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## G = C42⋊18D10order 320 = 26·5

### 18th semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊18D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C2×D4×D5 — C42⋊18D10
 Lower central C5 — C2×C10 — C42⋊18D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C4218D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=a2b-1, dbd=b-1, dcd=c-1 >

Subgroups: 1598 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×30], C5, C2×C4, C2×C4 [×4], C2×C4 [×11], D4 [×22], Q8 [×2], C23 [×2], C23 [×13], D5 [×6], C10, C10 [×2], C10 [×2], C42, C42, C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×18], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×22], C2×C10, C2×C10 [×6], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4, C4.4D4, C41D4 [×2], C22×D4, C2×C4○D4, C4×D5 [×4], C4×D5 [×4], D20 [×12], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×4], C22×D5 [×8], C22×C10 [×2], C22.29C24, C4×Dic5, C10.D4 [×2], D10⋊C4 [×6], C4×C20, C5×C22⋊C4 [×4], C2×C4×D5, C2×C4×D5 [×2], C2×D20 [×2], C2×D20 [×6], D4×D5 [×4], Q82D5 [×4], C2×C5⋊D4 [×6], D4×C10, Q8×C10, C23×D5 [×2], C42⋊D5, C204D4, C22⋊D20 [×4], D10⋊D4 [×4], C20⋊D4, C20.23D4, C5×C4.4D4, C2×D4×D5, C2×Q82D5, C4218D10
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ 1+4 [×2], C22×D5 [×7], C22.29C24, D4×D5 [×2], C23×D5, C2×D4×D5, D48D10 [×2], C4218D10

Smallest permutation representation of C4218D10
On 80 points
Generators in S80
```(1 53 16 58)(2 59 17 54)(3 55 18 60)(4 51 19 56)(5 57 20 52)(6 64 14 69)(7 70 15 65)(8 66 11 61)(9 62 12 67)(10 68 13 63)(21 76 50 36)(22 37 41 77)(23 78 42 38)(24 39 43 79)(25 80 44 40)(26 31 45 71)(27 72 46 32)(28 33 47 73)(29 74 48 34)(30 35 49 75)
(1 74 10 39)(2 80 6 35)(3 76 7 31)(4 72 8 37)(5 78 9 33)(11 77 19 32)(12 73 20 38)(13 79 16 34)(14 75 17 40)(15 71 18 36)(21 65 26 60)(22 56 27 61)(23 67 28 52)(24 58 29 63)(25 69 30 54)(41 51 46 66)(42 62 47 57)(43 53 48 68)(44 64 49 59)(45 55 50 70)
(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 19)(2 18)(3 17)(4 16)(5 20)(6 15)(7 14)(8 13)(9 12)(10 11)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 48)(42 47)(43 46)(44 45)(49 50)(51 53)(54 60)(55 59)(56 58)(61 63)(64 70)(65 69)(66 68)(71 80)(72 79)(73 78)(74 77)(75 76)```

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

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

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

50 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 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20L 20M 20N 20O 20P order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 1 1 4 4 10 10 20 20 20 20 2 2 4 4 4 4 10 10 20 20 2 2 2 ··· 2 8 8 8 8 4 ··· 4 8 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 D10 D10 D10 D10 2+ 1+4 D4×D5 D4⋊8D10 kernel C42⋊18D10 C42⋊D5 C20⋊4D4 C22⋊D20 D10⋊D4 C20⋊D4 C20.23D4 C5×C4.4D4 C2×D4×D5 C2×Q8⋊2D5 C4×D5 C4.4D4 C42 C22⋊C4 C2×D4 C2×Q8 C10 C4 C2 # reps 1 1 1 4 4 1 1 1 1 1 4 2 2 8 2 2 2 4 8

Matrix representation of C4218D10 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 39 0 0 0 0 0 40 0 0 39 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 9 0 0 0 0 0 0 18 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 40 40 0
,
 0 34 0 0 0 0 0 0 6 35 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 23 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 40 0 0 0 0 0 40 0 0 40
,
 6 34 0 0 0 0 0 0 5 35 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 18 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1

`G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,40,0,0,1,0,0,0,0,39,0,0,1,0,0,0,0,0,39,1,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,9,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[0,6,0,0,0,0,0,0,34,35,0,0,0,0,0,0,0,0,40,23,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[6,5,0,0,0,0,0,0,34,35,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1] >;`

C4218D10 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{18}D_{10}`
`% in TeX`

`G:=Group("C4^2:18D10");`
`// GroupNames label`

`G:=SmallGroup(320,1346);`
`// by ID`

`G=gap.SmallGroup(320,1346);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,570,297,192,12550]);`
`// Polycyclic`

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

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