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## G = C42⋊21D10order 320 = 26·5

### 21st semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊21D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — C42⋊21D10
 Lower central C5 — C2×C10 — C42⋊21D10
 Upper central C1 — C22 — C4.4D4

Generators and relations for C4221D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 950 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×18], C5, C2×C4, C2×C4 [×4], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], D5 [×4], C10, C10 [×2], C10 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic5 [×6], C20 [×5], D10 [×4], D10 [×8], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2 [×2], C4×D5 [×6], C2×Dic5 [×6], C2×Dic5, C5⋊D4 [×4], C2×C20, C2×C20 [×4], C5×D4, C5×Q8, C22×D5 [×2], C22×D5 [×5], C22×C10 [×2], C22.45C24, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5 [×2], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×C4×D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D4×C10, Q8×C10, C23×D5, C42⋊D5 [×2], C23.D10 [×2], D5×C22⋊C4 [×2], Dic54D4 [×2], D10.12D4 [×2], C23.18D10, C23⋊D10, D103Q8 [×2], C5×C4.4D4, C4221D10
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, C23×D5, D46D10, D5×C4○D4 [×2], C4221D10

Smallest permutation representation of C4221D10
On 80 points
Generators in S80
```(1 34 11 51)(2 40 12 57)(3 36 13 53)(4 32 14 59)(5 38 15 55)(6 39 16 56)(7 35 17 52)(8 31 18 58)(9 37 19 54)(10 33 20 60)(21 61 26 76)(22 72 27 67)(23 63 28 78)(24 74 29 69)(25 65 30 80)(41 79 46 64)(42 70 47 75)(43 71 48 66)(44 62 49 77)(45 73 50 68)
(1 41 6 29)(2 47 7 25)(3 43 8 21)(4 49 9 27)(5 45 10 23)(11 46 16 24)(12 42 17 30)(13 48 18 26)(14 44 19 22)(15 50 20 28)(31 61 36 71)(32 77 37 67)(33 63 38 73)(34 79 39 69)(35 65 40 75)(51 64 56 74)(52 80 57 70)(53 66 58 76)(54 72 59 62)(55 68 60 78)
(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 14)(2 13)(3 12)(4 11)(5 15)(6 19)(7 18)(8 17)(9 16)(10 20)(21 25)(22 24)(26 30)(27 29)(31 52)(32 51)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 49)(42 48)(43 47)(44 46)(61 65)(62 64)(66 70)(67 69)(71 75)(72 74)(76 80)(77 79)```

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

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

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

53 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 4M 4N 4O 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 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 10 10 2 2 2 2 4 4 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 8 8 8 8 4 ··· 4 8 8 8 8

53 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 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4⋊6D10 D5×C4○D4 kernel C42⋊21D10 C42⋊D5 C23.D10 D5×C22⋊C4 Dic5⋊4D4 D10.12D4 C23.18D10 C23⋊D10 D10⋊3Q8 C5×C4.4D4 C4.4D4 D10 C42 C22⋊C4 C2×D4 C2×Q8 C10 C2 C2 # reps 1 2 2 2 2 2 1 1 2 1 2 8 2 8 2 2 1 4 8

Matrix representation of C4221D10 in GL6(𝔽41)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 0 0 0 32
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 6 7 0 0 0 0 35 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 35 1 0 0 0 0 6 6 0 0 0 0 0 0 40 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;`

C4221D10 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{21}D_{10}`
`% in TeX`

`G:=Group("C4^2:21D10");`
`// GroupNames label`

`G:=SmallGroup(320,1351);`
`// by ID`

`G=gap.SmallGroup(320,1351);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,387,100,346,136,12550]);`
`// 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*b^2,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;`
`// generators/relations`

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