direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D20, C42⋊13D10, C10.1022+ (1+4), C5⋊1(D42), C4⋊4(D4×D5), (C5×D4)⋊9D4, C20⋊1(C2×D4), C4⋊1(C2×D20), C4⋊C4⋊46D10, D10⋊5(C2×D4), (C4×D4)⋊11D5, (C4×D20)⋊27C2, (D4×C20)⋊13C2, C20⋊7D4⋊7C2, C22⋊1(C2×D20), C22⋊D20⋊5C2, C4⋊2D20⋊14C2, C4⋊D20⋊11C2, (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, C10.15(C22×D4), C2.17(C22×D20), (C23×D5)⋊5C22, D10⋊C4⋊4C22, (C2×C20).158C23, C2.14(D4⋊8D10), (D4×C10).256C22, (C2×Dic5).39C23, C23.171(C22×D5), C22.118(C23×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
Subgroups: 2086 in 428 conjugacy classes, 123 normal (29 characteristic)
C1, C2 [×3], C2 [×12], C4 [×4], C4 [×5], C22, C22 [×4], C22 [×40], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×10], D4 [×4], D4 [×30], C23 [×2], C23 [×26], D5 [×8], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4, C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×31], C24 [×4], Dic5 [×2], C20 [×4], C20 [×3], D10 [×4], D10 [×32], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×D4, C4×D4, C22≀C2 [×4], C4⋊D4 [×4], C4⋊1D4, C22×D4 [×4], C4×D5 [×4], D20 [×4], D20 [×18], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×D5 [×6], C22×D5 [×20], C22×C10 [×2], D42, C4⋊Dic5, D10⋊C4 [×6], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5 [×2], C2×D20, C2×D20 [×10], C2×D20 [×8], D4×D5 [×8], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C23×D5 [×4], C4×D20, C4⋊D20, C22⋊D20 [×4], C4⋊2D20 [×2], C20⋊7D4 [×2], D4×C20, C22×D20 [×2], C2×D4×D5 [×2], D4×D20
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C24, D10 [×7], C22×D4 [×2], 2+ (1+4), D20 [×4], C22×D5 [×7], D42, C2×D20 [×6], D4×D5 [×2], C23×D5, C22×D20, C2×D4×D5, D4⋊8D10, D4×D20
Generators and relations
G = < a,b,c,d | a4=b2=c20=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 80 points
(1 40 55 64)(2 21 56 65)(3 22 57 66)(4 23 58 67)(5 24 59 68)(6 25 60 69)(7 26 41 70)(8 27 42 71)(9 28 43 72)(10 29 44 73)(11 30 45 74)(12 31 46 75)(13 32 47 76)(14 33 48 77)(15 34 49 78)(16 35 50 79)(17 36 51 80)(18 37 52 61)(19 38 53 62)(20 39 54 63)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 41)(37 42)(38 43)(39 44)(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 43)(44 60)(45 59)(46 58)(47 57)(48 56)(49 55)(50 54)(51 53)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)
G:=sub<Sym(80)| (1,40,55,64)(2,21,56,65)(3,22,57,66)(4,23,58,67)(5,24,59,68)(6,25,60,69)(7,26,41,70)(8,27,42,71)(9,28,43,72)(10,29,44,73)(11,30,45,74)(12,31,46,75)(13,32,47,76)(14,33,48,77)(15,34,49,78)(16,35,50,79)(17,36,51,80)(18,37,52,61)(19,38,53,62)(20,39,54,63), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,41)(37,42)(38,43)(39,44)(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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,43)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)>;
G:=Group( (1,40,55,64)(2,21,56,65)(3,22,57,66)(4,23,58,67)(5,24,59,68)(6,25,60,69)(7,26,41,70)(8,27,42,71)(9,28,43,72)(10,29,44,73)(11,30,45,74)(12,31,46,75)(13,32,47,76)(14,33,48,77)(15,34,49,78)(16,35,50,79)(17,36,51,80)(18,37,52,61)(19,38,53,62)(20,39,54,63), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,41)(37,42)(38,43)(39,44)(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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,43)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)(51,53)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72) );
G=PermutationGroup([(1,40,55,64),(2,21,56,65),(3,22,57,66),(4,23,58,67),(5,24,59,68),(6,25,60,69),(7,26,41,70),(8,27,42,71),(9,28,43,72),(10,29,44,73),(11,30,45,74),(12,31,46,75),(13,32,47,76),(14,33,48,77),(15,34,49,78),(16,35,50,79),(17,36,51,80),(18,37,52,61),(19,38,53,62),(20,39,54,63)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57),(33,58),(34,59),(35,60),(36,41),(37,42),(38,43),(39,44),(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(34,40),(35,39),(36,38),(41,43),(44,60),(45,59),(46,58),(47,57),(48,56),(49,55),(50,54),(51,53),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72)])
Matrix representation ►G ⊆ 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] >;
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 | C4⋊D20 | C22⋊D20 | C4⋊2D20 | 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 |
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