metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.11D20, Q8.11D20, D40⋊11C22, C20.61C24, C40.10C23, M4(2)⋊20D10, D20.24C23, Dic20⋊10C22, Dic10.24C23, C8○D4⋊3D5, (C2×C8)⋊6D10, (C2×C40)⋊9C22, (C5×D4).23D4, C20.73(C2×D4), C4.27(C2×D20), C5⋊1(D4○SD16), (C5×Q8).23D4, D4⋊8D10⋊3C2, C8⋊D10⋊11C2, C4○D4.38D10, C4○D20⋊1C22, D40⋊7C2⋊11C2, C4.58(C23×D5), C8.55(C22×D5), C22.3(C2×D20), C8.D10⋊11C2, C40⋊C2⋊11C22, C2.30(C22×D20), C10.28(C22×D4), D4.10D10⋊3C2, (C2×C20).515C23, (C2×Dic10)⋊35C22, (C2×D20).181C22, (C5×M4(2))⋊22C22, (C5×C8○D4)⋊3C2, (C2×C40⋊C2)⋊6C2, (C2×C10).8(C2×D4), (C5×C4○D4).45C22, (C2×C4).226(C22×D5), SmallGroup(320,1423)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1094 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C5, C8, C8 [×3], C2×C4 [×3], C2×C4 [×9], D4 [×3], D4 [×13], Q8, Q8 [×7], C23 [×3], D5 [×4], C10, C10 [×3], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×4], C4○D4, C4○D4 [×10], Dic5 [×4], C20, C20 [×3], D10 [×7], C2×C10 [×3], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), C40, C40 [×3], Dic10, Dic10 [×3], Dic10 [×3], C4×D5 [×6], D20, D20 [×3], D20 [×3], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×3], D4○SD16, C40⋊C2, C40⋊C2 [×9], D40 [×3], Dic20 [×3], C2×C40 [×3], C5×M4(2) [×3], C2×Dic10 [×3], C2×D20 [×3], C4○D20 [×6], D4×D5 [×3], D4⋊2D5 [×3], Q8×D5, Q8⋊2D5, C5×C4○D4, C2×C40⋊C2 [×3], D40⋊7C2 [×3], C8⋊D10 [×3], C8.D10 [×3], C5×C8○D4, D4⋊8D10, D4.10D10, D4.11D20
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, D20 [×4], C22×D5 [×7], D4○SD16, C2×D20 [×6], C23×D5, C22×D20, D4.11D20
Generators and relations
G = < a,b,c,d | a4=b2=1, c20=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c19 >
►On 80 points
(1 68 21 48)(2 69 22 49)(3 70 23 50)(4 71 24 51)(5 72 25 52)(6 73 26 53)(7 74 27 54)(8 75 28 55)(9 76 29 56)(10 77 30 57)(11 78 31 58)(12 79 32 59)(13 80 33 60)(14 41 34 61)(15 42 35 62)(16 43 36 63)(17 44 37 64)(18 45 38 65)(19 46 39 66)(20 47 40 67)
(1 48)(2 49)(3 50)(4 51)(5 52)(6 53)(7 54)(8 55)(9 56)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
(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 21 40)(2 39 22 19)(3 18 23 38)(4 37 24 17)(5 16 25 36)(6 35 26 15)(7 14 27 34)(8 33 28 13)(9 12 29 32)(10 31 30 11)(41 54 61 74)(42 73 62 53)(43 52 63 72)(44 71 64 51)(45 50 65 70)(46 69 66 49)(47 48 67 68)(55 80 75 60)(56 59 76 79)(57 78 77 58)
G:=sub<Sym(80)| (1,68,21,48)(2,69,22,49)(3,70,23,50)(4,71,24,51)(5,72,25,52)(6,73,26,53)(7,74,27,54)(8,75,28,55)(9,76,29,56)(10,77,30,57)(11,78,31,58)(12,79,32,59)(13,80,33,60)(14,41,34,61)(15,42,35,62)(16,43,36,63)(17,44,37,64)(18,45,38,65)(19,46,39,66)(20,47,40,67), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (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), (1,20,21,40)(2,39,22,19)(3,18,23,38)(4,37,24,17)(5,16,25,36)(6,35,26,15)(7,14,27,34)(8,33,28,13)(9,12,29,32)(10,31,30,11)(41,54,61,74)(42,73,62,53)(43,52,63,72)(44,71,64,51)(45,50,65,70)(46,69,66,49)(47,48,67,68)(55,80,75,60)(56,59,76,79)(57,78,77,58)>;
