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## G = C42⋊20D10order 320 = 26·5

### 20th semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊20D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — C42⋊20D10
 Lower central C5 — C2×C10 — C42⋊20D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C4220D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >

Subgroups: 1070 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×20], C5, C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], D5 [×4], C10 [×3], C10 [×2], C42, C42, C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C24, Dic5 [×5], C20 [×5], D10 [×2], D10 [×12], C2×C10, C2×C10 [×6], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C4.4D4, C422C2 [×2], C4×D5 [×3], D20 [×3], C2×Dic5 [×5], C2×Dic5, C5⋊D4 [×5], C2×C20 [×5], C5×D4, C5×Q8, C22×D5 [×3], C22×D5 [×4], C22×C10 [×2], C22.32C24, C4×Dic5, C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×C4×D5 [×3], C2×D20 [×2], C22×Dic5, C2×C5⋊D4 [×4], D4×C10, Q8×C10, C23×D5, C4×D20, C422D5, C23.D10, D5×C22⋊C4, Dic54D4, C22⋊D20, D10.12D4, D10⋊D4 [×2], C22.D20, C23⋊D10, Dic5⋊D4, D103Q8, C20.23D4, C5×C4.4D4, C4220D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4 [×2], C22×D5 [×7], C22.32C24, C23×D5, D46D10, D5×C4○D4, D48D10, C4220D10

Smallest permutation representation of C4220D10
On 80 points
Generators in S80
```(1 51 10 77)(2 57 6 73)(3 53 7 79)(4 59 8 75)(5 55 9 71)(11 58 19 74)(12 54 20 80)(13 60 16 76)(14 56 17 72)(15 52 18 78)(21 47 26 63)(22 69 27 43)(23 49 28 65)(24 61 29 45)(25 41 30 67)(31 46 36 62)(32 68 37 42)(33 48 38 64)(34 70 39 44)(35 50 40 66)
(1 27 14 38)(2 23 15 34)(3 29 11 40)(4 25 12 36)(5 21 13 32)(6 28 18 39)(7 24 19 35)(8 30 20 31)(9 26 16 37)(10 22 17 33)(41 54 62 59)(42 71 63 76)(43 56 64 51)(44 73 65 78)(45 58 66 53)(46 75 67 80)(47 60 68 55)(48 77 69 72)(49 52 70 57)(50 79 61 74)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 17)(7 16)(8 20)(9 19)(10 18)(21 24)(22 23)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 40)(38 39)(41 46)(42 45)(43 44)(47 50)(48 49)(51 57)(52 56)(53 55)(58 60)(61 68)(62 67)(63 66)(64 65)(69 70)(71 79)(72 78)(73 77)(74 76)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 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 4 4 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 2 2 4 4 4 4 10 10 20 20 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 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4⋊6D10 D5×C4○D4 D4⋊8D10 kernel C42⋊20D10 C4×D20 C42⋊2D5 C23.D10 D5×C22⋊C4 Dic5⋊4D4 C22⋊D20 D10.12D4 D10⋊D4 C22.D20 C23⋊D10 Dic5⋊D4 D10⋊3Q8 C20.23D4 C5×C4.4D4 C4.4D4 D10 C42 C22⋊C4 C2×D4 C2×Q8 C10 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 4 2 8 2 2 2 4 4 4

Matrix representation of C4220D10 in GL6(𝔽41)

 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 4 5 0 0 0 0 38 37 0 0 0 0 0 0 11 9 0 0 0 0 32 30 0 0 0 0 0 0 11 9 0 0 0 0 32 30
,
 40 0 0 0 0 0 18 1 0 0 0 0 0 0 7 7 0 0 0 0 34 40 0 0 0 0 0 0 34 34 0 0 0 0 7 1
,
 1 0 0 0 0 0 23 40 0 0 0 0 0 0 7 7 0 0 0 0 40 34 0 0 0 0 0 0 34 34 0 0 0 0 1 7

`G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[4,38,0,0,0,0,5,37,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,0,0,0,0,11,32,0,0,0,0,9,30],[40,18,0,0,0,0,0,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,34,1,0,0,0,0,34,7] >;`

C4220D10 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{20}D_{10}`
`% in TeX`

`G:=Group("C4^2:20D10");`
`// GroupNames label`

`G:=SmallGroup(320,1350);`
`// by ID`

`G=gap.SmallGroup(320,1350);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,675,570,80,12550]);`
`// 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,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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