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## G = C42⋊25D10order 320 = 26·5

### 25th semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊25D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22⋊D20 — C42⋊25D10
 Lower central C5 — C2×C10 — C42⋊25D10
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for C4225D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >

Subgroups: 1254 in 252 conjugacy classes, 91 normal (16 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×22], C5, C2×C4 [×6], C2×C4 [×6], D4 [×12], C23, C23 [×8], D5 [×5], C10 [×3], C10, C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×3], C2×D4 [×12], C24, Dic5 [×3], C20 [×6], D10 [×19], C2×C10, C2×C10 [×3], C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C422C2, C422C2, C41D4, C4×D5 [×3], D20 [×9], C2×Dic5 [×3], C5⋊D4 [×3], C2×C20 [×6], C22×D5 [×2], C22×D5 [×3], C22×D5 [×3], C22×C10, C22.54C24, C10.D4 [×3], D10⋊C4 [×9], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×C4×D5 [×3], C2×D20 [×9], C2×C5⋊D4 [×3], C23×D5, C204D4, C422D5, C22⋊D20 [×3], D10⋊D4 [×3], D10.13D4 [×3], C4⋊D20 [×3], C5×C422C2, C4225D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4 [×3], C22×D5 [×7], C22.54C24, C23×D5, D48D10 [×3], C4225D10

Smallest permutation representation of C4225D10
On 80 points
Generators in S80
```(1 52 6 28)(2 58 7 24)(3 54 8 30)(4 60 9 26)(5 56 10 22)(11 59 18 25)(12 55 19 21)(13 51 20 27)(14 57 16 23)(15 53 17 29)(31 42 36 71)(32 77 37 48)(33 44 38 73)(34 79 39 50)(35 46 40 75)(41 64 80 69)(43 66 72 61)(45 68 74 63)(47 70 76 65)(49 62 78 67)
(1 40 16 64)(2 36 17 70)(3 32 18 66)(4 38 19 62)(5 34 20 68)(6 35 14 69)(7 31 15 65)(8 37 11 61)(9 33 12 67)(10 39 13 63)(21 78 60 73)(22 50 51 45)(23 80 52 75)(24 42 53 47)(25 72 54 77)(26 44 55 49)(27 74 56 79)(28 46 57 41)(29 76 58 71)(30 48 59 43)
(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 6)(2 10)(3 9)(4 8)(5 7)(11 19)(12 18)(13 17)(14 16)(15 20)(21 30)(22 29)(23 28)(24 27)(25 26)(31 63)(32 62)(33 61)(34 70)(35 69)(36 68)(37 67)(38 66)(39 65)(40 64)(42 50)(43 49)(44 48)(45 47)(51 58)(52 57)(53 56)(54 55)(59 60)(71 79)(72 78)(73 77)(74 76)```

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D5 D10 D10 D10 2+ 1+4 D4⋊8D10 kernel C42⋊25D10 C20⋊4D4 C42⋊2D5 C22⋊D20 D10⋊D4 D10.13D4 C4⋊D20 C5×C42⋊2C2 C42⋊2C2 C42 C22⋊C4 C4⋊C4 C10 C2 # reps 1 1 1 3 3 3 3 1 2 2 6 6 3 12

Matrix representation of C4225D10 in GL8(𝔽41)

 2 28 10 12 0 0 0 0 13 39 10 10 0 0 0 0 0 0 2 13 0 0 0 0 0 0 28 39 0 0 0 0 0 0 0 0 11 32 0 0 0 0 0 0 9 30 0 0 0 0 0 0 0 0 11 32 0 0 0 0 0 0 9 30
,
 1 0 28 24 0 0 0 0 0 1 13 28 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 11 32 39 0 0 0 0 0 9 30 0 39 0 0 0 0 0 0 30 9 0 0 0 0 0 0 32 11
,
 40 35 0 0 0 0 0 0 6 35 0 0 0 0 0 0 32 14 6 6 0 0 0 0 14 32 35 1 0 0 0 0 0 0 0 0 40 7 0 0 0 0 0 0 34 7 0 0 0 0 0 0 11 14 1 34 0 0 0 0 27 27 7 34
,
 1 0 0 0 0 0 0 0 35 40 0 0 0 0 0 0 9 27 35 35 0 0 0 0 27 30 40 6 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 34 1 0 0 0 0 0 0 11 32 40 0 0 0 0 0 27 30 34 1

`G:=sub<GL(8,GF(41))| [2,13,0,0,0,0,0,0,28,39,0,0,0,0,0,0,10,10,2,28,0,0,0,0,12,10,13,39,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,32,30,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,32,30],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,28,13,40,0,0,0,0,0,24,28,0,40,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,32,30,0,0,0,0,0,0,39,0,30,32,0,0,0,0,0,39,9,11],[40,6,32,14,0,0,0,0,35,35,14,32,0,0,0,0,0,0,6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,40,34,11,27,0,0,0,0,7,7,14,27,0,0,0,0,0,0,1,7,0,0,0,0,0,0,34,34],[1,35,9,27,0,0,0,0,0,40,27,30,0,0,0,0,0,0,35,40,0,0,0,0,0,0,35,6,0,0,0,0,0,0,0,0,40,34,11,27,0,0,0,0,0,1,32,30,0,0,0,0,0,0,40,34,0,0,0,0,0,0,0,1] >;`

C4225D10 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{25}D_{10}`
`% in TeX`

`G:=Group("C4^2:25D10");`
`// GroupNames label`

`G:=SmallGroup(320,1383);`
`// by ID`

`G=gap.SmallGroup(320,1383);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,184,1571,570,192,12550]);`
`// 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*b^2,d*a*d=a*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;`
`// generators/relations`

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