77),(27,68,37,78),(28,69,38,79),(29,70,39,80),(30,71,40,61)]])
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 | D4.10D20 |
| kernel | D4.10D20 | D20⋊4C4 | C4.12D20 | 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 D4.10D20 ►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] >;
D4.10D20 in GAP, Magma, Sage, TeX
D_4._{10}D_{20} % in TeX
G:=Group("D4.10D20"); // 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