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## G = C42⋊28D10order 320 = 26·5

### 28th semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊28D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C23⋊D10 — C42⋊28D10
 Lower central C5 — C2×C10 — C42⋊28D10
 Upper central C1 — C22 — C4⋊1D4

Generators and relations for C4228D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=b-1, dbd=a2b, dcd=c-1 >

Subgroups: 1062 in 252 conjugacy classes, 91 normal (12 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×22], C5, C2×C4 [×3], C2×C4 [×9], D4 [×12], C23, C23 [×3], C23 [×5], D5 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×3], C2×D4 [×6], C2×D4 [×6], C24, Dic5 [×6], C20 [×3], D10 [×10], C2×C10, C2×C10 [×12], C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C422C2 [×2], C41D4, C2×Dic5 [×6], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×6], C22×D5 [×2], C22×D5 [×3], C22×C10, C22×C10 [×3], C22.54C24, C10.D4 [×6], D10⋊C4 [×6], C23.D5 [×6], C4×C20, C22×Dic5 [×3], C2×C5⋊D4 [×6], D4×C10 [×6], C23×D5, C422D5 [×2], C23.18D10 [×3], C23⋊D10 [×3], Dic5⋊D4 [×6], C5×C41D4, C4228D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4 [×3], C22×D5 [×7], C22.54C24, C23×D5, D46D10 [×3], C4228D10

