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## G = C8.30D8order 128 = 27

### 7th non-split extension by C8 of D8 acting via D8/D4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C8.30D8
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×C8 — C8.12D4 — C8.30D8
 Lower central C1 — C2 — C2×C4 — C2×C8 — C8.30D8
 Upper central C1 — C22 — C42 — C4×C8 — C8.30D8
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C8.30D8

Generators and relations for C8.30D8
G = < a,b,c | a8=b8=1, c2=a, bab-1=a3, ac=ca, cbc-1=ab-1 >

Character table of C8.30D8

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 16 2 2 2 2 4 16 2 2 2 2 4 4 8 8 8 8 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -i -i i i -i i -i -i i i i -i linear of order 4 ρ6 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 i i -i -i i -i i i -i -i -i i linear of order 4 ρ7 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -i -i i i i -i i i -i -i -i i linear of order 4 ρ8 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 i i -i -i -i i -i -i i i i -i linear of order 4 ρ9 2 2 2 2 0 -2 2 2 -2 -2 0 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 2 2 2 2 2 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 √2 √2 √2 -√2 -√2 √2 -√2 -√2 orthogonal lifted from D8 ρ12 2 -2 2 -2 0 0 2 -2 0 0 0 2 -2 -2 2 0 0 -√2 √2 √2 -√2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ13 2 2 2 2 0 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 -√2 -√2 -√2 √2 √2 -√2 √2 √2 orthogonal lifted from D8 ρ14 2 -2 2 -2 0 0 2 -2 0 0 0 2 -2 -2 2 0 0 √2 -√2 -√2 √2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ15 2 -2 2 -2 0 0 2 -2 0 0 0 -2 2 2 -2 0 0 √-2 -√-2 √-2 -√-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ16 2 2 2 2 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ17 2 2 2 2 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ18 2 -2 2 -2 0 0 2 -2 0 0 0 -2 2 2 -2 0 0 -√-2 √-2 -√-2 √-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ19 2 -2 -2 2 0 2i 0 0 -2i 0 0 √-2 -√-2 √-2 -√-2 -√2 √2 0 0 0 0 ζ1615+ζ1613 ζ163+ζ16 ζ167+ζ165 ζ1611+ζ16 ζ1615+ζ165 ζ1611+ζ169 ζ1613+ζ167 ζ169+ζ163 complex lifted from D8.C4 ρ20 2 -2 -2 2 0 -2i 0 0 2i 0 0 -√-2 √-2 -√-2 √-2 -√2 √2 0 0 0 0 ζ163+ζ16 ζ1615+ζ1613 ζ1611+ζ169 ζ1615+ζ165 ζ1611+ζ16 ζ167+ζ165 ζ169+ζ163 ζ1613+ζ167 complex lifted from D8.C4 ρ21 2 -2 -2 2 0 -2i 0 0 2i 0 0 √-2 -√-2 √-2 -√-2 √2 -√2 0 0 0 0 ζ1613+ζ167 ζ169+ζ163 ζ1615+ζ165 ζ163+ζ16 ζ1615+ζ1613 ζ1611+ζ16 ζ167+ζ165 ζ1611+ζ169 complex lifted from D8.C4 ρ22 2 -2 -2 2 0 -2i 0 0 2i 0 0 √-2 -√-2 √-2 -√-2 √2 -√2 0 0 0 0 ζ1615+ζ165 ζ1611+ζ16 ζ1613+ζ167 ζ1611+ζ169 ζ167+ζ165 ζ169+ζ163 ζ1615+ζ1613 ζ163+ζ16 complex lifted from D8.C4 ρ23 2 -2 -2 2 0 -2i 0 0 2i 0 0 -√-2 √-2 -√-2 √-2 -√2 √2 0 0 0 0 ζ1611+ζ169 ζ167+ζ165 ζ163+ζ16 ζ1613+ζ167 ζ169+ζ163 ζ1615+ζ1613 ζ1611+ζ16 ζ1615+ζ165 complex lifted from D8.C4 ρ24 2 -2 -2 2 0 2i 0 0 -2i 0 0 -√-2 √-2 -√-2 √-2 √2 -√2 0 0 0 0 ζ169+ζ163 ζ1613+ζ167 ζ1611+ζ16 ζ1615+ζ1613 ζ163+ζ16 ζ1615+ζ165 ζ1611+ζ169 ζ167+ζ165 complex lifted from D8.C4 ρ25 2 -2 -2 2 0 2i 0 0 -2i 0 0 -√-2 √-2 -√-2 √-2 √2 -√2 0 0 0 0 ζ1611+ζ16 ζ1615+ζ165 ζ169+ζ163 ζ167+ζ165 ζ1611+ζ169 ζ1613+ζ167 ζ163+ζ16 ζ1615+ζ1613 complex lifted from D8.C4 ρ26 2 -2 -2 2 0 2i 0 0 -2i 0 0 √-2 -√-2 √-2 -√-2 -√2 √2 0 0 0 0 ζ167+ζ165 ζ1611+ζ169 ζ1615+ζ1613 ζ169+ζ163 ζ1613+ζ167 ζ163+ζ16 ζ1615+ζ165 ζ1611+ζ16 complex lifted from D8.C4 ρ27 4 -4 4 -4 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ28 4 4 -4 -4 0 0 0 0 0 0 0 2√-2 2√-2 -2√-2 -2√-2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊2C4 ρ29 4 4 -4 -4 0 0 0 0 0 0 0 -2√-2 -2√-2 2√-2 2√-2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from D8⋊2C4

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

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

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

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

Matrix representation of C8.30D8 in GL4(𝔽17) generated by

 12 12 0 0 5 12 0 0 0 0 1 0 0 0 0 1
,
 15 14 0 0 14 2 0 0 0 0 3 3 0 0 14 3
,
 15 14 0 0 3 15 0 0 0 0 3 3 0 0 3 14
`G:=sub<GL(4,GF(17))| [12,5,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[15,14,0,0,14,2,0,0,0,0,3,14,0,0,3,3],[15,3,0,0,14,15,0,0,0,0,3,3,0,0,3,14] >;`

C8.30D8 in GAP, Magma, Sage, TeX

`C_8._{30}D_8`
`% in TeX`

`G:=Group("C8.30D8");`
`// GroupNames label`

`G:=SmallGroup(128,92);`
`// by ID`

`G=gap.SmallGroup(128,92);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,387,520,1690,192,2804,1411,172,4037,2028,124]);`
`// Polycyclic`

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

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