metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊10D4, C42⋊19D10, C10.1252+ 1+4, C4.70(D4×D5), (C4×D20)⋊43C2, C5⋊9(D4⋊5D4), (C2×Q8)⋊19D10, C20.63(C2×D4), C20⋊2D4⋊33C2, (C4×C20)⋊23C22, C22⋊C4⋊33D10, D10.47(C2×D4), C4.4D4⋊10D5, D10⋊11(C4○D4), D10⋊D4⋊40C2, C22⋊D20⋊24C2, D10⋊3Q8⋊28C2, (C2×D4).173D10, (C2×D20)⋊28C22, C4⋊Dic5⋊60C22, (Q8×C10)⋊13C22, C10.90(C22×D4), (C2×C20).601C23, (C2×C10).220C24, D10.12D4⋊42C2, C23.D5⋊33C22, C2.49(D4⋊8D10), D10⋊C4⋊54C22, C23.42(C22×D5), (D4×C10).155C22, C10.D4⋊26C22, (C22×C10).50C23, (C23×D5).64C22, C22.241(C23×D5), (C2×Dic5).115C23, (C22×D5).225C23, (C2×D4×D5)⋊17C2, C2.63(C2×D4×D5), C2.76(D5×C4○D4), (C2×C4×D5)⋊26C22, (D5×C22⋊C4)⋊17C2, (C2×Q8⋊2D5)⋊11C2, C10.187(C2×C4○D4), (C5×C4.4D4)⋊12C2, (C2×C5⋊D4)⋊23C22, (C5×C22⋊C4)⋊29C22, (C2×C4).195(C22×D5), SmallGroup(320,1348)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊10D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a9, cbc-1=a10b, dbd=a18b, dcd=c-1 >
Subgroups: 1478 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×D5, C22×D5, C22×C10, D4⋊5D4, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, D4×D5, Q8⋊2D5, C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, C4×D20, D5×C22⋊C4, C22⋊D20, D10.12D4, D10⋊D4, C20⋊2D4, D10⋊3Q8, C5×C4.4D4, C2×D4×D5, C2×Q8⋊2D5, D20⋊10D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2+ 1+4, C22×D5, D4⋊5D4, D4×D5, C23×D5, C2×D4×D5, D5×C4○D4, D4⋊8D10, D20⋊10D4
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 80)
(1 65 38 54)(2 66 39 55)(3 67 40 56)(4 68 21 57)(5 69 22 58)(6 70 23 59)(7 71 24 60)(8 72 25 41)(9 73 26 42)(10 74 27 43)(11 75 28 44)(12 76 29 45)(13 77 30 46)(14 78 31 47)(15 79 32 48)(16 80 33 49)(17 61 34 50)(18 62 35 51)(19 63 36 52)(20 64 37 53)
(1 75)(2 64)(3 73)(4 62)(5 71)(6 80)(7 69)(8 78)(9 67)(10 76)(11 65)(12 74)(13 63)(14 72)(15 61)(16 70)(17 79)(18 68)(19 77)(20 66)(21 51)(22 60)(23 49)(24 58)(25 47)(26 56)(27 45)(28 54)(29 43)(30 52)(31 41)(32 50)(33 59)(34 48)(35 57)(36 46)(37 55)(38 44)(39 53)(40 42)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80), (1,65,38,54)(2,66,39,55)(3,67,40,56)(4,68,21,57)(5,69,22,58)(6,70,23,59)(7,71,24,60)(8,72,25,41)(9,73,26,42)(10,74,27,43)(11,75,28,44)(12,76,29,45)(13,77,30,46)(14,78,31,47)(15,79,32,48)(16,80,33,49)(17,61,34,50)(18,62,35,51)(19,63,36,52)(20,64,37,53), (1,75)(2,64)(3,73)(4,62)(5,71)(6,80)(7,69)(8,78)(9,67)(10,76)(11,65)(12,74)(13,63)(14,72)(15,61)(16,70)(17,79)(18,68)(19,77)(20,66)(21,51)(22,60)(23,49)(24,58)(25,47)(26,56)(27,45)(28,54)(29,43)(30,52)(31,41)(32,50)(33,59)(34,48)(35,57)(36,46)(37,55)(38,44)(39,53)(40,42)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80), (1,65,38,54)(2,66,39,55)(3,67,40,56)(4,68,21,57)(5,69,22,58)(6,70,23,59)(7,71,24,60)(8,72,25,41)(9,73,26,42)(10,74,27,43)(11,75,28,44)(12,76,29,45)(13,77,30,46)(14,78