Smallest permutation representation of C4228D10
On 80 points
Generators in S80
```(1 66 18 61)(2 62 19 67)(3 68 20 63)(4 64 16 69)(5 70 17 65)(6 54 13 59)(7 60 14 55)(8 56 15 51)(9 52 11 57)(10 58 12 53)(21 49 75 38)(22 39 76 50)(23 41 77 40)(24 31 78 42)(25 43 79 32)(26 33 80 44)(27 45 71 34)(28 35 72 46)(29 47 73 36)(30 37 74 48)
(1 23 8 28)(2 29 9 24)(3 25 10 30)(4 21 6 26)(5 27 7 22)(11 78 19 73)(12 74 20 79)(13 80 16 75)(14 76 17 71)(15 72 18 77)(31 62 47 52)(32 53 48 63)(33 64 49 54)(34 55 50 65)(35 66 41 56)(36 57 42 67)(37 68 43 58)(38 59 44 69)(39 70 45 60)(40 51 46 61)
(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 12)(2 11)(3 15)(4 14)(5 13)(6 17)(7 16)(8 20)(9 19)(10 18)(21 22)(23 30)(24 29)(25 28)(26 27)(31 42)(32 41)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(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)| (1,66,18,61)(2,62,19,67)(3,68,20,63)(4,64,16,69)(5,70,17,65)(6,54,13,59)(7,60,14,55)(8,56,15,51)(9,52,11,57)(10,58,12,53)(21,49,75,38)(22,39,76,50)(23,41,77,40)(24,31,78,42)(25,43,79,32)(26,33,80,44)(27,45,71,34)(28,35,72,46)(29,47,73,36)(30,37,74,48), (1,23,8,28)(2,29,9,24)(3,25,10,30)(4,21,6,26)(5,27,7,22)(11,78,19,73)(12,74,20,79)(13,80,16,75)(14,76,17,71)(15,72,18,77)(31,62,47,52)(32,53,48,63)(33,64,49,54)(34,55,50,65)(35,66,41,56)(36,57,42,67)(37,68,43,58)(38,59,44,69)(39,70,45,60)(40,51,46,61), (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,17)(7,16)(8,20)(9,19)(10,18)(21,22)(23,30)(24,29)(25,28)(26,27)(31,42)(32,41)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(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( (1,66,18,61)(2,62,19,67)(3,68,20,63)(4,64,16,69)(5,70,17,65)(6,54,13,59)(7,60,14,55)(8,56,15,51)(9,52,11,57)(10,58,12,53)(21,49,75,38)(22,39,76,50)(23,41,77,40)(24,31,78,42)(25,43,79,32)(26,33,80,44)(27,45,71,34)(28,35,72,46)(29,47,73,36)(30,37,74,48), (1,23,8,28)(2,29,9,24)(3,25,10,30)(4,21,6,26)(5,27,7,22)(11,78,19,73)(12,74,20,79)(13,80,16,75)(14,76,17,71)(15,72,18,77)(31,62,47,52)(32,53,48,63)(33,64,49,54)(34,55,50,65)(35,66,41,56)(36,57,42,67)(37,68,43,58)(38,59,44,69)(39,70,45,60)(40,51,46,61), (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,17)(7,16)(8,20)(9,19)(10,18)(21,22)(23,30)(24,29)(25,28)(26,27)(31,42)(32,41)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(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([(1,66,18,61),(2,62,19,67),(3,68,20,63),(4,64,16,69),(5,70,17,65),(6,54,13,59),(7,60,14,55),(8,56,15,51),(9,52,11,57),(10,58,12,53),(21,49,75,38),(22,39,76,50),(23,41,77,40),(24,31,78,42),(25,43,79,32),(26,33,80,44),(27,45,71,34),(28,35,72,46),(29,47,73,36),(30,37,74,48)], [(1,23,8,28),(2,29,9,24),(3,25,10,30),(4,21,6,26),(5,27,7,22),(11,78,19,73),(12,74,20,79),(13,80,16,75),(14,76,17,71),(15,72,18,77),(31,62,47,52),(32,53,48,63),(33,64,49,54),(34,55,50,65),(35,66,41,56),(36,57,42,67),(37,68,43,58),(38,59,44,69),(39,70,45,60),(40,51,46,61)], [(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)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,17),(7,16),(8,20),(9,19),(10,18),(21,22),(23,30),(24,29),(25,28),(26,27),(31,42),(32,41),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(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)])`

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D5 D10 D10 2+ 1+4 D4⋊6D10 kernel C42⋊28D10 C42⋊2D5 C23.18D10 C23⋊D10 Dic5⋊D4 C5×C4⋊1D4 C4⋊1D4 C42 C2×D4 C10 C2 # reps 1 2 3 3 6 1 2 2 12 3 12

Matrix representation of C4228D10 in GL8(𝔽41)

 18 6 0 0 0 0 0 0 35 23 0 0 0 0 0 0 0 0 18 6 0 0 0 0 0 0 35 23 0 0 0 0 0 0 0 0 26 0 7 7 0 0 0 0 39 30 36 12 0 0 0 0 13 12 13 2 0 0 0 0 25 29 2 13
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 24 34 38 3 0 0 0 0 6 17 38 0 0 0 0 0 0 28 18 6 0 0 0 0 13 28 35 23
,
 1 6 0 0 0 0 0 0 35 6 0 0 0 0 0 0 0 0 40 35 0 0 0 0 0 0 6 35 0 0 0 0 0 0 0 0 0 35 3 16 0 0 0 0 7 35 16 28 0 0 0 0 0 0 40 7 0 0 0 0 0 0 34 7
,
 6 1 0 0 0 0 0 0 6 35 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 6 35 0 0 0 0 0 0 0 0 6 35 1 40 0 0 0 0 40 35 1 0 0 0 0 0 0 0 7 40 0 0 0 0 0 0 7 34

`G:=sub<GL(8,GF(41))| [18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,26,39,13,25,0,0,0,0,0,30,12,29,0,0,0,0,7,36,13,2,0,0,0,0,7,12,2,13],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24,6,0,13,0,0,0,0,34,17,28,28,0,0,0,0,38,38,18,35,0,0,0,0,3,0,6,23],[1,35,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,35,0,0,0,0,0,0,3,16,40,34,0,0,0,0,16,28,7,7],[6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,35,35,0,0,0,0,0,0,1,1,7,7,0,0,0,0,40,0,40,34] >;`

C4228D10 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{28}D_{10}`
`% in TeX`

`G:=Group("C4^2:28D10");`
`// GroupNames label`

`G:=SmallGroup(320,1392);`
`// by ID`

`G=gap.SmallGroup(320,1392);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,297,136,12550]);`
`// Polycyclic`

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

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