Copied to
clipboard

## G = C42⋊10D10order 320 = 26·5

### 10th semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊10D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — C42⋊10D10
 Lower central C5 — C2×C10 — C42⋊10D10
 Upper central C1 — C22 — C42⋊C2

Generators and relations for C4210D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 998 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×2], C22 [×16], C5, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×8], D5 [×4], C10 [×3], C10 [×2], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, C24, Dic5 [×5], C20 [×6], D10 [×2], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C22⋊C4 [×2], C42⋊C2, C42⋊C2, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2 [×2], Dic10, C4×D5 [×4], D20 [×3], C2×Dic5 [×5], C2×Dic5, C5⋊D4 [×2], C2×C20 [×6], C2×C20 [×2], C22×D5 [×3], C22×D5 [×5], C22×C10, C22.45C24, C4×Dic5, C10.D4 [×5], C4⋊Dic5, D10⋊C4 [×11], C23.D5, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×3], C2×D20 [×2], C22×Dic5, C2×C5⋊D4, C22×C20, C23×D5, C42⋊D5, C4×D20, C4.D20, C422D5, Dic5.14D4, D5×C22⋊C4, Dic54D4, C22⋊D20, D10.13D4 [×2], D10⋊Q8, C4⋊C4⋊D5, C2×D10⋊C4, C23.23D10, C5×C42⋊C2, C4210D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.45C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D48D10, C4210D10

Smallest permutation representation of C4210D10
On 80 points
Generators in S80
```(1 53 13 41)(2 59 14 47)(3 55 15 43)(4 51 11 49)(5 57 12 45)(6 56 16 44)(7 52 17 50)(8 58 18 46)(9 54 19 42)(10 60 20 48)(21 79 31 61)(22 75 32 67)(23 71 33 63)(24 77 34 69)(25 73 35 65)(26 74 36 66)(27 80 37 62)(28 76 38 68)(29 72 39 64)(30 78 40 70)
(1 28 8 23)(2 29 9 24)(3 30 10 25)(4 26 6 21)(5 27 7 22)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 68 46 63)(42 69 47 64)(43 70 48 65)(44 61 49 66)(45 62 50 67)(51 74 56 79)(52 75 57 80)(53 76 58 71)(54 77 59 72)(55 78 60 73)
(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 7)(2 6)(3 10)(4 9)(5 8)(11 19)(12 18)(13 17)(14 16)(15 20)(21 39)(22 38)(23 37)(24 36)(25 40)(26 34)(27 33)(28 32)(29 31)(30 35)(41 45)(42 44)(46 50)(47 49)(51 59)(52 58)(53 57)(54 56)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 80)(69 79)(70 78)```

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

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

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

65 conjugacy classes

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

65 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 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 D10 D10 C4○D20 2+ 1+4 D5×C4○D4 D4⋊8D10 kernel C42⋊10D10 C42⋊D5 C4×D20 C4.D20 C42⋊2D5 Dic5.14D4 D5×C22⋊C4 Dic5⋊4D4 C22⋊D20 D10.13D4 D10⋊Q8 C4⋊C4⋊D5 C2×D10⋊C4 C23.23D10 C5×C42⋊C2 C42⋊C2 D10 C2×C10 C42 C22⋊C4 C4⋊C4 C22×C4 C22 C10 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 4 4 4 4 4 2 16 1 4 4

Matrix representation of C4210D10 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 1 21 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 7 4 0 0 0 0 29 34 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 40 7 0 0 0 0 34 7 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 37 40
,
 40 0 0 0 0 0 34 1 0 0 0 0 0 0 1 0 0 0 0 0 17 40 0 0 0 0 0 0 40 0 0 0 0 0 4 1

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,21,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,29,0,0,0,0,4,34,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,37,0,0,0,0,0,40],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,40,0,0,0,0,0,0,40,4,0,0,0,0,0,1] >;`

C4210D10 in GAP, Magma, Sage, TeX

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

`G:=Group("C4^2:10D10");`
`// GroupNames label`

`G:=SmallGroup(320,1199);`
`// by ID`

`G=gap.SmallGroup(320,1199);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,675,136,12550]);`
`// 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*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

׿
×
𝔽