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## G = C24.D10order 320 = 26·5

### 1st non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C23×C10 — C2×C23.D5 — C24.D10
 Lower central C5 — C10 — C2×C10 — C24.D10
 Upper central C1 — C22 — C24 — C2×C22⋊C4

Generators and relations for C24.D10
G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=abc, ab=ba, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bcd-1 >

Subgroups: 518 in 142 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C22×C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C22⋊C4, C2×Dic5, C2×C20, C22×C10, C22×C10, C22×C10, C23.9D4, C23.D5, C23.D5, C5×C22⋊C4, C22×Dic5, C22×C20, C23×C10, C2×C23.D5, C10×C22⋊C4, C24.D10
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, C23⋊C4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C23.9D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C10.10C42, C23⋊Dic5, C24.D10

Smallest permutation representation of C24.D10
On 80 points
Generators in S80
```(2 70)(4 72)(6 74)(8 76)(10 78)(12 80)(14 62)(16 64)(18 66)(20 68)(21 58)(23 60)(25 42)(27 44)(29 46)(31 48)(33 50)(35 52)(37 54)(39 56)
(1 59)(2 60)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 58)(22 59)(23 60)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)
(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 48 22 10)(2 9 60 30)(3 46 24 8)(4 7 42 28)(5 44 26 6)(11 58 32 20)(12 19 50 40)(13 56 34 18)(14 17 52 38)(15 54 36 16)(21 49 68 79)(23 47 70 77)(25 45 72 75)(27 43 74 73)(29 41 76 71)(31 59 78 69)(33 57 80 67)(35 55 62 65)(37 53 64 63)(39 51 66 61)```

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4L 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 4 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + - + - + - + + image C1 C2 C2 C4 C4 D4 Q8 D5 Dic5 D10 Dic10 C4×D5 D20 C5⋊D4 C23⋊C4 C23⋊Dic5 kernel C24.D10 C2×C23.D5 C10×C22⋊C4 C23.D5 C22×C20 C22×C10 C22×C10 C2×C22⋊C4 C22×C4 C24 C23 C23 C23 C23 C10 C2 # reps 1 2 1 8 4 3 1 2 4 2 4 8 4 8 2 8

Matrix representation of C24.D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 24 40 0 0 0 0 1 17 0 0 0 0 0 0 24 40 0 0 0 0 1 17
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 0 34 1 0 0 0 0 40 0 0 0 34 1 0 0 0 0 40 0 0 0
,
 0 32 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 40 24 0 0 0 0 17 1 0 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,1,0,0,0,0,40,17,0,0,0,0,0,0,24,1,0,0,0,0,40,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,34,40,0,0,0,0,1,0,0,0,34,40,0,0,0,0,1,0,0,0],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,0,0,40,17,0,0,0,0,24,1,0,0,0,1,0,0,0,0,1,0,0,0] >;`

C24.D10 in GAP, Magma, Sage, TeX

`C_2^4.D_{10}`
`% in TeX`

`G:=Group("C2^4.D10");`
`// GroupNames label`

`G:=SmallGroup(320,84);`
`// by ID`

`G=gap.SmallGroup(320,84);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1684,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=a*b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^-1>;`
`// generators/relations`

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