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## G = C4⋊C4⋊21D10order 320 = 26·5

### 4th semidirect product of C4⋊C4 and D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C4⋊C4⋊21D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C4⋊C4⋊21D10
 Lower central C5 — C2×C10 — C4⋊C4⋊21D10
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C4⋊C421D10
G = < a,b,c,d | a4=b4=c10=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b-1, dbd=a2b, dcd=c-1 >

Subgroups: 1198 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×24], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×24], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], D5 [×4], C10 [×3], C10 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×11], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×3], D10 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×8], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4, C4⋊D4, C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], C4×D5 [×6], C2×Dic5 [×2], C2×Dic5 [×4], C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×6], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×2], C22.19C24, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×4], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×4], C2×C4×D5 [×4], D42D5 [×4], C22×Dic5, C22×Dic5 [×2], C2×C5⋊D4 [×4], C22×C20, D4×C10, D4×C10 [×2], C23×D5, Dic54D4 [×2], D10.12D4 [×2], C4⋊C47D5, D102Q8, C20.48D4, D4×Dic5 [×2], C23⋊D10 [×2], C202D4, C5×C4⋊D4, D5×C22×C4, C2×D42D5, C4⋊C421D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C22×D5 [×7], C22.19C24, D4×D5 [×2], D42D5 [×2], C23×D5, C2×D4×D5, C2×D42D5, D5×C4○D4, C4⋊C421D10

Smallest permutation representation of C4⋊C421D10
On 80 points
Generators in S80
```(1 48 13 58)(2 49 14 59)(3 50 15 60)(4 41 16 51)(5 42 17 52)(6 43 18 53)(7 44 19 54)(8 45 20 55)(9 46 11 56)(10 47 12 57)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 78 18 67)(2 68 19 79)(3 80 20 69)(4 70 11 71)(5 72 12 61)(6 62 13 73)(7 74 14 63)(8 64 15 75)(9 76 16 65)(10 66 17 77)(21 52 32 47)(22 48 33 53)(23 54 34 49)(24 50 35 55)(25 56 36 41)(26 42 37 57)(27 58 38 43)(28 44 39 59)(29 60 40 45)(30 46 31 51)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 20)(9 19)(10 18)(21 27)(22 26)(23 25)(28 30)(31 39)(32 38)(33 37)(34 36)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(50 60)(61 67)(62 66)(63 65)(68 70)(71 79)(72 78)(73 77)(74 76)```

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 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 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 C4○D4 D10 D10 D10 D10 D4×D5 D4⋊2D5 D5×C4○D4 kernel C4⋊C4⋊21D10 Dic5⋊4D4 D10.12D4 C4⋊C4⋊7D5 D10⋊2Q8 C20.48D4 D4×Dic5 C23⋊D10 C20⋊2D4 C5×C4⋊D4 D5×C22×C4 C2×D4⋊2D5 C4×D5 C4⋊D4 D10 C2×C10 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C4 C22 C2 # reps 1 2 2 1 1 1 2 2 1 1 1 1 4 2 4 4 4 2 2 6 4 4 4

Matrix representation of C4⋊C421D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 0 9
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 9 0 0 0 0 32 0
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 35 35 0 0 0 0 6 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 35 35 0 0 0 0 40 6 0 0 0 0 0 0 40 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,0,0,0,0,9,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;`

C4⋊C421D10 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4\rtimes_{21}D_{10}`
`% in TeX`

`G:=Group("C4:C4:21D10");`
`// GroupNames label`

`G:=SmallGroup(320,1278);`
`// by ID`

`G=gap.SmallGroup(320,1278);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,1123,794,297,12550]);`
`// Polycyclic`

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

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