direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D20, C42⋊13D10, C10.1022+ 1+4, C5⋊1D42, C4⋊4(D4×D5), (C5×D4)⋊9D4, C4⋊1(C2×D20), C20⋊1(C2×D4), C4⋊C4⋊46D10, D10⋊5(C2×D4), (C4×D4)⋊11D5, (D4×C20)⋊13C2, (C4×D20)⋊27C2, C20⋊7D4⋊7C2, C22⋊1(C2×D20), C22⋊D20⋊5C2, C20⋊4D4⋊11C2, C4⋊D20⋊14C2, (C4×C20)⋊18C22, C22⋊C4⋊45D10, (C22×D20)⋊8C2, (C22×C4)⋊11D10, (C2×D4).247D10, (C2×D20)⋊16C22, (C2×C10).93C24, C4⋊Dic5⋊58C22, (C22×C20)⋊9C22, C2.17(C22×D20), C10.15(C22×D4), (C23×D5)⋊5C22, D10⋊C4⋊4C22, (C2×C20).158C23, C2.14(D4⋊8D10), (D4×C10).256C22, (C2×Dic5).39C23, C22.118(C23×D5), C23.171(C22×D5), (C22×C10).163C23, (C22×D5).181C23, (C2×D4×D5)⋊3C2, C2.21(C2×D4×D5), (C2×C10)⋊1(C2×D4), (C2×C4×D5)⋊2C22, (C5×C4⋊C4)⋊58C22, (C2×C5⋊D4)⋊2C22, (C5×C22⋊C4)⋊49C22, (C2×C4).157(C22×D5), SmallGroup(320,1221)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D20
G = < a,b,c,d | a4=b2=c20=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 2086 in 428 conjugacy classes, 123 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C4×D4, C4×D4, C22≀C2, C4⋊D4, C4⋊1D4, C22×D4, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, D42, C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20, C2×D20, D4×D5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C4×D20, C20⋊4D4, C22⋊D20, C4⋊D20, C20⋊7D4, D4×C20, C22×D20, C2×D4×D5, D4×D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, D20, C22×D5, D42, C2×D20, D4×D5, C23×D5, C22×D20, C2×D4×D5, D4⋊8D10, D4×D20
►On 80 points
(1 60 29 69)(2 41 30 70)(3 42 31 71)(4 43 32 72)(5 44 33 73)(6 45 34 74)(7 46 35 75)(8 47 36 76)(9 48 37 77)(10 49 38 78)(11 50 39 79)(12 51 40 80)(13 52 21 61)(14 53 22 62)(15 54 23 63)(16 55 24 64)(17 56 25 65)(18 57 26 66)(19 58 27 67)(20 59 28 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 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 41)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 31)(22 30)(23 29)(24 28)(25 27)(32 40)(33 39)(34 38)(35 37)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)
G:=sub<Sym(80)| (1,60,29,69)(2,41,30,70)(3,42,31,71)(4,43,32,72)(5,44,33,73)(6,45,34,74)(7,46,35,75)(8,47,36,76)(9,48,37,77)(10,49,38,78)(11,50,39,79)(12,51,40,80)(13,52,21,61)(14,53,22,62)(15,54,23,63)(16,55,24,64)(17,56,25,65)(18,57,26,66)(19,58,27,67)(20,59,28,68), (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)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,41), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,31)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)>;
G:=Group( (1,60,29,69)(2,41,30,70)(3,42,31,71)(4,43,32,72)(5,44,33,73)(6,45,34,74)(7,46,35,75)(8,47,36,76)(9,48,37,77)(10,49,38,78)(11,50,39,79)(12,51,40,80)(13,52,21,61)(14,53,22,62)(15,54,23,63)(16,55,24,64)(17,56,25,65)(18,57,26,66)(19,58,27,67)(20,59,28,68), (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)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,41), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,31)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77) );
G=PermutationGroup([[(1,60,29,69),(2,41,30,70),(3,42,31,71),(4,43,32,72),(5,44,33,73),(6,45,34,74),(7,46,35,75),(8,47,36,76),(9,48,37,77),(10,49,38,78),(11,50,39,79),(12,51,40,80),(13,52,21,61),(14,53,22,62),(15,54,23,63),(16,55,24,64),(17,56,25,65),(18,57,26,66),(19,58,27,67),(20,59,28,68)], [(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),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,58),(38,59),(39,60),(40,41)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,31),(22,30),(23,29),(24,28),(25,27),(32,40),(33,39),(34,38),(35,37),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(54,60),(55,59),(56,58),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | D20 | 2+ 1+4 | D4×D5 | D4⋊8D10 |
| kernel | D4×D20 | C4×D20 | C20⋊4D4 | C22⋊D20 | C4⋊D20 | C20⋊7D4 | D4×C20 | C22×D20 | C2×D4×D5 | D20 | C5×D4 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of D4×D20 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 37 |
| 0 | 0 | 21 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 37 |
| 0 | 0 | 0 | 40 |
| 2 | 30 | 0 | 0 |
| 27 | 16 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,21,0,0,37,40],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,37,40],[2,27,0,0,30,16,0,0,0,0,40,0,0,0,0,40],[40,5,0,0,0,1,0,0,0,0,1,0,0,0,0,1] >;
D4×D20 in GAP, Magma, Sage, TeX
D_4\times D_{20} % in TeX
G:=Group("D4xD20"); // GroupNames label
G:=SmallGroup(320,1221);
// by ID
G=gap.SmallGroup(320,1221);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations