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## G = C42⋊11D10order 320 = 26·5

### 11st semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42⋊11D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — C42⋊11D10
 Lower central C5 — C10 — C42⋊11D10
 Upper central C1 — C22 — C4×D4

Generators and relations for C4211D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 1294 in 338 conjugacy classes, 151 normal (43 characteristic)
C1, C2 [×3], C2 [×10], C4 [×2], C4 [×10], C22, C22 [×4], C22 [×24], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×17], D4 [×4], D4 [×12], C23 [×2], C23 [×15], D5 [×6], C10 [×3], C10 [×4], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×7], C2×D4, C2×D4 [×11], C24 [×2], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×6], D10 [×14], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C22⋊C4 [×4], C42⋊C2 [×2], C4×D4, C4×D4 [×7], C22×D4, C4×D5 [×4], C4×D5 [×4], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5, C22×D5 [×10], C22×D5 [×4], C22×C10 [×2], C22.11C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×C4×D5 [×4], C2×D20, D4×D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C23×D5 [×2], C42⋊D5, C4×D20, D5×C22⋊C4 [×2], Dic54D4 [×2], C4⋊C47D5, D208C4, C2×D10⋊C4 [×2], C4×C5⋊D4 [×2], D4×Dic5, D4×C20, C2×D4×D5, C4211D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2+ 1+4 [×2], C4×D5 [×4], C22×D5 [×7], C22.11C24, C2×C4×D5 [×6], C23×D5, D5×C22×C4, D46D10, D48D10, C4211D10

Smallest permutation representation of C4211D10
On 80 points
Generators in S80
```(1 70 30 53)(2 54 21 61)(3 62 22 55)(4 56 23 63)(5 64 24 57)(6 58 25 65)(7 66 26 59)(8 60 27 67)(9 68 28 51)(10 52 29 69)(11 45 73 32)(12 33 74 46)(13 47 75 34)(14 35 76 48)(15 49 77 36)(16 37 78 50)(17 41 79 38)(18 39 80 42)(19 43 71 40)(20 31 72 44)
(1 37 6 32)(2 38 7 33)(3 39 8 34)(4 40 9 35)(5 31 10 36)(11 70 78 58)(12 61 79 59)(13 62 80 60)(14 63 71 51)(15 64 72 52)(16 65 73 53)(17 66 74 54)(18 67 75 55)(19 68 76 56)(20 69 77 57)(21 41 26 46)(22 42 27 47)(23 43 28 48)(24 44 29 49)(25 45 30 50)
(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 72)(12 71)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 28)(22 27)(23 26)(24 25)(29 30)(31 45)(32 44)(33 43)(34 42)(35 41)(36 50)(37 49)(38 48)(39 47)(40 46)(51 54)(52 53)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 70)```

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H ··· 2M 4A ··· 4J 4K ··· 4T 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 10 ··· 10 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 D5 D10 D10 D10 D10 D10 C4×D5 2+ 1+4 D4⋊6D10 D4⋊8D10 kernel C42⋊11D10 C42⋊D5 C4×D20 D5×C22⋊C4 Dic5⋊4D4 C4⋊C4⋊7D5 D20⋊8C4 C2×D10⋊C4 C4×C5⋊D4 D4×Dic5 D4×C20 C2×D4×D5 D4×D5 C4×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C10 C2 C2 # reps 1 1 1 2 2 1 1 2 2 1 1 1 16 2 2 4 2 4 2 16 2 4 4

Matrix representation of C4211D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 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
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 2 13 0 0 0 0 28 39 0 0 0 0 0 0 2 13 0 0 0 0 28 39
,
 40 7 0 0 0 0 34 7 0 0 0 0 0 0 6 6 0 0 0 0 35 1 0 0 0 0 0 0 35 35 0 0 0 0 6 40
,
 1 0 0 0 0 0 7 40 0 0 0 0 0 0 6 6 0 0 0 0 1 35 0 0 0 0 0 0 35 35 0 0 0 0 40 6

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,40,0,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,2,28,0,0,0,0,13,39],[40,34,0,0,0,0,7,7,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,35,6,0,0,0,0,35,40],[1,7,0,0,0,0,0,40,0,0,0,0,0,0,6,1,0,0,0,0,6,35,0,0,0,0,0,0,35,40,0,0,0,0,35,6] >;`

C4211D10 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{11}D_{10}`
`% in TeX`

`G:=Group("C4^2:11D10");`
`// GroupNames label`

`G:=SmallGroup(320,1217);`
`// by ID`

`G=gap.SmallGroup(320,1217);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,80,12550]);`
`// Polycyclic`

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

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