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## G = C42⋊5D10order 320 = 26·5

### 5th semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42⋊5D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4⋊6D10 — C42⋊5D10
 Lower central C5 — C10 — C2×C20 — C42⋊5D10
 Upper central C1 — C2 — C2×C4 — C4.4D4

Generators and relations for C425D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1b2, dad=a-1b, cbc-1=b-1, bd=db, dcd=c-1 >

Subgroups: 686 in 152 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C52C8, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, D4.9D4, C4.Dic5, Q8⋊D5, C5⋊Q16, C4×C20, C5×C22⋊C4, C4○D20, D4×D5, D42D5, C2×C5⋊D4, D4×C10, Q8×C10, D204C4, C20.D4, C20.C23, C5×C4.4D4, D46D10, C425D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, D4.9D4, D4×D5, C2×C5⋊D4, C23⋊D10, C425D10

Smallest permutation representation of C425D10
On 80 points
Generators in S80
```(1 31 23 70)(2 79 24 57)(3 33 25 62)(4 71 26 59)(5 35 27 64)(6 73 28 51)(7 37 29 66)(8 75 30 53)(9 39 21 68)(10 77 22 55)(11 72 44 60)(12 36 45 65)(13 74 46 52)(14 38 47 67)(15 76 48 54)(16 40 49 69)(17 78 50 56)(18 32 41 61)(19 80 42 58)(20 34 43 63)
(1 28 50 12)(2 13 41 29)(3 30 42 14)(4 15 43 21)(5 22 44 16)(6 17 45 23)(7 24 46 18)(8 19 47 25)(9 26 48 20)(10 11 49 27)(31 51 56 36)(32 37 57 52)(33 53 58 38)(34 39 59 54)(35 55 60 40)(61 66 79 74)(62 75 80 67)(63 68 71 76)(64 77 72 69)(65 70 73 78)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 30)(20 29)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 48)(42 47)(43 46)(44 45)(49 50)(61 63)(64 70)(65 69)(66 68)(71 79)(72 78)(73 77)(74 76)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 D10 D10 D10 C5⋊D4 D4.9D4 D4×D5 C42⋊5D10 kernel C42⋊5D10 D20⋊4C4 C20.D4 C20.C23 C5×C4.4D4 D4⋊6D10 Dic10 D20 C22×C10 C4.4D4 C42 C2×D4 C2×Q8 C23 C5 C4 C1 # reps 1 2 1 2 1 1 2 2 2 2 2 2 2 8 2 4 8

Matrix representation of C425D10 in GL6(𝔽41)

 17 40 0 0 0 0 1 24 0 0 0 0 0 0 16 16 25 16 0 0 16 16 16 25 0 0 16 25 16 16 0 0 25 16 16 16
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 0 0 0 0 0 0 40 0 0
,
 7 7 0 0 0 0 34 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
,
 7 7 0 0 0 0 40 34 0 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 1 0 0 0

`G:=sub<GL(6,GF(41))| [17,1,0,0,0,0,40,24,0,0,0,0,0,0,16,16,16,25,0,0,16,16,25,16,0,0,25,16,16,16,0,0,16,25,16,16],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[7,34,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1,0,0,0] >;`

C425D10 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_5D_{10}`
`% in TeX`

`G:=Group("C4^2:5D10");`
`// GroupNames label`

`G:=SmallGroup(320,688);`
`// by ID`

`G=gap.SmallGroup(320,688);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,184,1123,570,297,136,1684,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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