metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊11D4, C42⋊27D10, C10.772+ 1+4, C5⋊4D42, C4⋊2(D4×D5), C20⋊3(C2×D4), D10⋊8(C2×D4), C4⋊1D4⋊6D5, (C4×D20)⋊49C2, (C2×D4)⋊26D10, C20⋊2D4⋊36C2, (C4×C20)⋊27C22, C23⋊D10⋊27C2, (D4×C10)⋊33C22, C4⋊Dic5⋊74C22, C10.95(C22×D4), (C2×C10).261C24, (C2×C20).509C23, (C23×D5)⋊13C22, C2.81(D4⋊6D10), C23.D5⋊37C22, D10⋊C4⋊70C22, C23.67(C22×D5), (C2×D20).277C22, (C22×C10).75C23, C22.282(C23×D5), (C2×Dic5).136C23, (C22×D5).239C23, (C2×D4×D5)⋊20C2, C2.68(C2×D4×D5), (C5×C4⋊1D4)⋊8C2, (C2×C4×D5)⋊29C22, (C2×C5⋊D4)⋊27C22, (C2×C4).214(C22×D5), SmallGroup(320,1389)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊11D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a10b, bd=db, dcd=c-1 >
Subgroups: 1974 in 428 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C4×D4, C22≀C2, C4⋊D4, C4⋊1D4, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, D42, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, C4×D20, C23⋊D10, C20⋊2D4, C5×C4⋊1D4, C2×D4×D5, D20⋊11D4
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⋊6D10, D20⋊11D4
►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 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 62)(22 61)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)
(1 70 51 24)(2 71 52 25)(3 72 53 26)(4 73 54 27)(5 74 55 28)(6 75 56 29)(7 76 57 30)(8 77 58 31)(9 78 59 32)(10 79 60 33)(11 80 41 34)(12 61 42 35)(13 62 43 36)(14 63 44 37)(15 64 45 38)(16 65 46 39)(17 66 47 40)(18 67 48 21)(19 68 49 22)(20 69 50 23)
(1 56)(2 47)(3 58)(4 49)(5 60)(6 51)(7 42)(8 53)(9 44)(10 55)(11 46)(12 57)(13 48)(14 59)(15 50)(16 41)(17 52)(18 43)(19 54)(20 45)(21 36)(22 27)(23 38)(24 29)(25 40)(26 31)(28 33)(30 35)(32 37)(34 39)(61 76)(62 67)(63 78)(64 69)(65 80)(66 71)(68 73)(70 75)(72 77)(74 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), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,62)(22,61)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63), (1,70,51,24)(2,71,52,25)(3,72,53,26)(4,73,54,27)(5,74,55,28)(6,75,56,29)(7,76,57,30)(8,77,58,31)(9,78,59,32)(10,79,60,33)(11,80,41,34)(12,61,42,35)(13,62,43,36)(14,63,44,37)(15,64,45,38)(16,65,46,39)(17,66,47,40)(18,67,48,21)(19,68,49,22)(20,69,50,23), (1,56)(2,47)(3,58)(4,49)(5,60)(6,51)(7,42)(8,53)(9,44)(10,55)(11,46)(12,57)(13,48)(14,59)(15,50)(16,41)(17,52)(18,43)(19,54)(20,45)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39)(61,76)(62,67)(63,78)(64,69)(65,80)(66,71)(68,73)(70,75)(72,77)(74,79)>;
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), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,62)(22,61)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63), (1,70,51,24)(2,71,52,25)(3,72,53,26)(4,73,54,27)(5,74,55,28)(6,75,56,29)(7,76,57,30)(8,77,58,31)(9,78,59,32)(10,79,60,33)(11,80,41,34)(12,61,42,35)(13,62,43,36)(14,63,44,37)(15,64,45,38)(16,65,46,39)(17,66,47,40)(18,67,48,21)(19,68,49,22)(20,69,50,23), (1,56)(2,47)(3,58)(4,49)(5,60)(6,51)(7,42)(8,53)(9,44)(10,55)(11,46)(12,57)(13,48)(14,59)(15,50)(16,41)(17,52)(18,43)(19,54)(20,45)(21,36)(22,27)(23,38)(24,29)(25,40)(26,31)(28,33)(30,35)(32,37)(34,39)(61,76)(62,67)(63,78)(64,69)(65,80)(66,71)(68,73)(70,75)(72,77)(74,79) );
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)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,60),(12,59),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,62),(22,61),(23,80),(24,79),(25,78),(26,77),(27,76),(28,75),(29,74),(30,73),(31,72),(32,71),(33,70),(34,69),(35,68),(36,67),(37,66),(38,65),(39,64),(40,63)], [(1,70,51,24),(2,71,52,25),(3,72,53,26),(4,73,54,27),(5,74,55,28),(6,75,56,29),(7,76,57,30),(8,77,58,31),(9,78,59,32),(10,79,60,33),(11,80,41,34),(12,61,42,35),(13,62,43,36),(14,63,44,37),(15,64,45,38),(16,65,46,39),(17,66,47,40),(18,67,48,21),(19,68,49,22),(20,69,50,23)], [(1,56),(2,47),(3,58),(4,49),(5,60),(6,51),(7,42),(8,53),(9,44),(10,55),(11,46),(12,57),(13,48),(14,59),(15,50),(16,41),(17,52),(18,43),(19,54),(20,45),(21,36),(22,27),(23,38),(24,29),(25,40),(26,31),(28,33),(30,35),(32,37),(34,39),(61,76),(62,67),(63,78),(64,69),(65,80),(66,71),(68,73),(70,75),(72,77),(74,79)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 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 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | 2+ 1+4 | D4×D5 | D4⋊6D10 |
| kernel | D20⋊11D4 | C4×D20 | C23⋊D10 | C20⋊2D4 | C5×C4⋊1D4 | C2×D4×D5 | D20 | C4⋊1D4 | C42 | C2×D4 | C10 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 1 | 4 | 8 | 2 | 2 | 12 | 1 | 8 | 4 |
Matrix representation of D20⋊11D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 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 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 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 | 39 |
| 0 | 0 | 0 | 0 | 1 | 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 | 39 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,2,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1] >;
D20⋊11D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{11}D_4 % in TeX
G:=Group("D20:11D4"); // GroupNames label
G:=SmallGroup(320,1389);
// by ID
G=gap.SmallGroup(320,1389);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,570,297,136,12550]);
// 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^11,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations