metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊19D4, C10.1142+ 1+4, C5⋊3D42, C4⋊3(D4×D5), C20⋊6(C2×D4), C5⋊D4⋊1D4, C4⋊C4⋊23D10, D10⋊7(C2×D4), C22⋊2(D4×D5), (C2×D4)⋊23D10, C4⋊D4⋊12D5, Dic5⋊4(C2×D4), C20⋊D4⋊17C2, C4⋊D20⋊22C2, C22⋊C4⋊11D10, (C22×C4)⋊17D10, C23⋊D10⋊10C2, D20⋊8C4⋊20C2, D10⋊D4⋊20C2, C22⋊D20⋊13C2, (C2×D20)⋊46C22, (D4×C10)⋊13C22, (C22×D20)⋊15C2, (C2×C20).40C23, C10.68(C22×D4), (C2×C10).153C24, (C22×C20)⋊21C22, (C4×Dic5)⋊22C22, (C23×D5)⋊10C22, C23.D5⋊51C22, C2.28(D4⋊8D10), D10⋊C4⋊53C22, C10.D4⋊52C22, (C22×D5).64C23, C23.181(C22×D5), C22.174(C23×D5), (C22×C10).188C23, (C2×Dic5).237C23, (C2×D4×D5)⋊11C2, C2.41(C2×D4×D5), (C2×C10)⋊3(C2×D4), (C4×C5⋊D4)⋊16C2, (C2×C4×D5)⋊14C22, (C5×C4⋊D4)⋊15C2, (C5×C4⋊C4)⋊11C22, (C2×C5⋊D4)⋊15C22, (C5×C22⋊C4)⋊13C22, (C2×C4).176(C22×D5), SmallGroup(320,1281)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊19D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=dbd=a10b, dcd=c-1 >
Subgroups: 2038 in 428 conjugacy classes, 115 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C4⋊1D4, C22×D4, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C22×C10, D42, C4×Dic5, C10.D4, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C2×D20, D4×D5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×D5, C22⋊D20, D10⋊D4, D20⋊8C4, C4⋊D20, C4×C5⋊D4, C23⋊D10, C20⋊D4, C5×C4⋊D4, C22×D20, C2×D4×D5, C2×D4×D5, D20⋊19D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, C22×D5, D42, D4×D5, C23×D5, C2×D4×D5, D4⋊8D10, D20⋊19D4
►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 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 68)(62 67)(63 66)(64 65)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)
(1 75 47 24)(2 66 48 35)(3 77 49 26)(4 68 50 37)(5 79 51 28)(6 70 52 39)(7 61 53 30)(8 72 54 21)(9 63 55 32)(10 74 56 23)(11 65 57 34)(12 76 58 25)(13 67 59 36)(14 78 60 27)(15 69 41 38)(16 80 42 29)(17 71 43 40)(18 62 44 31)(19 73 45 22)(20 64 46 33)
(1 57)(2 48)(3 59)(4 50)(5 41)(6 52)(7 43)(8 54)(9 45)(10 56)(11 47)(12 58)(13 49)(14 60)(15 51)(16 42)(17 53)(18 44)(19 55)(20 46)(22 32)(24 34)(26 36)(28 38)(30 40)(61 71)(63 73)(65 75)(67 77)(69 79)
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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75), (1,75,47,24)(2,66,48,35)(3,77,49,26)(4,68,50,37)(5,79,51,28)(6,70,52,39)(7,61,53,30)(8,72,54,21)(9,63,55,32)(10,74,56,23)(11,65,57,34)(12,76,58,25)(13,67,59,36)(14,78,60,27)(15,69,41,38)(16,80,42,29)(17,71,43,40)(18,62,44,31)(19,73,45,22)(20,64,46,33), (1,57)(2,48)(3,59)(4,50)(5,41)(6,52)(7,43)(8,54)(9,45)(10,56)(11,47)(12,58)(13,49)(14,60)(15,51)(16,42)(17,53)(18,44)(19,55)(20,46)(22,32)(24,34)(26,36)(28,38)(30,40)(61,71)(63,73)(65,75)(67,77)(69,79)>;
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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75), (1,75,47,24)(2,66,48,35)(3,77,49,26)(4,68,50,37)(5,79,51,28)(6,70,52,39)(7,61,53,30)(8,72,54,21)(9,63,55,32)(10,74,56,23)(11,65,57,34)(12,76,58,25)(13,67,59,36)(14,78,60,27)(15,69,41,38)(16,80,42,29)(17,71,43,40)(18,62,44,31)(19,73,45,22)(20,64,46,33), (1,57)(2,48)(3,59)(4,50)(5,41)(6,52)(7,43)(8,54)(9,45)(10,56)(11,47)(12,58)(13,49)(14,60)(15,51)(16,42)(17,53)(18,44)(19,55)(20,46)(22,32)(24,34)(26,36)(28,38)(30,40)(61,71)(63,73)(65,75)(67,77)(69,79) );
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),(21,26),(22,25),(23,24),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,60),(54,59),(55,58),(56,57),(61,68),(62,67),(63,66),(64,65),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75)], [(1,75,47,24),(2,66,48,35),(3,77,49,26),(4,68,50,37),(5,79,51,28),(6,70,52,39),(7,61,53,30),(8,72,54,21),(9,63,55,32),(10,74,56,23),(11,65,57,34),(12,76,58,25),(13,67,59,36),(14,78,60,27),(15,69,41,38),(16,80,42,29),(17,71,43,40),(18,62,44,31),(19,73,45,22),(20,64,46,33)], [(1,57),(2,48),(3,59),(4,50),(5,41),(6,52),(7,43),(8,54),(9,45),(10,56),(11,47),(12,58),(13,49),(14,60),(15,51),(16,42),(17,53),(18,44),(19,55),(20,46),(22,32),(24,34),(26,36),(28,38),(30,40),(61,71),(63,73),(65,75),(67,77),(69,79)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | ··· | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 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 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 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 | D4 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4×D5 | D4×D5 | D4⋊8D10 |
| kernel | D20⋊19D4 | C22⋊D20 | D10⋊D4 | D20⋊8C4 | C4⋊D20 | C4×C5⋊D4 | C23⋊D10 | C20⋊D4 | C5×C4⋊D4 | C22×D20 | C2×D4×D5 | D20 | C5⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 4 | 4 | 2 | 4 | 2 | 2 | 6 | 1 | 4 | 4 | 4 |
Matrix representation of D20⋊19D4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;
D20⋊19D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{19}D_4 % in TeX
G:=Group("D20:19D4"); // GroupNames label
G:=SmallGroup(320,1281);
// by ID
G=gap.SmallGroup(320,1281);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations