metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊13D4, C22⋊2D40, C23.37D20, (C2×C10)⋊1D8, (C2×C8)⋊1D10, (C2×D40)⋊2C2, C22⋊C8⋊3D5, C2.7(C2×D40), C10.5(C2×D8), C20⋊7D4⋊1C2, C5⋊1(C22⋊D8), (C2×C40)⋊1C22, (C2×C4).32D20, (C2×C20).43D4, C4.120(D4×D5), D20⋊5C4⋊4C2, C10.8C22≀C2, C20.332(C2×D4), (C2×D20)⋊1C22, (C22×D20)⋊2C2, C4⋊Dic5⋊2C22, C10.9(C8⋊C22), (C22×C4).82D10, (C22×C10).52D4, C2.12(C8⋊D10), (C2×C20).742C23, C22.105(C2×D20), C2.11(C22⋊D20), (C22×C20).51C22, (C5×C22⋊C8)⋊5C2, (C2×C10).125(C2×D4), (C2×C4).687(C22×D5), SmallGroup(320,359)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊13D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a3b, bd=db, dcd=c-1 >
Subgroups: 1214 in 198 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C40, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C22⋊D8, D40, C4⋊Dic5, D10⋊C4, C2×C40, C2×D20, C2×D20, C2×D20, C2×C5⋊D4, C22×C20, C23×D5, D20⋊5C4, C5×C22⋊C8, C2×D40, C20⋊7D4, C22×D20, D20⋊13D4
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C22≀C2, C2×D8, C8⋊C22, D20, C22×D5, C22⋊D8, D40, C2×D20, D4×D5, C22⋊D20, C2×D40, C8⋊D10, D20⋊13D4
►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 31)(22 30)(23 29)(24 28)(25 27)(32 40)(33 39)(34 38)(35 37)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)
(1 65 57 34)(2 64 58 33)(3 63 59 32)(4 62 60 31)(5 61 41 30)(6 80 42 29)(7 79 43 28)(8 78 44 27)(9 77 45 26)(10 76 46 25)(11 75 47 24)(12 74 48 23)(13 73 49 22)(14 72 50 21)(15 71 51 40)(16 70 52 39)(17 69 53 38)(18 68 54 37)(19 67 55 36)(20 66 56 35)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,31)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78), (1,65,57,34)(2,64,58,33)(3,63,59,32)(4,62,60,31)(5,61,41,30)(6,80,42,29)(7,79,43,28)(8,78,44,27)(9,77,45,26)(10,76,46,25)(11,75,47,24)(12,74,48,23)(13,73,49,22)(14,72,50,21)(15,71,51,40)(16,70,52,39)(17,69,53,38)(18,68,54,37)(19,67,55,36)(20,66,56,35), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,31)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,60)(54,59)(55,58)(56,57)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78), (1,65,57,34)(2,64,58,33)(3,63,59,32)(4,62,60,31)(5,61,41,30)(6,80,42,29)(7,79,43,28)(8,78,44,27)(9,77,45,26)(10,76,46,25)(11,75,47,24)(12,74,48,23)(13,73,49,22)(14,72,50,21)(15,71,51,40)(16,70,52,39)(17,69,53,38)(18,68,54,37)(19,67,55,36)(20,66,56,35), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,31),(22,30),(23,29),(24,28),(25,27),(32,40),(33,39),(34,38),(35,37),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,60),(54,59),(55,58),(56,57),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78)], [(1,65,57,34),(2,64,58,33),(3,63,59,32),(4,62,60,31),(5,61,41,30),(6,80,42,29),(7,79,43,28),(8,78,44,27),(9,77,45,26),(10,76,46,25),(11,75,47,24),(12,74,48,23),(13,73,49,22),(14,72,50,21),(15,71,51,40),(16,70,52,39),(17,69,53,38),(18,68,54,37),(19,67,55,36),(20,66,56,35)], [(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 40 | 2 | 2 | 4 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D8 | D10 | D10 | D20 | D20 | D40 | C8⋊C22 | D4×D5 | C8⋊D10 |
| kernel | D20⋊13D4 | D20⋊5C4 | C5×C22⋊C8 | C2×D40 | C20⋊7D4 | C22×D20 | D20 | C2×C20 | C22×C10 | C22⋊C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C10 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of D20⋊13D4 ►in GL4(𝔽41) generated by
| 16 | 30 | 0 | 0 |
| 27 | 2 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 39 | 30 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 14 | 1 |
| 18 | 39 | 0 | 0 |
| 18 | 23 | 0 | 0 |
| 0 | 0 | 28 | 4 |
| 0 | 0 | 19 | 13 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 14 | 1 |
G:=sub<GL(4,GF(41))| [16,27,0,0,30,2,0,0,0,0,40,0,0,0,0,40],[39,4,0,0,30,2,0,0,0,0,40,14,0,0,0,1],[18,18,0,0,39,23,0,0,0,0,28,19,0,0,4,13],[40,0,0,0,0,40,0,0,0,0,40,14,0,0,0,1] >;
D20⋊13D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{13}D_4 % in TeX
G:=Group("D20:13D4"); // GroupNames label
G:=SmallGroup(320,359);
// by ID
G=gap.SmallGroup(320,359);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,226,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations