metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊5D20, C42⋊15D10, C10.172+ 1+4, C4⋊C4⋊48D10, (C5×D4)⋊10D4, (C4×D4)⋊16D5, (C4×D20)⋊30C2, (D4×C20)⋊18C2, C20⋊7D4⋊9C2, C5⋊3(D4⋊5D4), C20.54(C2×D4), C4.22(C2×D20), D10⋊7(C4○D4), C22⋊D20⋊6C2, C4⋊D20⋊15C2, (C4×C20)⋊20C22, C22⋊C4⋊47D10, (C2×D20)⋊6C22, (C22×C4)⋊13D10, C4⋊Dic5⋊8C22, C22.1(C2×D20), D10⋊2Q8⋊14C2, (C2×D4).248D10, C4.D20⋊18C2, (C2×C10).98C24, C10.16(C22×D4), C2.18(C22×D20), (C2×C20).159C23, (C22×C20)⋊10C22, C22.D20⋊5C2, C2.18(D4⋊6D10), D10⋊C4⋊52C22, (C2×Dic10)⋊17C22, (D4×C10).259C22, (C2×Dic5).42C23, (C22×Dic5)⋊9C22, (C22×D5).33C23, (C23×D5).40C22, C22.123(C23×D5), C23.172(C22×D5), (C22×C10).168C23, (C2×D4×D5)⋊4C2, (C2×C4×D5)⋊3C22, C2.22(D5×C4○D4), (C2×C10).1(C2×D4), (C2×D4⋊2D5)⋊3C2, (C5×C4⋊C4)⋊60C22, (C2×C5⋊D4)⋊4C22, C10.139(C2×C4○D4), (C2×D10⋊C4)⋊21C2, (C5×C22⋊C4)⋊50C22, (C2×C4).160(C22×D5), SmallGroup(320,1226)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊5D20
G = < a,b,c,d | a4=b2=c20=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 1462 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, D4⋊5D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C2×D20, D4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C4×D20, C4.D20, C22⋊D20, C22.D20, C4⋊D20, D10⋊2Q8, C2×D10⋊C4, C20⋊7D4, D4×C20, C2×D4×D5, C2×D4⋊2D5, D4⋊5D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2+ 1+4, D20, C22×D5, D4⋊5D4, C2×D20, C23×D5, C22×D20, D4⋊6D10, D5×C4○D4, D4⋊5D20
►On 80 points
(1 66 25 50)(2 51 26 67)(3 68 27 52)(4 53 28 69)(5 70 29 54)(6 55 30 71)(7 72 31 56)(8 57 32 73)(9 74 33 58)(10 59 34 75)(11 76 35 60)(12 41 36 77)(13 78 37 42)(14 43 38 79)(15 80 39 44)(16 45 40 61)(17 62 21 46)(18 47 22 63)(19 64 23 48)(20 49 24 65)
(1 50)(2 67)(3 52)(4 69)(5 54)(6 71)(7 56)(8 73)(9 58)(10 75)(11 60)(12 77)(13 42)(14 79)(15 44)(16 61)(17 46)(18 63)(19 48)(20 65)(21 62)(22 47)(23 64)(24 49)(25 66)(26 51)(27 68)(28 53)(29 70)(30 55)(31 72)(32 57)(33 74)(34 59)(35 76)(36 41)(37 78)(38 43)(39 80)(40 45)
(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 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 61)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)
G:=sub<Sym(80)| (1,66,25,50)(2,51,26,67)(3,68,27,52)(4,53,28,69)(5,70,29,54)(6,55,30,71)(7,72,31,56)(8,57,32,73)(9,74,33,58)(10,59,34,75)(11,76,35,60)(12,41,36,77)(13,78,37,42)(14,43,38,79)(15,80,39,44)(16,45,40,61)(17,62,21,46)(18,47,22,63)(19,64,23,48)(20,49,24,65), (1,50)(2,67)(3,52)(4,69)(5,54)(6,71)(7,56)(8,73)(9,58)(10,75)(11,60)(12,77)(13,42)(14,79)(15,44)(16,61)(17,46)(18,63)(19,48)(20,65)(21,62)(22,47)(23,64)(24,49)(25,66)(26,51)(27,68)(28,53)(29,70)(30,55)(31,72)(32,57)(33,74)(34,59)(35,76)(36,41)(37,78)(38,43)(39,80)(40,45), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)>;
G:=Group( (1,66,25,50)(2,51,26,67)(3,68,27,52)(4,53,28,69)(5,70,29,54)(6,55,30,71)(7,72,31,56)(8,57,32,73)(9,74,33,58)(10,59,34,75)(11,76,35,60)(12,41,36,77)(13,78,37,42)(14,43,38,79)(15,80,39,44)(16,45,40,61)(17,62,21,46)(18,47,22,63)(19,64,23,48)(20,49,24,65), (1,50)(2,67)(3,52)(4,69)(5,54)(6,71)(7,56)(8,73)(9,58)(10,75)(11,60)(12,77)(13,42)(14,79)(15,44)(16,61)(17,46)(18,63)(19,48)(20,65)(21,62)(22,47)(23,64)(24,49)(25,66)(26,51)(27,68)(28,53)(29,70)(30,55)(31,72)(32,57)(33,74)(34,59)(35,76)(36,41)(37,78)(38,43)(39,80)(40,45), (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70) );
G=PermutationGroup([[(1,66,25,50),(2,51,26,67),(3,68,27,52),(4,53,28,69),(5,70,29,54),(6,55,30,71),(7,72,31,56),(8,57,32,73),(9,74,33,58),(10,59,34,75),(11,76,35,60),(12,41,36,77),(13,78,37,42),(14,43,38,79),(15,80,39,44),(16,45,40,61),(17,62,21,46),(18,47,22,63),(19,64,23,48),(20,49,24,65)], [(1,50),(2,67),(3,52),(4,69),(5,54),(6,71),(7,56),(8,73),(9,58),(10,75),(11,60),(12,77),(13,42),(14,79),(15,44),(16,61),(17,46),(18,63),(19,48),(20,65),(21,62),(22,47),(23,64),(24,49),(25,66),(26,51),(27,68),(28,53),(29,70),(30,55),(31,72),(32,57),(33,74),(34,59),(35,76),(36,41),(37,78),(38,43),(39,80),(40,45)], [(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)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,60),(11,59),(12,58),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,69),(22,68),(23,67),(24,66),(25,65),(26,64),(27,63),(28,62),(29,61),(30,80),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,70)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | D20 | 2+ 1+4 | D4⋊6D10 | D5×C4○D4 |
| kernel | D4⋊5D20 | C4×D20 | C4.D20 | C22⋊D20 | C22.D20 | C4⋊D20 | D10⋊2Q8 | C2×D10⋊C4 | C20⋊7D4 | D4×C20 | C2×D4×D5 | C2×D4⋊2D5 | C5×D4 | C4×D4 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of D4⋊5D20 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 8 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 32 | 0 | 0 |
| 0 | 0 | 23 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 8 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 32 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 23 |
| 0 | 0 | 0 | 0 | 9 | 9 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1],[1,8,0,0,0,0,40,34,0,0,0,0,0,0,1,23,0,0,0,0,32,40,0,0,0,0,0,0,32,9,0,0,0,0,0,9],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,32,40,0,0,0,0,0,0,32,9,0,0,0,0,23,9] >;
D4⋊5D20 in GAP, Magma, Sage, TeX
D_4\rtimes_5D_{20} % in TeX
G:=Group("D4:5D20"); // GroupNames label
G:=SmallGroup(320,1226);
// by ID
G=gap.SmallGroup(320,1226);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations