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## G = C8.24D20order 320 = 26·5

### 10th non-split extension by C8 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C8.24D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D20.3C4 — C8.24D20
 Lower central C5 — C10 — C2×C20 — C8.24D20
 Upper central C1 — C2 — C2×C4 — C8.C4

Generators and relations for C8.24D20
G = < a,b,c | a40=1, b4=c2=a20, bab-1=a11, cac-1=a19, cbc-1=b3 >

Subgroups: 510 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C52C8, C40, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, D4.3D4, C8×D5, C8⋊D5, C40⋊C2, D40, Dic20, C4.Dic5, C2×C40, C5×M4(2), C2×Dic10, C2×D20, C4○D20, C20.46D4, C4.12D20, C5×C8.C4, D20.3C4, C2×C40⋊C2, C8⋊D10, C8.D10, C8.24D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, D20, C22×D5, D4.3D4, C2×D20, D4×D5, Q82D5, C4⋊D20, C8.24D20

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 D20 D4.3D4 D4×D5 Q8⋊2D5 C8.24D20 kernel C8.24D20 C20.46D4 C4.12D20 C5×C8.C4 D20.3C4 C2×C40⋊C2 C8⋊D10 C8.D10 C40 Dic10 D20 C8.C4 C2×C10 C2×C8 M4(2) C8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 2 1 1 2 2 2 4 8 2 2 2 8

Matrix representation of C8.24D20 in GL4(𝔽41) generated by

 39 14 0 0 27 37 0 0 0 0 27 2 0 0 39 15
,
 0 0 1 0 0 0 0 1 2 28 0 0 13 39 0 0
,
 15 39 0 0 31 26 0 0 0 0 4 14 0 0 31 37
`G:=sub<GL(4,GF(41))| [39,27,0,0,14,37,0,0,0,0,27,39,0,0,2,15],[0,0,2,13,0,0,28,39,1,0,0,0,0,1,0,0],[15,31,0,0,39,26,0,0,0,0,4,31,0,0,14,37] >;`

C8.24D20 in GAP, Magma, Sage, TeX

`C_8._{24}D_{20}`
`% in TeX`

`G:=Group("C8.24D20");`
`// GroupNames label`

`G:=SmallGroup(320,525);`
`// by ID`

`G=gap.SmallGroup(320,525);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,58,1123,136,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=1,b^4=c^2=a^20,b*a*b^-1=a^11,c*a*c^-1=a^19,c*b*c^-1=b^3>;`
`// generators/relations`

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