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## G = C42⋊24D10order 320 = 26·5

### 24th semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊24D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22⋊C4 — C42⋊24D10
 Lower central C5 — C2×C10 — C42⋊24D10
 Upper central C1 — C22 — C42⋊2C2

Generators and relations for C4224D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >

Subgroups: 1014 in 248 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×18], C5, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×8], D5 [×5], C10 [×3], C10, C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×3], C4⋊C4 [×5], C22×C4 [×5], C2×D4 [×3], C2×Q8, C24, Dic5 [×5], C20 [×6], D10 [×4], D10 [×11], C2×C10, C2×C10 [×3], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2, C422C2, Dic10, C4×D5 [×7], D20 [×4], C2×Dic5 [×5], C5⋊D4, C2×C20 [×6], C22×D5 [×3], C22×D5 [×5], C22×C10, C22.45C24, C4×Dic5 [×2], C10.D4 [×3], C4⋊Dic5 [×2], D10⋊C4 [×9], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10, C2×C4×D5 [×5], C2×D20 [×2], C2×C5⋊D4, C23×D5, C42⋊D5, C4×D20, C23.D10, D5×C22⋊C4 [×2], C22⋊D20, D10.12D4, Dic5.5D4, C4⋊C47D5, D208C4, D10.13D4 [×2], D10⋊Q8, D102Q8, C5×C422C2, C4224D10
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, D5×C4○D4 [×2], D48D10, C4224D10

Smallest permutation representation of C4224D10
On 80 points
Generators in S80
```(1 57 7 49)(2 53 8 45)(3 59 9 41)(4 55 10 47)(5 51 6 43)(11 58 16 50)(12 54 17 46)(13 60 18 42)(14 56 19 48)(15 52 20 44)(21 69 75 64)(22 31 76 36)(23 61 77 66)(24 33 78 38)(25 63 79 68)(26 35 80 40)(27 65 71 70)(28 37 72 32)(29 67 73 62)(30 39 74 34)
(1 32 20 66)(2 38 16 62)(3 34 17 68)(4 40 18 64)(5 36 19 70)(6 31 14 65)(7 37 15 61)(8 33 11 67)(9 39 12 63)(10 35 13 69)(21 55 26 42)(22 48 27 51)(23 57 28 44)(24 50 29 53)(25 59 30 46)(41 74 54 79)(43 76 56 71)(45 78 58 73)(47 80 60 75)(49 72 52 77)
(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 2)(3 5)(6 9)(7 8)(11 15)(12 14)(16 20)(17 19)(22 30)(23 29)(24 28)(25 27)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 70)(40 69)(41 48)(42 47)(43 46)(44 45)(49 50)(51 54)(52 53)(55 60)(56 59)(57 58)(71 79)(72 78)(73 77)(74 76)```

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

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

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

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 20A ··· 20L 20M ··· 20R 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 20 ··· 20 20 ··· 20 size 1 1 1 1 4 10 10 10 10 20 2 2 2 2 4 4 4 4 10 10 10 10 20 20 20 2 2 2 ··· 2 8 8 4 ··· 4 8 ··· 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 2+ 1+4 D5×C4○D4 D4⋊8D10 kernel C42⋊24D10 C42⋊D5 C4×D20 C23.D10 D5×C22⋊C4 C22⋊D20 D10.12D4 Dic5.5D4 C4⋊C4⋊7D5 D20⋊8C4 D10.13D4 D10⋊Q8 D10⋊2Q8 C5×C42⋊2C2 C42⋊2C2 D10 C42 C22⋊C4 C4⋊C4 C10 C2 C2 # reps 1 1 1 1 2 1 1 1 1 1 2 1 1 1 2 8 2 6 6 1 8 4

Matrix representation of C4224D10 in GL6(𝔽41)

 0 9 0 0 0 0 9 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
,
 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 0 1 0 0 0 0 40 0
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 0 0 0 0 1 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 34 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(41))| [0,9,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C4224D10 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_{24}D_{10}`
`% in TeX`

`G:=Group("C4^2:24D10");`
`// GroupNames label`

`G:=SmallGroup(320,1377);`
`// by ID`

`G=gap.SmallGroup(320,1377);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,570,192,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*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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