metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.6D4, Dic10.6D4, M4(2).5D10, C20.99(C2×D4), C4.152(D4×D5), C5⋊2C8.42D4, (C2×Q8).7D10, C8⋊D10.1C2, C4.10D4⋊3D5, C5⋊2(D4.3D4), D20.2C4⋊7C2, C20.46D4⋊8C2, C20.C23⋊1C2, C20.53D4⋊3C2, (C2×C20).11C23, C4○D20.7C22, (Q8×C10).9C22, C10.11(C4⋊D4), (C2×D20).46C22, C2.14(D10⋊D4), C4.Dic5.6C22, C22.15(C4○D20), (C5×M4(2)).14C22, (C2×Q8⋊D5)⋊1C2, (C5×C4.10D4)⋊1C2, (C2×C5⋊2C8).3C22, (C2×C4).11(C22×D5), (C2×C10).32(C4○D4), SmallGroup(320,381)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.6D4
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a15, bab=a-1, cac-1=a11, ad=da, cbc-1=a10b, dbd-1=a15b, dcd-1=a15c3 >
Subgroups: 478 in 104 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C5⋊2C8, C5⋊2C8, C40, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C2×C20, C5×Q8, C22×D5, D4.3D4, C8×D5, C8⋊D5, C40⋊C2, D40, C2×C5⋊2C8, C4.Dic5, Q8⋊D5, C5⋊Q16, C5×M4(2), C2×D20, C4○D20, Q8×C10, C20.53D4, C20.46D4, C5×C4.10D4, D20.2C4, C8⋊D10, C2×Q8⋊D5, C20.C23, D20.6D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C22×D5, D4.3D4, C4○D20, D4×D5, D10⋊D4, D20.6D4
►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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 80)(19 79)(20 78)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 60)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)
(1 48 6 43 11 58 16 53)(2 59 7 54 12 49 17 44)(3 50 8 45 13 60 18 55)(4 41 9 56 14 51 19 46)(5 52 10 47 15 42 20 57)(21 71 26 66 31 61 36 76)(22 62 27 77 32 72 37 67)(23 73 28 68 33 63 38 78)(24 64 29 79 34 74 39 69)(25 75 30 70 35 65 40 80)
(1 73 16 68 11 63 6 78)(2 74 17 69 12 64 7 79)(3 75 18 70 13 65 8 80)(4 76 19 71 14 66 9 61)(5 77 20 72 15 67 10 62)(21 56 36 51 31 46 26 41)(22 57 37 52 32 47 27 42)(23 58 38 53 33 48 28 43)(24 59 39 54 34 49 29 44)(25 60 40 55 35 50 30 45)
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,80)(19,79)(20,78)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50), (1,48,6,43,11,58,16,53)(2,59,7,54,12,49,17,44)(3,50,8,45,13,60,18,55)(4,41,9,56,14,51,19,46)(5,52,10,47,15,42,20,57)(21,71,26,66,31,61,36,76)(22,62,27,77,32,72,37,67)(23,73,28,68,33,63,38,78)(24,64,29,79,34,74,39,69)(25,75,30,70,35,65,40,80), (1,73,16,68,11,63,6,78)(2,74,17,69,12,64,7,79)(3,75,18,70,13,65,8,80)(4,76,19,71,14,66,9,61)(5,77,20,72,15,67,10,62)(21,56,36,51,31,46,26,41)(22,57,37,52,32,47,27,42)(23,58,38,53,33,48,28,43)(24,59,39,54,34,49,29,44)(25,60,40,55,35,50,30,45)>;
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,80)(19,79)(20,78)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50), (1,48,6,43,11,58,16,53)(2,59,7,54,12,49,17,44)(3,50,8,45,13,60,18,55)(4,41,9,56,14,51,19,46)(5,52,10,47,15,42,20,57)(21,71,26,66,31,61,36,76)(22,62,27,77,32,72,37,67)(23,73,28,68,33,63,38,78)(24,64,29,79,34,74,39,69)(25,75,30,70,35,65,40,80), (1,73,16,68,11,63,6,78)(2,74,17,69,12,64,7,79)(3,75,18,70,13,65,8,80)(4,76,19,71,14,66,9,61)(5,77,20,72,15,67,10,62)(21,56,36,51,31,46,26,41)(22,57,37,52,32,47,27,42)(23,58,38,53,33,48,28,43)(24,59,39,54,34,49,29,44)(25,60,40,55,35,50,30,45) );
