metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊11D4, C42⋊27D10, C10.772+ (1+4), C5⋊4(D42), 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×C20).509C23, (C2×C10).261C24, (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
Subgroups: 1974 in 428 conjugacy classes, 115 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×4], C4 [×5], C22, C22 [×44], C5, C2×C4, C2×C4 [×2], C2×C4 [×12], D4 [×34], C23 [×4], C23 [×24], D5 [×8], C10, C10 [×2], C10 [×4], C42, C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×4], C2×D4 [×6], C2×D4 [×26], C24 [×4], Dic5 [×4], C20 [×4], C20, D10 [×8], D10 [×24], C2×C10, C2×C10 [×12], C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C4⋊1D4, C22×D4 [×4], C4×D5 [×8], D20 [×8], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20, C2×C20 [×2], C5×D4 [×10], C22×D5 [×4], C22×D5 [×20], C22×C10 [×4], D42, C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×4], C4×C20, C2×C4×D5 [×4], C2×D20 [×2], D4×D5 [×16], C2×C5⋊D4 [×8], D4×C10 [×6], C23×D5 [×4], C4×D20 [×2], C23⋊D10 [×4], C20⋊2D4 [×4], C5×C4⋊1D4, C2×D4×D5 [×4], D20⋊11D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C24, D10 [×7], C22×D4 [×2], 2+ (1+4), C22×D5 [×7], D42, D4×D5 [×4], C23×D5, C2×D4×D5 [×2], D4⋊6D10, D20⋊11D4
Generators and relations
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 >
►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 24)(2 23)(3 22)(4 21)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)
(1 59 25 61)(2 60 26 62)(3 41 27 63)(4 42 28 64)(5 43 29 65)(6 44 30 66)(7 45 31 67)(8 46 32 68)(9 47 33 69)(10 48 34 70)(11 49 35 71)(12 50 36 72)(13 51 37 73)(14 52 38 74)(15 53 39 75)(16 54 40 76)(17 55 21 77)(18 56 22 78)(19 57 23 79)(20 58 24 80)
(1 30)(2 21)(3 32)(4 23)(5 34)(6 25)(7 36)(8 27)(9 38)(10 29)(11 40)(12 31)(13 22)(14 33)(15 24)(16 35)(17 26)(18 37)(19 28)(20 39)(41 46)(42 57)(43 48)(44 59)(45 50)(47 52)(49 54)(51 56)(53 58)(55 60)(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), (1,24)(2,23)(3,22)(4,21)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69), (1,59,25,61)(2,60,26,62)(3,41,27,63)(4,42,28,64)(5,43,29,65)(6,44,30,66)(7,45,31,67)(8,46,32,68)(9,47,33,69)(10,48,34,70)(11,49,35,71)(12,50,36,72)(13,51,37,73)(14,52,38,74)(15,53,39,75)(16,54,40,76)(17,55,21,77)(18,56,22,78)(19,57,23,79)(20,58,24,80), (1,30)(2,21)(3,32)(4,23)(5,34)(6,25)(7,36)(8,27)(9,38)(10,29)(11,40)(12,31)(13,22)(14,33)(15,24)(16,35)(17,26)(18,37)(19,28)(20,39)(41,46)(42,57)(43,48)(44,59)(45,50)(47,52)(49,54)(51,56)(53,58)(55,60)(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), (1,24)(2,23)(3,22)(4,21)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69), (1,59,25,61)(2,60,26,62)(3,41,27,63)(4,42,28,64)(5,43,29,65)(6,44,30,66)(7,45,31,67)(8,46,32,68)(9,47,33,69)(10,48,34,70)(11,49,35,71)(12,50,36,72)(13,51,37,73)(14,52,38,74)(15,53,39,75)(16,54,40,76)(17,55,21,77)(18,56,22,78)(19,57,23,79)(20,58,24,80), (1,30)(2,21)(3,32)(4,23)(5,34)(6,25)(7,36)(8,27)(9,38)(10,29)(11,40)(12,31)(13,22)(14,33)(15,24)(16,35)(17,26)(18,37)(19,28)(20,39)(41,46)(42,57)(43,48)(44,59)(45,50)(47,52)(49,54)(51,56)(53,58)(55,60)(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)], [(1,24),(2,23),(3,22),(4,21),(5,40),(6,39),(7,38),(8,37),(9,36),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,26),(20,25),(41,68),(42,67),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61),(49,80),(50,79),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,72),(58,71),(59,70),(60,69)], [(1,59,25,61),(2,60,26,62),(3,41,27,63),(4,42,28,64),(5,43,29,65),(6,44,30,66),(7,45,31,67),(8,46,32,68),(9,47,33,69),(10,48,34,70),(11,49,35,71),(12,50,36,72),(13,51,37,73),(14,52,38,74),(15,53,39,75),(16,54,40,76),(17,55,21,77),(18,56,22,78),(19,57,23,79),(20,58,24,80)], [(1,30),(2,21),(3,32),(4,23),(5,34),(6,25),(7,36),(8,27),(9,38),(10,29),(11,40),(12,31),(13,22),(14,33),(15,24),(16,35),(17,26),(18,37),(19,28),(20,39),(41,46),(42,57),(43,48),(44,59),(45,50),(47,52),(49,54),(51,56),(53,58),(55,60),(61,66),(62,77),(63,68),(64,79),(65,70),(67,72),(69,74),(71,76),(73,78),(75,80)])
Matrix representation ►G ⊆ 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] >;
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 |
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