metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.29D4, C20.9C24, SD16⋊13D10, D40⋊21C22, C40.36C23, D20.5C23, Dic10.29D4, Dic20⋊18C22, Dic10.5C23, (C2×C8)⋊11D10, C4.76(D4×D5), C5⋊D4.9D4, D40⋊C2⋊5C2, (C2×C40)⋊6C22, D4⋊D5⋊2C22, (C2×Q8)⋊11D10, (D5×SD16)⋊5C2, (C2×SD16)⋊6D5, C5⋊2(D4○SD16), C20.84(C2×D4), (C8×D5)⋊9C22, Q8⋊D5⋊1C22, D40⋊7C2⋊8C2, D4⋊6D10⋊6C2, (Q8×D5)⋊1C22, C4.9(C23×D5), (C10×SD16)⋊2C2, D10.50(C2×D4), SD16⋊D5⋊5C2, C4○D20⋊4C22, C5⋊2C8.3C23, D4.D5⋊2C22, D4.7(C22×D5), (C4×D5).5C23, C5⋊Q16⋊1C22, (C5×D4).7C23, (D4×D5).1C22, C22.21(D4×D5), C8.12(C22×D5), SD16⋊3D5⋊5C2, D20.3C4⋊5C2, (C2×D4).116D10, D4.D10⋊8C2, Q8.3(C22×D5), (C5×Q8).3C23, C20.C23⋊7C2, C40⋊C2⋊19C22, C8⋊D5⋊10C22, Dic5.56(C2×D4), Q8⋊2D5⋊1C22, (Q8×C10)⋊19C22, (C2×C20).526C23, Q8.10D10⋊3C2, (C5×SD16)⋊14C22, D4⋊2D5.1C22, C10.110(C22×D4), C4.Dic5⋊29C22, (D4×C10).167C22, C2.83(C2×D4×D5), (C2×C10).399(C2×D4), (C2×C4).230(C22×D5), SmallGroup(320,1434)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.29D4
G = < a,b,c,d | a20=b2=d2=1, c4=a10, bab=a-1, ac=ca, dad=a11, bc=cb, dbd=a10b, dcd=c3 >
Subgroups: 1014 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C8○D4, C2×SD16, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×C10, D4○SD16, C8×D5, C8⋊D5, C40⋊C2, D40, Dic20, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C5×SD16, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, Q8×D5, Q8⋊2D5, Q8⋊2D5, C2×C5⋊D4, D4×C10, Q8×C10, D20.3C4, D40⋊7C2, D5×SD16, D40⋊C2, SD16⋊D5, SD16⋊3D5, D4.D10, C20.C23, C10×SD16, D4⋊6D10, Q8.10D10, D20.29D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, D4○SD16, D4×D5, C23×D5, C2×D4×D5, D20.29D4
►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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 25)(22 24)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)
(1 31 54 62 11 21 44 72)(2 32 55 63 12 22 45 73)(3 33 56 64 13 23 46 74)(4 34 57 65 14 24 47 75)(5 35 58 66 15 25 48 76)(6 36 59 67 16 26 49 77)(7 37 60 68 17 27 50 78)(8 38 41 69 18 28 51 79)(9 39 42 70 19 29 52 80)(10 40 43 71 20 30 53 61)
(1 49)(2 60)(3 51)(4 42)(5 53)(6 44)(7 55)(8 46)(9 57)(10 48)(11 59)(12 50)(13 41)(14 52)(15 43)(16 54)(17 45)(18 56)(19 47)(20 58)(21 26)(22 37)(23 28)(24 39)(25 30)(27 32)(29 34)(31 36)(33 38)(35 40)(61 66)(62 77)(63 68)(64 79)(65 70)(67 72)(69 74)(71 76)(73 78)(75 80)
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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75), (1,31,54,62,11,21,44,72)(2,32,55,63,12,22,45,73)(3,33,56,64,13,23,46,74)(4,34,57,65,14,24,47,75)(5,35,58,66,15,25,48,76)(6,36,59,67,16,26,49,77)(7,37,60,68,17,27,50,78)(8,38,41,69,18,28,51,79)(9,39,42,70,19,29,52,80)(10,40,43,71,20,30,53,61), (1,49)(2,60)(3,51)(4,42)(5,53)(6,44)(7,55)(8,46)(9,57)(10,48)(11,59)(12,50)(13,41)(14,52)(15,43)(16,54)(17,45)(18,56)(19,47)(20,58)(21,26)(22,37)(23,28)(24,39)(25,30)(27,32)(29,34)(31,36)(33,38)(35,40)(61,66)(62,77)(63,68)(64,79)(65,70)(67,72)(69,74)(71,76)(73,78)(75,80)>;