G:=Group( (1,68,21,48)(2,69,22,49)(3,70,23,50)(4,71,24,51)(5,72,25,52)(6,73,26,53)(7,74,27,54)(8,75,28,55)(9,76,29,56)(10,77,30,57)(11,78,31,58)(12,79,32,59)(13,80,33,60)(14,41,34,61)(15,42,35,62)(16,43,36,63)(17,44,37,64)(18,45,38,65)(19,46,39,66)(20,47,40,67), (1,48)(2,49)(3,50)(4,51)(5,52)(6,53)(7,54)(8,55)(9,56)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,78)(32,79)(33,80)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (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), (1,20,21,40)(2,39,22,19)(3,18,23,38)(4,37,24,17)(5,16,25,36)(6,35,26,15)(7,14,27,34)(8,33,28,13)(9,12,29,32)(10,31,30,11)(41,54,61,74)(42,73,62,53)(43,52,63,72)(44,71,64,51)(45,50,65,70)(46,69,66,49)(47,48,67,68)(55,80,75,60)(56,59,76,79)(57,78,77,58) );
G=PermutationGroup([(1,68,21,48),(2,69,22,49),(3,70,23,50),(4,71,24,51),(5,72,25,52),(6,73,26,53),(7,74,27,54),(8,75,28,55),(9,76,29,56),(10,77,30,57),(11,78,31,58),(12,79,32,59),(13,80,33,60),(14,41,34,61),(15,42,35,62),(16,43,36,63),(17,44,37,64),(18,45,38,65),(19,46,39,66),(20,47,40,67)], [(1,48),(2,49),(3,50),(4,51),(5,52),(6,53),(7,54),(8,55),(9,56),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,78),(32,79),(33,80),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47)], [(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)], [(1,20,21,40),(2,39,22,19),(3,18,23,38),(4,37,24,17),(5,16,25,36),(6,35,26,15),(7,14,27,34),(8,33,28,13),(9,12,29,32),(10,31,30,11),(41,54,61,74),(42,73,62,53),(43,52,63,72),(44,71,64,51),(45,50,65,70),(46,69,66,49),(47,48,67,68),(55,80,75,60),(56,59,76,79),(57,78,77,58)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 23 | 16 | 0 | 0 |
| 25 | 12 | 0 | 0 |
| 0 | 0 | 23 | 16 |
| 0 | 0 | 25 | 12 |
| 12 | 14 | 0 | 0 |
| 16 | 29 | 0 | 0 |
| 0 | 0 | 12 | 14 |
| 0 | 0 | 16 | 29 |
G:=sub<GL(4,GF(41))| [0,0,40,0,0,0,0,40,1,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[23,25,0,0,16,12,0,0,0,0,23,25,0,0,16,12],[12,16,0,0,14,29,0,0,0,0,12,16,0,0,14,29] >;
62 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 | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | ··· | 40H | 40I | ··· | 40T |
| 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 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D20 | D4○SD16 | D4.11D20 |
| kernel | D4.11D20 | C2×C40⋊C2 | D40⋊7C2 | C8⋊D10 | C8.D10 | C5×C8○D4 | D4⋊8D10 | D4.10D10 | C5×D4 | C5×Q8 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 3 | 1 | 2 | 6 | 6 | 2 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
D_4._{11}D_{20} % in TeX
G:=Group("D4.11D20"); // GroupNames label
G:=SmallGroup(320,1423);
// by ID
G=gap.SmallGroup(320,1423);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,80,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^20=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^19>;
// generators/relations