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, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8, C22×D5, D4.8D4, C40⋊C2, D40, C4×Dic5, D10⋊C4, C5×M4(2), C2×D20, C4○D20, C4○D20, Q8×D5, Q8⋊2D5, Q8×C10, D20⋊7C4, C5×C4.10D4, C8⋊D10, C20.23D4, Q8.10D10, D20.5D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, D4.8D4, C2×D20, D4×D5, C22⋊D20, D20.5D4
►On 80 points
Generators in S80
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 40)(17 39)(18 38)(19 37)(20 36)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 70)
(1 31 6 26 11 21 16 36)(2 22 7 37 12 32 17 27)(3 33 8 28 13 23 18 38)(4 24 9 39 14 34 19 29)(5 35 10 30 15 25 20 40)(41 63 46 78 51 73 56 68)(42 74 47 69 52 64 57 79)(43 65 48 80 53 75 58 70)(44 76 49 71 54 66 59 61)(45 67 50 62 55 77 60 72)
(1 79 6 64 11 69 16 74)(2 80 7 65 12 70 17 75)(3 61 8 66 13 71 18 76)(4 62 9 67 14 72 19 77)(5 63 10 68 15 73 20 78)(21 42 26 47 31 52 36 57)(22 43 27 48 32 53 37 58)(23 44 28 49 33 54 38 59)(24 45 29 50 34 55 39 60)(25 46 30 51 35 56 40 41)
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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,40)(17,39)(18,38)(19,37)(20,36)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,70), (1,31,6,26,11,21,16,36)(2,22,7,37,12,32,17,27)(3,33,8,28,13,23,18,38)(4,24,9,39,14,34,19,29)(5,35,10,30,15,25,20,40)(41,63,46,78,51,73,56,68)(42,74,47,69,52,64,57,79)(43,65,48,80,53,75,58,70)(44,76,49,71,54,66,59,61)(45,67,50,62,55,77,60,72), (1,79,6,64,11,69,16,74)(2,80,7,65,12,70,17,75)(3,61,8,66,13,71,18,76)(4,62,9,67,14,72,19,77)(5,63,10,68,15,73,20,78)(21,42,26,47,31,52,36,57)(22,43,27,48,32,53,37,58)(23,44,28,49,33,54,38,59)(24,45,29,50,34,55,39,60)(25,46,30,51,35,56,40,41)>;
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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,40)(17,39)(18,38)(19,37)(20,36)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,70), (1,31,6,26,11,21,16,36)(2,22,7,37,12,32,17,27)(3,33,8,28,13,23,18,38)(4,24,9,39,14,34,19,29)(5,35,10,30,15,25,20,40)(41,63,46,78,51,73,56,68)(42,74,47,69,52,64,57,79)(43,65,48,80,53,75,58,70)(44,76,49,71,54,66,59,61)(45,67,50,62,55,77,60,72), (1,79,6,64,11,69,16,74)(2,80,7,65,12,70,17,75)(3,61,8,66,13,71,18,76)(4,62,9,67,14,72,19,77)(5,63,10,68,15,73,20,78)(21,42,26,47,31,52,36,57)(22,43,27,48,32,53,37,58)(23,44,28,49,33,54,38,59)(24,45,29,50,34,55,39,60)(25,46,30,51,35,56,40,41) );
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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,40),(17,39),(18,38),(19,37),(20,36),(41,69),(42,68),(43,67),(44,66),(45,65),(46,64),(47,63),(48,62),(49,61),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,72),(59,71),(60,70)], [(1,31,6,26,11,21,16,36),(2,22,7,37,12,32,17,27),(3,33,8,28,13,23,18,38),(4,24,9,39,14,34,19,29),(5,35,10,30,15,25,20,40),(41,63,46,78,51,73,56,68),(42,74,47,69,52,64,57,79),(43,65,48,80,53,75,58,70),(44,76,49,71,54,66,59,61),(45,67,50,62,55,77,60,72)], [(1,79,6,64,11,69,16,74),(2,80,7,65,12,70,17,75),(3,61,8,66,13,71,18,76),(4,62,9,67,14,72,19,77),(5,63,10,68,15,73,20,78),(21,42,26,47,31,52,36,57),(22,43,27,48,32,53,37,58),(23,44,28,49,33,54,38,59),(24,45,29,50,34,55,39,60),(25,46,30,51,35,56,40,41)]])
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