metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.5D4, Dic10.5D4, M4(2).4D10, (C2×C4).7D20, (C2×C20).9D4, C4.81(D4×D5), C8⋊D10⋊7C2, C20.98(C2×D4), (C2×Q8).6D10, D20⋊7C4⋊4C2, C4.10D4⋊2D5, C5⋊1(D4.8D4), C10.18C22≀C2, C20.23D4⋊1C2, (C2×C20).10C23, C4○D20.6C22, C22.13(C2×D20), (Q8×C10).8C22, Q8.10D10⋊1C2, (C2×D20).45C22, C2.21(C22⋊D20), (C4×Dic5).8C22, (C5×M4(2)).3C22, (C2×C10).23(C2×D4), (C5×C4.10D4)⋊4C2, (C2×C4).10(C22×D5), SmallGroup(320,380)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.5D4
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, dbd-1=a5b, dcd-1=a5c3 >
Subgroups: 718 in 146 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×5], C22, C22 [×5], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×7], D4 [×8], Q8 [×6], C23, D5 [×3], C10, C10, C42, C22⋊C4 [×2], M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C4○D4 [×6], Dic5 [×3], C20 [×2], C20 [×2], D10 [×5], C2×C10, C4.10D4, C4≀C2 [×2], C4.4D4, C8⋊C22 [×2], 2- 1+4, C40 [×2], Dic10 [×2], Dic10 [×2], C4×D5 [×6], D20 [×2], D20 [×4], C2×Dic5, C5⋊D4 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, D4.8D4, C40⋊C2 [×2], D40 [×2], C4×Dic5, D10⋊C4 [×2], C5×M4(2) [×2], C2×D20, C4○D20 [×2], C4○D20 [×2], Q8×D5 [×2], Q8⋊2D5 [×2], Q8×C10, D20⋊7C4 [×2], C5×C4.10D4, C8⋊D10 [×2], C20.23D4, Q8.10D10, D20.5D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, D20 [×2], C22×D5, D4.8D4, C2×D20, D4×D5 [×2], C22⋊D20, D20.5D4
►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 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 80)(58 79)(59 78)(60 77)
(1 30 6 25 11 40 16 35)(2 21 7 36 12 31 17 26)(3 32 8 27 13 22 18 37)(4 23 9 38 14 33 19 28)(5 34 10 29 15 24 20 39)(41 78 46 73 51 68 56 63)(42 69 47 64 52 79 57 74)(43 80 48 75 53 70 58 65)(44 71 49 66 54 61 59 76)(45 62 50 77 55 72 60 67)
(1 65 6 70 11 75 16 80)(2 66 7 71 12 76 17 61)(3 67 8 72 13 77 18 62)(4 68 9 73 14 78 19 63)(5 69 10 74 15 79 20 64)(21 44 26 49 31 54 36 59)(22 45 27 50 32 55 37 60)(23 46 28 51 33 56 38 41)(24 47 29 52 34 57 39 42)(25 48 30 53 35 58 40 43)
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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,80)(58,79)(59,78)(60,77), (1,30,6,25,11,40,16,35)(2,21,7,36,12,31,17,26)(3,32,8,27,13,22,18,37)(4,23,9,38,14,33,19,28)(5,34,10,29,15,24,20,39)(41,78,46,73,51,68,56,63)(42,69,47,64,52,79,57,74)(43,80,48,75,53,70,58,65)(44,71,49,66,54,61,59,76)(45,62,50,77,55,72,60,67), (1,65,6,70,11,75,16,80)(2,66,7,71,12,76,17,61)(3,67,8,72,13,77,18,62)(4,68,9,73,14,78,19,63)(5,69,10,74,15,79,20,64)(21,44,26,49,31,54,36,59)(22,45,27,50,32,55,37,60)(23,46,28,51,33,56,38,41)(24,47,29,52,34,57,39,42)(25,48,30,53,35,58,40,43)>;
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), (1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,80)(58,79)(59,78)(60,77), (1,30,6,25,11,40,16,35)(2,21,7,36,12,31,17,26)(3,32,8,27,13,22,18,37)(4,23,9,38,14,33,19,28)(5,34,10,29,15,24,20,39)(41,78,46,73,51,68,56,63)(42,69,47,64,52,79,57,74)(43,80,48,75,53,70,58,65)(44,71,49,66,54,61,59,76)(45,62,50,77,55,72,60,67), (1,65,6,70,11,75,16,80)(2,66,7,71,12,76,17,61)(3,67,8,72,13,77,18,62)(4,68,9,73,14,78,19,63)(5,69,10,74,15,79,20,64)(21,44,26,49,31,54,36,59)(22,45,27,50,32,55,37,60)(23,46,28,51,33,56,38,41)(24,47,29,52,34,57,39,42)(25,48,30,53,35,58,40,43) );
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)], [(1,34),(2,33),(3,32),(4,31),(5,30),(6,29),(7,28),(8,27),(9,26),(10,25),(11,24),(12,23),(13,22),(14,21),(15,40),(16,39),(17,38),(18,37),(19,36),(20,35),(41,76),(42,75),(43,74),(44,73),(45,72),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,80),(58,79),(59,78),(60,77)], [(1,30,6,25,11,40,16,35),(2,21,7,36,12,31,17,26),(3,32,8,27,13,22,18,37),(4,23,9,38,14,33,19,28),(5,34,10,29,15,24,20,39),(41,78,46,73,51,68,56,63),(42,69,47,64,52,79,57,74),(43,80,48,75,53,70,58,65),(44,71,49,66,54,61,59,76),(45,62,50,77,55,72,60,67)], [(1,65,6,70,11,75,16,80),(2,66,7,71,12,76,17,61),(3,67,8,72,13,77,18,62),(4,68,9,73,14,78,19,63),(5,69,10,74,15,79,20,64),(21,44,26,49,31,54,36,59),(22,45,27,50,32,55,37,60),(23,46,28,51,33,56,38,41),(24,47,29,52,34,57,39,42),(25,48,30,53,35,58,40,43)])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D20 | D4.8D4 | D4×D5 | D20.5D4 |
| kernel | D20.5D4 | D20⋊7C4 | C5×C4.10D4 | C8⋊D10 | C20.23D4 | Q8.10D10 | Dic10 | D20 | C2×C20 | C4.10D4 | M4(2) | C2×Q8 | C2×C4 | C5 | C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 8 | 2 | 4 | 2 |
Matrix representation of D20.5D4 ►in GL8(𝔽41)
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 35 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 32 | 0 | 32 |
| 0 | 0 | 0 | 0 | 0 | 9 | 9 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 40 | 0 | 40 |
| 0 | 0 | 0 | 0 | 23 | 1 | 32 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 2 | 0 | 32 | 40 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 1 |
| 0 | 0 | 0 | 0 | 18 | 0 | 0 | 40 |
G:=sub<GL(8,GF(41))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,35,0,0,0,0,0,0,40,0,35,0,0,0,0,0,0,0,0,32,25,16,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32],[0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,32,25,16,0,0,0,0,0,5,9,32,9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,23,20,23,0,0,0,0,37,1,40,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,2,0,18,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,4,40,1,40] >;
D20.5D4 in GAP, Magma, Sage, TeX
D_{20}._5D_4 % in TeX
G:=Group("D20.5D4"); // GroupNames label
G:=SmallGroup(320,380);
// by ID
G=gap.SmallGroup(320,380);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,226,1123,570,136,1684,438,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,d*b*d^-1=a^5*b,d*c*d^-1=a^5*c^3>;
// generators/relations