,31,47)(15,79,32,48)(16,80,33,49)(17,61,34,50)(18,62,35,51)(19,63,36,52)(20,64,37,53), (1,75)(2,64)(3,73)(4,62)(5,71)(6,80)(7,69)(8,78)(9,67)(10,76)(11,65)(12,74)(13,63)(14,72)(15,61)(16,70)(17,79)(18,68)(19,77)(20,66)(21,51)(22,60)(23,49)(24,58)(25,47)(26,56)(27,45)(28,54)(29,43)(30,52)(31,41)(32,50)(33,59)(34,48)(35,57)(36,46)(37,55)(38,44)(39,53)(40,42) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(57,60),(58,59),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(79,80)], [(1,65,38,54),(2,66,39,55),(3,67,40,56),(4,68,21,57),(5,69,22,58),(6,70,23,59),(7,71,24,60),(8,72,25,41),(9,73,26,42),(10,74,27,43),(11,75,28,44),(12,76,29,45),(13,77,30,46),(14,78,31,47),(15,79,32,48),(16,80,33,49),(17,61,34,50),(18,62,35,51),(19,63,36,52),(20,64,37,53)], [(1,75),(2,64),(3,73),(4,62),(5,71),(6,80),(7,69),(8,78),(9,67),(10,76),(11,65),(12,74),(13,63),(14,72),(15,61),(16,70),(17,79),(18,68),(19,77),(20,66),(21,51),(22,60),(23,49),(24,58),(25,47),(26,56),(27,45),(28,54),(29,43),(30,52),(31,41),(32,50),(33,59),(34,48),(35,57),(36,46),(37,55),(38,44),(39,53),(40,42)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 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 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | ··· | 10 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4×D5 | D5×C4○D4 | D4⋊8D10 |
| kernel | D20⋊10D4 | C4×D20 | D5×C22⋊C4 | C22⋊D20 | D10.12D4 | D10⋊D4 | C20⋊2D4 | D10⋊3Q8 | C5×C4.4D4 | C2×D4×D5 | C2×Q8⋊2D5 | D20 | C4.4D4 | D10 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 8 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of D20⋊10D4 ►in GL6(𝔽41)
| 7 | 1 | 0 | 0 | 0 | 0 |
| 33 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 40 |
| 0 | 0 | 0 | 0 | 11 | 25 |
| 40 | 40 | 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 | 16 | 40 |
| 0 | 0 | 0 | 0 | 9 | 25 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 39 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 9 |
| 0 | 0 | 0 | 0 | 24 | 21 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 33 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 32 |
| 0 | 0 | 0 | 0 | 17 | 20 |
G:=sub<GL(6,GF(41))| [7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,11,0,0,0,0,40,25],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,9,0,0,0,0,40,25],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,20,24,0,0,0,0,9,21],[7,33,0,0,0,0,6,34,0,0,0,0,0,0,40,0,0,0,0,0,2,1,0,0,0,0,0,0,21,17,0,0,0,0,32,20] >;
D20⋊10D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{10}D_4 % in TeX
G:=Group("D20:10D4"); // GroupNames label
G:=SmallGroup(320,1348);
// by ID
G=gap.SmallGroup(320,1348);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,570,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,c*b*c^-1=a^10*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations