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## G = D40⋊7C2order 160 = 25·5

### The semidirect product of D40 and C2 acting through Inn(D40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D40⋊7C2
 Chief series C1 — C5 — C10 — C20 — D20 — C4○D20 — D40⋊7C2
 Lower central C5 — C10 — C20 — D40⋊7C2
 Upper central C1 — C4 — C2×C4 — C2×C8

Generators and relations for D407C2
G = < a,b,c | a40=b2=c2=1, bab=a-1, ac=ca, cbc=a20b >

Subgroups: 232 in 62 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, D10, C2×C10, C4○D8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C40⋊C2, D40, Dic20, C2×C40, C4○D20, D407C2
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4○D8, D20, C22×D5, C2×D20, D407C2

Smallest permutation representation of D407C2
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(41 51)(42 50)(43 49)(44 48)(45 47)(52 80)(53 79)(54 78)(55 77)(56 76)(57 75)(58 74)(59 73)(60 72)(61 71)(62 70)(63 69)(64 68)(65 67)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)```

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

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

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

46 conjugacy classes

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

46 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 C4○D8 D20 D20 D40⋊7C2 kernel D40⋊7C2 C40⋊C2 D40 Dic20 C2×C40 C4○D20 C20 C2×C10 C2×C8 C8 C2×C4 C5 C4 C22 C1 # reps 1 2 1 1 1 2 1 1 2 4 2 4 4 4 16

Matrix representation of D407C2 in GL2(𝔽41) generated by

 26 21 23 6
,
 27 5 2 14
,
 24 34 6 17
`G:=sub<GL(2,GF(41))| [26,23,21,6],[27,2,5,14],[24,6,34,17] >;`

D407C2 in GAP, Magma, Sage, TeX

`D_{40}\rtimes_7C_2`
`% in TeX`

`G:=Group("D40:7C2");`
`// GroupNames label`

`G:=SmallGroup(160,125);`
`// by ID`

`G=gap.SmallGroup(160,125);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,50,579,69,4613]);`
`// Polycyclic`

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

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