Copied to
clipboard

## G = D40⋊C22order 320 = 26·5

### 3rd semidirect product of D40 and C22 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D40⋊C22
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — D5×C4○D4 — D40⋊C22
 Lower central C5 — C10 — C20 — D40⋊C22
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for D40⋊C22
G = < a,b,c,d | a40=b2=c2=d2=1, bab=a-1, cac=a21, dad=a29, cbc=a20b, dbd=a8b, cd=dc >

Subgroups: 1038 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×11], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4, D4 [×13], Q8, Q8 [×2], Q8 [×3], C23 [×3], D5 [×5], C10, C10 [×2], C2×C8 [×2], M4(2), M4(2) [×3], D8 [×4], SD16 [×2], SD16 [×6], Q16 [×2], Q16 [×2], C22×C4 [×3], C2×D4 [×4], C2×Q8, C2×Q8, C4○D4, C4○D4 [×11], Dic5 [×2], Dic5, C20 [×2], C20 [×3], D10 [×2], D10 [×8], C2×C10, C2×C10, C2×M4(2), C4○D8 [×4], C8⋊C22 [×4], C8.C22, C8.C22 [×3], C2×C4○D4 [×2], C52C8 [×2], C40 [×2], Dic10, Dic10, C4×D5 [×4], C4×D5 [×7], D20, D20 [×2], D20 [×6], C2×Dic5, C2×Dic5, C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C5×Q8, C22×D5, C22×D5 [×2], D8⋊C22, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40 [×2], C4.Dic5, D4⋊D5 [×2], Q8⋊D5 [×4], C5⋊Q16 [×2], C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, Q82D5 [×4], Q82D5 [×2], Q8×C10, C5×C4○D4, D5×M4(2), C8⋊D10, D40⋊C2 [×2], SD163D5 [×2], Q16⋊D5 [×2], Q8.D10 [×2], C20.C23, D4⋊D10, C5×C8.C22, C2×Q82D5, D5×C4○D4, D40⋊C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D8⋊C22, D4×D5 [×2], C23×D5, C2×D4×D5, D40⋊C22

Smallest permutation representation of D40⋊C22
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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(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 60)(58 59)(77 80)(78 79)
(1 21)(3 23)(5 25)(7 27)(9 29)(11 31)(13 33)(15 35)(17 37)(19 39)(41 61)(43 63)(45 65)(47 67)(49 69)(51 71)(53 73)(55 75)(57 77)(59 79)
(1 49)(2 78)(3 67)(4 56)(5 45)(6 74)(7 63)(8 52)(9 41)(10 70)(11 59)(12 48)(13 77)(14 66)(15 55)(16 44)(17 73)(18 62)(19 51)(20 80)(21 69)(22 58)(23 47)(24 76)(25 65)(26 54)(27 43)(28 72)(29 61)(30 50)(31 79)(32 68)(33 57)(34 46)(35 75)(36 64)(37 53)(38 42)(39 71)(40 60)```

`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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(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)(57,60)(58,59)(77,80)(78,79), (1,21)(3,23)(5,25)(7,27)(9,29)(11,31)(13,33)(15,35)(17,37)(19,39)(41,61)(43,63)(45,65)(47,67)(49,69)(51,71)(53,73)(55,75)(57,77)(59,79), (1,49)(2,78)(3,67)(4,56)(5,45)(6,74)(7,63)(8,52)(9,41)(10,70)(11,59)(12,48)(13,77)(14,66)(15,55)(16,44)(17,73)(18,62)(19,51)(20,80)(21,69)(22,58)(23,47)(24,76)(25,65)(26,54)(27,43)(28,72)(29,61)(30,50)(31,79)(32,68)(33,57)(34,46)(35,75)(36,64)(37,53)(38,42)(39,71)(40,60)>;`

`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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(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)(57,60)(58,59)(77,80)(78,79), (1,21)(3,23)(5,25)(7,27)(9,29)(11,31)(13,33)(15,35)(17,37)(19,39)(41,61)(43,63)(45,65)(47,67)(49,69)(51,71)(53,73)(55,75)(57,77)(59,79), (1,49)(2,78)(3,67)(4,56)(5,45)(6,74)(7,63)(8,52)(9,41)(10,70)(11,59)(12,48)(13,77)(14,66)(15,55)(16,44)(17,73)(18,62)(19,51)(20,80)(21,69)(22,58)(23,47)(24,76)(25,65)(26,54)(27,43)(28,72)(29,61)(30,50)(31,79)(32,68)(33,57)(34,46)(35,75)(36,64)(37,53)(38,42)(39,71)(40,60) );`

