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## G = C42⋊12D10order 320 = 26·5

### 12nd semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊12D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C42⋊12D10
 Lower central C5 — C2×C10 — C42⋊12D10
 Upper central C1 — C2×C4 — C4×D4

Generators and relations for C4212D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1246 in 330 conjugacy classes, 109 normal (91 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×24], C5, C2×C4 [×5], C2×C4 [×23], D4 [×14], Q8 [×2], C23 [×2], C23 [×9], D5 [×5], C10 [×3], C10 [×3], C42, C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×5], C22×C4 [×2], C22×C4 [×10], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×4], D10 [×15], C2×C10, C2×C10 [×2], C2×C10 [×5], C42⋊C2, C4×D4, C4×D4 [×3], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×5], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×D5 [×3], C22×D5 [×6], C22×C10 [×2], C22.19C24, C4×Dic5, C10.D4 [×4], C4⋊Dic5, D10⋊C4 [×6], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×5], C2×C4×D5 [×4], C2×D20 [×2], C4○D20 [×4], C22×Dic5, C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C23×D5, C42⋊D5, C4×D20, Dic5.14D4, C22⋊D20, D10.12D4, D10⋊D4, D10.13D4, D10⋊Q8, C4×C5⋊D4 [×2], C23⋊D10, Dic5⋊D4, D4×C20, D5×C22×C4, C2×C4○D20, C4212D10
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, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D5×C4○D4, C4212D10

Smallest permutation representation of C4212D10
On 80 points
Generators in S80
```(1 33 25 45)(2 46 26 34)(3 35 27 47)(4 48 28 36)(5 37 29 49)(6 50 30 38)(7 39 21 41)(8 42 22 40)(9 31 23 43)(10 44 24 32)(11 60 73 65)(12 66 74 51)(13 52 75 67)(14 68 76 53)(15 54 77 69)(16 70 78 55)(17 56 79 61)(18 62 80 57)(19 58 71 63)(20 64 72 59)
(1 55 6 60)(2 56 7 51)(3 57 8 52)(4 58 9 53)(5 59 10 54)(11 45 78 38)(12 46 79 39)(13 47 80 40)(14 48 71 31)(15 49 72 32)(16 50 73 33)(17 41 74 34)(18 42 75 35)(19 43 76 36)(20 44 77 37)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)
(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 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 80)(19 79)(20 78)(21 28)(22 27)(23 26)(24 25)(29 30)(31 41)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(51 58)(52 57)(53 56)(54 55)(59 60)(61 68)(62 67)(63 66)(64 65)(69 70)```

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

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

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

68 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 ··· 10N 20A ··· 20H 20I ··· 20X 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 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 10 10 10 10 20 1 1 1 1 2 2 4 4 4 10 10 10 10 20 20 20 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 C4○D4 D10 D10 D10 D10 D10 C4○D20 D4×D5 D5×C4○D4 kernel C42⋊12D10 C42⋊D5 C4×D20 Dic5.14D4 C22⋊D20 D10.12D4 D10⋊D4 D10.13D4 D10⋊Q8 C4×C5⋊D4 C23⋊D10 Dic5⋊D4 D4×C20 D5×C22×C4 C2×C4○D20 C4×D5 C4×D4 D10 C2×C10 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C22 C4 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 4 2 4 4 2 4 2 4 2 16 4 4

Matrix representation of C4212D10 in GL4(𝔽41) generated by

 1 40 0 0 2 40 0 0 0 0 18 35 0 0 6 23
,
 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 40 1 0 0 0 1 0 0 0 0 6 6 0 0 35 1
,
 40 0 0 0 0 40 0 0 0 0 6 6 0 0 1 35
`G:=sub<GL(4,GF(41))| [1,2,0,0,40,40,0,0,0,0,18,6,0,0,35,23],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[40,0,0,0,1,1,0,0,0,0,6,35,0,0,6,1],[40,0,0,0,0,40,0,0,0,0,6,1,0,0,6,35] >;`

C4212D10 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{12}D_{10}`
`% in TeX`

`G:=Group("C4^2:12D10");`
`// GroupNames label`

`G:=SmallGroup(320,1219);`
`// by ID`

`G=gap.SmallGroup(320,1219);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,675,570,12550]);`
`// 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,d*a*d=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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