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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,25)(22,24)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75), (1,31,54,62,11,21,44,72)(2,32,55,63,12,22,45,73)(3,33,56,64,13,23,46,74)(4,34,57,65,14,24,47,75)(5,35,58,66,15,25,48,76)(6,36,59,67,16,26,49,77)(7,37,60,68,17,27,50,78)(8,38,41,69,18,28,51,79)(9,39,42,70,19,29,52,80)(10,40,43,71,20,30,53,61), (1,49)(2,60)(3,51)(4,42)(5,53)(6,44)(7,55)(8,46)(9,57)(10,48)(11,59)(12,50)(13,41)(14,52)(15,43)(16,54)(17,45)(18,56)(19,47)(20,58)(21,26)(22,37)(23,28)(24,39)(25,30)(27,32)(29,34)(31,36)(33,38)(35,40)(61,66)(62,77)(63,68)(64,79)(65,70)(67,72)(69,74)(71,76)(73,78)(75,80) );
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)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(21,25),(22,24),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,51),(42,50),(43,49),(44,48),(45,47),(52,60),(53,59),(54,58),(55,57),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75)], [(1,31,54,62,11,21,44,72),(2,32,55,63,12,22,45,73),(3,33,56,64,13,23,46,74),(4,34,57,65,14,24,47,75),(5,35,58,66,15,25,48,76),(6,36,59,67,16,26,49,77),(7,37,60,68,17,27,50,78),(8,38,41,69,18,28,51,79),(9,39,42,70,19,29,52,80),(10,40,43,71,20,30,53,61)], [(1,49),(2,60),(3,51),(4,42),(5,53),(6,44),(7,55),(8,46),(9,57),(10,48),(11,59),(12,50),(13,41),(14,52),(15,43),(16,54),(17,45),(18,56),(19,47),(20,58),(21,26),(22,37),(23,28),(24,39),(25,30),(27,32),(29,34),(31,36),(33,38),(35,40),(61,66),(62,77),(63,68),(64,79),(65,70),(67,72),(69,74),(71,76),(73,78),(75,80)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D4○SD16 | D4×D5 | D4×D5 | D20.29D4 |
| kernel | D20.29D4 | D20.3C4 | D40⋊7C2 | D5×SD16 | D40⋊C2 | SD16⋊D5 | SD16⋊3D5 | D4.D10 | C20.C23 | C10×SD16 | D4⋊6D10 | Q8.10D10 | Dic10 | D20 | C5⋊D4 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of D20.29D4 ►in GL4(𝔽41) generated by
| 0 | 0 | 1 | 24 |
| 0 | 0 | 17 | 38 |
| 40 | 17 | 0 | 0 |
| 24 | 3 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 26 | 0 | 15 | 0 |
| 0 | 26 | 0 | 15 |
| 26 | 0 | 26 | 0 |
| 0 | 26 | 0 | 26 |
| 17 | 40 | 0 | 0 |
| 1 | 24 | 0 | 0 |
| 0 | 0 | 24 | 1 |
| 0 | 0 | 40 | 17 |
G:=sub<GL(4,GF(41))| [0,0,40,24,0,0,17,3,1,17,0,0,24,38,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[26,0,26,0,0,26,0,26,15,0,26,0,0,15,0,26],[17,1,0,0,40,24,0,0,0,0,24,40,0,0,1,17] >;
D20.29D4 in GAP, Magma, Sage, TeX
D_{20}._{29}D_4 % in TeX
G:=Group("D20.29D4"); // GroupNames label
G:=SmallGroup(320,1434);
// by ID
G=gap.SmallGroup(320,1434);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=d^2=1,c^4=a^10,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,b*c=c*b,d*b*d=a^10*b,d*c*d=c^3>;
// generators/relations