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## G = C8.21D8order 128 = 27

### 12nd non-split extension by C8 of D8 acting via D8/C8=C2

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C8.21D8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C4×C16 — C8.21D8
 Lower central C1 — C2 — C4 — C2×C8 — C8.21D8
 Upper central C1 — C22 — C42 — C4×C8 — C8.21D8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C8.21D8

Generators and relations for C8.21D8
G = < a,b,c | a8=c2=1, b8=a4, ab=ba, cac=a3, cbc=a4b7 >

Subgroups: 264 in 92 conjugacy classes, 36 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×4], Q8 [×4], C23 [×2], C16 [×4], C42, C22⋊C4 [×4], C2×C8 [×2], D8 [×4], SD16 [×4], Q16 [×4], C2×D4 [×2], C2×Q8 [×2], C4×C8, C2×C16 [×2], D16 [×2], SD32 [×4], Q32 [×2], C4.4D4 [×2], C2×D8 [×2], C2×SD16 [×2], C2×Q16 [×2], C4×C16, C8.12D4 [×2], C2×D16, C2×SD32 [×2], C2×Q32, C8.21D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], C41D4, C2×D8 [×2], C84D4, C4○D16 [×2], C8.21D8

Smallest permutation representation of C8.21D8
On 64 points
Generators in S64
```(1 18 51 46 9 26 59 38)(2 19 52 47 10 27 60 39)(3 20 53 48 11 28 61 40)(4 21 54 33 12 29 62 41)(5 22 55 34 13 30 63 42)(6 23 56 35 14 31 64 43)(7 24 57 36 15 32 49 44)(8 25 58 37 16 17 50 45)
(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)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 48)(32 47)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 64)(62 63)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4F 4G 4H 8A ··· 8H 16A ··· 16P order 1 2 2 2 2 2 4 ··· 4 4 4 8 ··· 8 16 ··· 16 size 1 1 1 1 16 16 2 ··· 2 16 16 2 ··· 2 2 ··· 2

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D4 D8 D8 C4○D16 kernel C8.21D8 C4×C16 C8.12D4 C2×D16 C2×SD32 C2×Q32 C16 C42 C2×C8 C8 C2×C4 C2 # reps 1 1 2 1 2 1 4 1 1 4 4 16

Matrix representation of C8.21D8 in GL4(𝔽17) generated by

 16 15 0 0 1 1 0 0 0 0 10 7 0 0 5 0
,
 16 0 0 0 0 16 0 0 0 0 7 12 0 0 11 2
,
 16 0 0 0 1 1 0 0 0 0 7 12 0 0 13 10
`G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,10,5,0,0,7,0],[16,0,0,0,0,16,0,0,0,0,7,11,0,0,12,2],[16,1,0,0,0,1,0,0,0,0,7,13,0,0,12,10] >;`

C8.21D8 in GAP, Magma, Sage, TeX

`C_8._{21}D_8`
`% in TeX`

`G:=Group("C8.21D8");`
`// GroupNames label`

`G:=SmallGroup(128,981);`
`// by ID`

`G=gap.SmallGroup(128,981);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,288,422,436,1123,360,4037,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=c^2=1,b^8=a^4,a*b=b*a,c*a*c=a^3,c*b*c=a^4*b^7>;`
`// generators/relations`

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