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## G = C4○D20⋊C4order 320 = 26·5

### 3rd semidirect product of C4○D20 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C4○D20⋊C4
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — D5⋊C8 — C2×D5⋊C8 — C4○D20⋊C4
 Lower central C5 — C10 — C20 — C4○D20⋊C4
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for C4○D20⋊C4
G = < a,b,c,d | a4=c2=d4=1, b10=a2, ab=ba, ac=ca, ad=da, cbc=a2b9, dbd-1=b7, dcd-1=a2bc >

Subgroups: 682 in 158 conjugacy classes, 50 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C5⋊C8, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×F5, C22×D5, C22×D5, C23.24D4, D5⋊C8, D5⋊C8, C4×F5, C4⋊F5, C2×C5⋊C8, C22⋊F5, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, D20⋊C4, Q8⋊F5, C2×D5⋊C8, D10.C23, D5×C4○D4, C4○D20⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, F5, C2×C22⋊C4, C4○D8, C2×F5, C23.24D4, C22⋊F5, C22×F5, C2×C22⋊F5, C4○D20⋊C4

Smallest permutation representation of C4○D20⋊C4
On 80 points
Generators in S80
```(1 58 11 48)(2 59 12 49)(3 60 13 50)(4 41 14 51)(5 42 15 52)(6 43 16 53)(7 44 17 54)(8 45 18 55)(9 46 19 56)(10 47 20 57)(21 67 31 77)(22 68 32 78)(23 69 33 79)(24 70 34 80)(25 71 35 61)(26 72 36 62)(27 73 37 63)(28 74 38 64)(29 75 39 65)(30 76 40 66)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(20 40)(41 62)(42 61)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)
(1 6)(2 9 10 13)(3 12 19 20)(4 15 8 7)(5 18 17 14)(11 16)(21 33 29 37)(22 36 38 24)(23 39 27 31)(26 28 34 32)(30 40)(41 52 45 44)(42 55 54 51)(43 58)(46 47 50 59)(48 53)(49 56 57 60)(62 64 70 68)(63 67 79 75)(65 73 77 69)(66 76)(72 74 80 78)```

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 D4 C4○D8 F5 C2×F5 C2×F5 C2×F5 C22⋊F5 C22⋊F5 C4○D20⋊C4 kernel C4○D20⋊C4 D20⋊C4 Q8⋊F5 C2×D5⋊C8 D10.C23 D5×C4○D4 C4○D20 D4⋊2D5 Q8⋊2D5 C5×C4○D4 C4×D5 C2×Dic5 C22×D5 D5 C4○D4 C2×C4 D4 Q8 C4 C22 C1 # reps 1 2 2 1 1 1 2 2 2 2 2 1 1 8 1 1 1 1 2 2 2

Matrix representation of C4○D20⋊C4 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 16 0 0 0 0 5 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 40 40 40
,
 0 28 0 0 0 0 22 0 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 1 1 1 1 0 0 0 0 0 40
,
 32 20 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 40 40 40 40 0 0 0 1 0 0

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,5,0,0,0,0,16,40,0,0,0,0,0,0,0,0,0,40,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,1,40],[0,22,0,0,0,0,28,0,0,0,0,0,0,0,0,40,1,0,0,0,40,0,1,0,0,0,0,0,1,0,0,0,0,0,1,40],[32,0,0,0,0,0,20,9,0,0,0,0,0,0,1,0,40,0,0,0,0,0,40,1,0,0,0,1,40,0,0,0,0,0,40,0] >;`

C4○D20⋊C4 in GAP, Magma, Sage, TeX

`C_4\circ D_{20}\rtimes C_4`
`% in TeX`

`G:=Group("C4oD20:C4");`
`// GroupNames label`

`G:=SmallGroup(320,1132);`
`// by ID`

`G=gap.SmallGroup(320,1132);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,297,136,438,102,6278,1595]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^2=d^4=1,b^10=a^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=a^2*b^9,d*b*d^-1=b^7,d*c*d^-1=a^2*b*c>;`
`// generators/relations`

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