`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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(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),(57,60),(58,59),(77,80),(78,79)], [(1,21),(3,23),(5,25),(7,27),(9,29),(11,31),(13,33),(15,35),(17,37),(19,39),(41,61),(43,63),(45,65),(47,67),(49,69),(51,71),(53,73),(55,75),(57,77),(59,79)], [(1,49),(2,78),(3,67),(4,56),(5,45),(6,74),(7,63),(8,52),(9,41),(10,70),(11,59),(12,48),(13,77),(14,66),(15,55),(16,44),(17,73),(18,62),(19,51),(20,80),(21,69),(22,58),(23,47),(24,76),(25,65),(26,54),(27,43),(28,72),(29,61),(30,50),(31,79),(32,68),(33,57),(34,46),(35,75),(36,64),(37,53),(38,42),(39,71),(40,60)])`

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 20A 20B 20C 20D 20E ··· 20J 40A 40B 40C 40D order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 10 10 10 10 10 20 20 20 20 20 ··· 20 40 40 40 40 size 1 1 2 4 10 10 20 20 20 2 2 4 4 4 5 5 10 20 2 2 4 4 20 20 2 2 4 4 8 8 4 4 4 4 8 ··· 8 8 8 8 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 D10 D10 D8⋊C22 D4×D5 D4×D5 D40⋊C22 kernel D40⋊C22 D5×M4(2) C8⋊D10 D40⋊C2 SD16⋊3D5 Q16⋊D5 Q8.D10 C20.C23 D4⋊D10 C5×C8.C22 C2×Q8⋊2D5 D5×C4○D4 C4×D5 C2×Dic5 C22×D5 C8.C22 M4(2) SD16 Q16 C2×Q8 C4○D4 C5 C4 C22 C1 # reps 1 1 1 2 2 2 2 1 1 1 1 1 2 1 1 2 2 4 4 2 2 2 2 2 2

Matrix representation of D40⋊C22 in GL8(𝔽41)

 28 13 0 27 0 0 0 0 28 19 14 0 0 0 0 0 32 19 22 28 0 0 0 0 0 32 13 13 0 0 0 0 0 0 0 0 3 3 25 0 0 0 0 0 2 2 33 9 0 0 0 0 1 1 36 0 0 0 0 0 0 9 23 0
,
 28 13 0 27 0 0 0 0 22 13 27 16 0 0 0 0 22 9 19 18 0 0 0 0 9 0 28 22 0 0 0 0 0 0 0 0 38 38 16 9 0 0 0 0 39 39 8 32 0 0 0 0 40 40 5 0 0 0 0 0 23 32 31 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 6 0 0 0 0 0 0 40 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 30 0 28 22 0 0 0 0 36 11 22 23 0 0 0 0 34 37 5 36 0 0 0 0 37 20 30 36 0 0 0 0 0 0 0 0 9 9 23 38 0 0 0 0 23 32 8 39 0 0 0 0 0 0 9 40 0 0 0 0 0 0 39 32

`G:=sub<GL(8,GF(41))| [28,28,32,0,0,0,0,0,13,19,19,32,0,0,0,0,0,14,22,13,0,0,0,0,27,0,28,13,0,0,0,0,0,0,0,0,3,2,1,0,0,0,0,0,3,2,1,9,0,0,0,0,25,33,36,23,0,0,0,0,0,9,0,0],[28,22,22,9,0,0,0,0,13,13,9,0,0,0,0,0,0,27,19,28,0,0,0,0,27,16,18,22,0,0,0,0,0,0,0,0,38,39,40,23,0,0,0,0,38,39,40,32,0,0,0,0,16,8,5,31,0,0,0,0,9,32,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,6,4,1,0,0,0,0,0,0,0,0,1],[30,36,34,37,0,0,0,0,0,11,37,20,0,0,0,0,28,22,5,30,0,0,0,0,22,23,36,36,0,0,0,0,0,0,0,0,9,23,0,0,0,0,0,0,9,32,0,0,0,0,0,0,23,8,9,39,0,0,0,0,38,39,40,32] >;`

D40⋊C22 in GAP, Magma, Sage, TeX

`D_{40}\rtimes C_2^2`
`% in TeX`

`G:=Group("D40:C2^2");`
`// GroupNames label`

`G:=SmallGroup(320,1449);`
`// by ID`

`G=gap.SmallGroup(320,1449);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,185,136,438,235,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^40=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^21,d*a*d=a^29,c*b*c=a^20*b,d*b*d=a^8*b,c*d=d*c>;`
`// generators/relations`

׿
×
𝔽