metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊2D4, D4⋊2D20, D10⋊4D8, C4⋊C4⋊1D10, (C5×D4)⋊1D4, (C2×C8)⋊2D10, C2.8(D5×D8), (C2×D40)⋊4C2, C4.84(D4×D5), C4.1(C2×D20), D4⋊C4⋊4D5, C4⋊D20⋊1C2, C5⋊2(C22⋊D8), (C2×C40)⋊2C22, C10.22(C2×D8), D10⋊1C8⋊4C2, D20⋊6C4⋊6C2, C20.107(C2×D4), C10.19C22≀C2, (C2×D4).134D10, (C2×D20)⋊12C22, (C2×Dic5).26D4, C22.171(D4×D5), C2.10(D40⋊C2), C10.55(C8⋊C22), (C2×C20).213C23, (C22×D5).107D4, (D4×C10).34C22, C2.22(C22⋊D20), (C2×D4×D5)⋊1C2, (C2×D4⋊D5)⋊2C2, (C5×C4⋊C4)⋊3C22, (C5×D4⋊C4)⋊4C2, (C2×C5⋊2C8)⋊2C22, (C2×C4×D5).11C22, (C2×C10).226(C2×D4), (C2×C4).320(C22×D5), SmallGroup(320,400)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D20
G = < a,b,c,d | a4=b2=c20=d2=1, bab=cac-1=dad=a-1, cbc-1=dbd=a-1b, dcd=c-1 >
Subgroups: 1166 in 198 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×D5, C22×C10, C22⋊D8, D40, C2×C5⋊2C8, D10⋊C4, D4⋊D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, D20⋊6C4, D10⋊1C8, C5×D4⋊C4, C4⋊D20, C2×D40, C2×D4⋊D5, C2×D4×D5, D4⋊D20
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C22≀C2, C2×D8, C8⋊C22, D20, C22×D5, C22⋊D8, C2×D20, D4×D5, C22⋊D20, D5×D8, D40⋊C2, D4⋊D20
►On 80 points
(1 22 78 53)(2 54 79 23)(3 24 80 55)(4 56 61 25)(5 26 62 57)(6 58 63 27)(7 28 64 59)(8 60 65 29)(9 30 66 41)(10 42 67 31)(11 32 68 43)(12 44 69 33)(13 34 70 45)(14 46 71 35)(15 36 72 47)(16 48 73 37)(17 38 74 49)(18 50 75 39)(19 40 76 51)(20 52 77 21)
(1 43)(2 12)(3 45)(4 14)(5 47)(6 16)(7 49)(8 18)(9 51)(10 20)(11 53)(13 55)(15 57)(17 59)(19 41)(21 42)(22 68)(23 44)(24 70)(25 46)(26 72)(27 48)(28 74)(29 50)(30 76)(31 52)(32 78)(33 54)(34 80)(35 56)(36 62)(37 58)(38 64)(39 60)(40 66)(61 71)(63 73)(65 75)(67 77)(69 79)
(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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 80)(19 79)(20 78)(21 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)
G:=sub<Sym(80)| (1,22,78,53)(2,54,79,23)(3,24,80,55)(4,56,61,25)(5,26,62,57)(6,58,63,27)(7,28,64,59)(8,60,65,29)(9,30,66,41)(10,42,67,31)(11,32,68,43)(12,44,69,33)(13,34,70,45)(14,46,71,35)(15,36,72,47)(16,48,73,37)(17,38,74,49)(18,50,75,39)(19,40,76,51)(20,52,77,21), (1,43)(2,12)(3,45)(4,14)(5,47)(6,16)(7,49)(8,18)(9,51)(10,20)(11,53)(13,55)(15,57)(17,59)(19,41)(21,42)(22,68)(23,44)(24,70)(25,46)(26,72)(27,48)(28,74)(29,50)(30,76)(31,52)(32,78)(33,54)(34,80)(35,56)(36,62)(37,58)(38,64)(39,60)(40,66)(61,71)(63,73)(65,75)(67,77)(69,79), (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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,80)(19,79)(20,78)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,60)(35,59)(36,58)(37,57)(38,56)(39,55)(40,54)>;
G:=Group( (1,22,78,53)(2,54,79,23)(3,24,80,55)(4,56,61,25)(5,26,62,57)(6,58,63,27)(7,28,64,59)(8,60,65,29)(9,30,66,41)(10,42,67,31)(11,32,68,43)(12,44,69,33)(13,34,70,45)(14,46,71,35)(15,36,72,47)(16,48,73,37)(17,38,74,49)(18,50,75,39)(19,40,76,51)(20,52,77,21), (1,43)(2,12)(3,45)(4,14)(5,47)(6,16)(7,49)(8,18)(9,51)(10,20)(11,53)(13,55)(15,57)(17,59)(19,41)(21,42)(22,68)(23,44)(24,70)(25,46)(26,72)(27,48)(28,74)(29,50)(30,76)(31,52)(32,78)(33,54)(34,80)(35,56)(36,62)(37,58)(38,64)(39,60)(40,66)(61,71)(63,73)(65,75)(67,77)(69,79), (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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,80)(19,79)(20,78)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,60)(35,59)(36,58)(37,57)(38,56)(39,55)(40,54) );
G=PermutationGroup([[(1,22,78,53),(2,54,79,23),(3,24,80,55),(4,56,61,25),(5,26,62,57),(6,58,63,27),(7,28,64,59),(8,60,65,29),(9,30,66,41),(10,42,67,31),(11,32,68,43),(12,44,69,33),(13,34,70,45),(14,46,71,35),(15,36,72,47),(16,48,73,37),(17,38,74,49),(18,50,75,39),(19,40,76,51),(20,52,77,21)], [(1,43),(2,12),(3,45),(4,14),(5,47),(6,16),(7,49),(8,18),(9,51),(10,20),(11,53),(13,55),(15,57),(17,59),(19,41),(21,42),(22,68),(23,44),(24,70),(25,46),(26,72),(27,48),(28,74),(29,50),(30,76),(31,52),(32,78),(33,54),(34,80),(35,56),(36,62),(37,58),(38,64),(39,60),(40,66),(61,71),(63,73),(65,75),(67,77),(69,79)], [(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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,80),(19,79),(20,78),(21,53),(22,52),(23,51),(24,50),(25,49),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,60),(35,59),(36,58),(37,57),(38,56),(39,55),(40,54)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 20 | 40 | 2 | 2 | 8 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D8 | D10 | D10 | D10 | D20 | C8⋊C22 | D4×D5 | D4×D5 | D5×D8 | D40⋊C2 |
| kernel | D4⋊D20 | D20⋊6C4 | D10⋊1C8 | C5×D4⋊C4 | C4⋊D20 | C2×D40 | C2×D4⋊D5 | C2×D4×D5 | D20 | C2×Dic5 | C5×D4 | C22×D5 | D4⋊C4 | D10 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D4⋊D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 25 | 11 | 0 | 0 |
| 14 | 39 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 12 | 29 |
| 39 | 30 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 29 | 29 |
| 0 | 0 | 29 | 12 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[25,14,0,0,11,39,0,0,0,0,12,12,0,0,12,29],[39,4,0,0,30,2,0,0,0,0,29,29,0,0,29,12] >;
D4⋊D20 in GAP, Magma, Sage, TeX
D_4\rtimes D_{20} % in TeX
G:=Group("D4:D20"); // GroupNames label
G:=SmallGroup(320,400);
// by ID
G=gap.SmallGroup(320,400);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations