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## G = C8.25D20order 320 = 26·5

### 11st non-split extension by C8 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C8.25D20
 Chief series C1 — C5 — C10 — C20 — C40 — C2×C40 — C2×C8⋊D5 — C8.25D20
 Lower central C5 — C10 — C2×C10 — C8.25D20
 Upper central C1 — C4 — C2×C8 — M5(2)

Generators and relations for C8.25D20
G = < a,b,c,d | a16=b2=c5=d2=1, bab=a9, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >

Subgroups: 214 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C16, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, D10, C2×C10, M5(2), M5(2), C2×M4(2), C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C23.C8, C52C16, C80, C8⋊D5, C2×C52C8, C2×C40, C2×C4×D5, C20.4C8, C5×M5(2), C2×C8⋊D5, C8.25D20
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C22⋊C4, C2×C8, M4(2), D10, C22⋊C8, C4×D5, D20, C5⋊D4, C23.C8, C8×D5, C8⋊D5, D10⋊C4, D101C8, C8.25D20

Smallest permutation representation of C8.25D20
On 80 points
Generators in S80
```(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)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)
(1 23 36 78 62)(2 24 37 79 63)(3 25 38 80 64)(4 26 39 65 49)(5 27 40 66 50)(6 28 41 67 51)(7 29 42 68 52)(8 30 43 69 53)(9 31 44 70 54)(10 32 45 71 55)(11 17 46 72 56)(12 18 47 73 57)(13 19 48 74 58)(14 20 33 75 59)(15 21 34 76 60)(16 22 35 77 61)
(1 62)(2 55)(3 56)(4 49)(5 50)(6 59)(7 60)(8 53)(9 54)(10 63)(11 64)(12 57)(13 58)(14 51)(15 52)(16 61)(17 80)(18 73)(19 74)(20 67)(21 68)(22 77)(23 78)(24 71)(25 72)(26 65)(27 66)(28 75)(29 76)(30 69)(31 70)(32 79)(33 41)(34 42)(37 45)(38 46)```

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 8F 10A 10B 10C 10D 16A 16B 16C 16D 16E 16F 16G 16H 20A 20B 20C 20D 20E 20F 40A ··· 40H 40I 40J 40K 40L 80A ··· 80P order 1 2 2 2 4 4 4 4 5 5 8 8 8 8 8 8 10 10 10 10 16 16 16 16 16 16 16 16 20 20 20 20 20 20 40 ··· 40 40 40 40 40 80 ··· 80 size 1 1 2 20 1 1 2 20 2 2 2 2 2 2 20 20 2 2 4 4 4 4 4 4 20 20 20 20 2 2 2 2 4 4 2 ··· 2 4 4 4 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 D4 D5 M4(2) D10 D20 C5⋊D4 C4×D5 C8⋊D5 C8×D5 C23.C8 C8.25D20 kernel C8.25D20 C20.4C8 C5×M5(2) C2×C8⋊D5 C2×C5⋊2C8 C2×C4×D5 C2×Dic5 C22×D5 C40 M5(2) C20 C2×C8 C8 C8 C2×C4 C4 C22 C5 C1 # reps 1 1 1 1 2 2 4 4 2 2 2 2 4 4 4 8 8 2 8

Matrix representation of C8.25D20 in GL4(𝔽241) generated by

 0 0 1 0 0 0 0 1 154 43 0 0 198 87 0 0
,
 1 0 0 0 0 1 0 0 0 0 240 0 0 0 0 240
,
 0 1 0 0 240 189 0 0 0 0 0 1 0 0 240 189
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(241))| [0,0,154,198,0,0,43,87,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[0,240,0,0,1,189,0,0,0,0,0,240,0,0,1,189],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;`

C8.25D20 in GAP, Magma, Sage, TeX

`C_8._{25}D_{20}`
`% in TeX`

`G:=Group("C8.25D20");`
`// GroupNames label`

`G:=SmallGroup(320,72);`
`// by ID`

`G=gap.SmallGroup(320,72);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,758,100,570,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^16=b^2=c^5=d^2=1,b*a*b=a^9,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;`
`// generators/relations`

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