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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,80),(19,79),(20,78),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(28,42),(29,41),(30,60),(31,59),(32,58),(33,57),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,50)], [(1,48,6,43,11,58,16,53),(2,59,7,54,12,49,17,44),(3,50,8,45,13,60,18,55),(4,41,9,56,14,51,19,46),(5,52,10,47,15,42,20,57),(21,71,26,66,31,61,36,76),(22,62,27,77,32,72,37,67),(23,73,28,68,33,63,38,78),(24,64,29,79,34,74,39,69),(25,75,30,70,35,65,40,80)], [(1,73,16,68,11,63,6,78),(2,74,17,69,12,64,7,79),(3,75,18,70,13,65,8,80),(4,76,19,71,14,66,9,61),(5,77,20,72,15,67,10,62),(21,56,36,51,31,46,26,41),(22,57,37,52,32,47,27,42),(23,58,38,53,33,48,28,43),(24,59,39,54,34,49,29,44),(25,60,40,55,35,50,30,45)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 40 | 2 | 2 | 8 | 20 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C4○D20 | D4.3D4 | D4×D5 | D20.6D4 |
| kernel | D20.6D4 | C20.53D4 | C20.46D4 | C5×C4.10D4 | D20.2C4 | C8⋊D10 | C2×Q8⋊D5 | C20.C23 | C5⋊2C8 | Dic10 | D20 | C4.10D4 | C2×C10 | M4(2) | C2×Q8 | C22 | C5 | C4 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 8 | 2 | 4 | 2 |
Matrix representation of D20.6D4 ►in GL6(𝔽41)
| 40 | 1 | 0 | 0 | 0 | 0 |
| 33 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 40 |
| 0 | 0 | 37 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 33 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 29 | 24 | 17 |
| 0 | 0 | 15 | 15 | 24 | 24 |
| 0 | 0 | 6 | 40 | 26 | 12 |
| 0 | 0 | 13 | 6 | 26 | 29 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 37 | 18 | 0 |
| 0 | 0 | 37 | 5 | 0 | 18 |
| 0 | 0 | 0 | 0 | 36 | 4 |
| 0 | 0 | 9 | 0 | 4 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 15 | 24 | 24 |
| 0 | 0 | 12 | 29 | 24 | 17 |
| 0 | 0 | 25 | 35 | 26 | 29 |
| 0 | 0 | 35 | 28 | 26 | 12 |
G:=sub<GL(6,GF(41))| [40,33,0,0,0,0,1,7,0,0,0,0,0,0,0,40,0,37,0,0,1,0,4,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,33,0,0,0,0,0,1,0,0,0,0,0,0,12,15,6,13,0,0,29,15,40,6,0,0,24,24,26,26,0,0,17,24,12,29],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,37,0,9,0,0,37,5,0,0,0,0,18,0,36,4,0,0,0,18,4,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,12,25,35,0,0,15,29,35,28,0,0,24,24,26,26,0,0,24,17,29,12] >;
D20.6D4 in GAP, Magma, Sage, TeX
D_{20}._6D_4 % in TeX
G:=Group("D20.6D4"); // GroupNames label
G:=SmallGroup(320,381);
// by ID
G=gap.SmallGroup(320,381);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,184,297,136,1684,851,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^15,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^10*b,d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^3>;
// generators/relations