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## G = C4.10D8order 64 = 26

### 2nd non-split extension by C4 of D8 acting via D8/D4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C4.10D8, C4.5Q16, C4.6SD16, C42.4C22, C4⋊C4.2C4, C4⋊C8.2C2, C4⋊Q8.1C2, (C2×C4).109D4, C2.5(D4⋊C4), C2.4(Q8⋊C4), C2.4(C4.10D4), C22.40(C22⋊C4), (C2×C4).13(C2×C4), SmallGroup(64,13)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.10D8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4⋊Q8 — C4.10D8
 Lower central C1 — C22 — C2×C4 — C4.10D8
 Upper central C1 — C22 — C42 — C4.10D8
 Jennings C1 — C22 — C22 — C42 — C4.10D8

Generators and relations for C4.10D8
G = < a,b,c | a4=b8=1, c2=bab-1=a-1, ac=ca, cbc-1=ab-1 >

Character table of C4.10D8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 2 2 4 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 trivial ρ2 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 linear of order 2 ρ4 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 -i -i i i -i i -i i linear of order 4 ρ6 1 1 1 1 -1 -1 -1 -1 1 -1 1 i i -i -i i -i i -i linear of order 4 ρ7 1 1 1 1 -1 -1 -1 -1 1 1 -1 -i i -i i -i -i i i linear of order 4 ρ8 1 1 1 1 -1 -1 -1 -1 1 1 -1 i -i i -i i i -i -i linear of order 4 ρ9 2 2 2 2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 2 0 0 -2 0 0 0 0 -√2 -√2 0 0 √2 √2 0 orthogonal lifted from D8 ρ12 2 -2 2 -2 2 0 0 -2 0 0 0 0 √2 √2 0 0 -√2 -√2 0 orthogonal lifted from D8 ρ13 2 -2 -2 2 0 -2 2 0 0 0 0 -√2 0 0 √2 √2 0 0 -√2 symplectic lifted from Q16, Schur index 2 ρ14 2 -2 -2 2 0 -2 2 0 0 0 0 √2 0 0 -√2 -√2 0 0 √2 symplectic lifted from Q16, Schur index 2 ρ15 2 -2 2 -2 -2 0 0 2 0 0 0 0 -√-2 √-2 0 0 -√-2 √-2 0 complex lifted from SD16 ρ16 2 -2 2 -2 -2 0 0 2 0 0 0 0 √-2 -√-2 0 0 √-2 -√-2 0 complex lifted from SD16 ρ17 2 -2 -2 2 0 2 -2 0 0 0 0 √-2 0 0 √-2 -√-2 0 0 -√-2 complex lifted from SD16 ρ18 2 -2 -2 2 0 2 -2 0 0 0 0 -√-2 0 0 -√-2 √-2 0 0 √-2 complex lifted from SD16 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2

Smallest permutation representation of C4.10D8
Regular action on 64 points
Generators in S64
```(1 25 11 45)(2 46 12 26)(3 27 13 47)(4 48 14 28)(5 29 15 41)(6 42 16 30)(7 31 9 43)(8 44 10 32)(17 35 61 55)(18 56 62 36)(19 37 63 49)(20 50 64 38)(21 39 57 51)(22 52 58 40)(23 33 59 53)(24 54 60 34)
(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 52 45 22 11 40 25 58)(2 21 26 51 12 57 46 39)(3 50 47 20 13 38 27 64)(4 19 28 49 14 63 48 37)(5 56 41 18 15 36 29 62)(6 17 30 55 16 61 42 35)(7 54 43 24 9 34 31 60)(8 23 32 53 10 59 44 33)```

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

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

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

Matrix representation of C4.10D8 in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 16 2 0 0 16 1
,
 3 3 0 0 14 3 0 0 0 0 16 2 0 0 0 1
,
 4 11 0 0 11 13 0 0 0 0 0 7 0 0 5 7
`G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,16,16,0,0,2,1],[3,14,0,0,3,3,0,0,0,0,16,0,0,0,2,1],[4,11,0,0,11,13,0,0,0,0,0,5,0,0,7,7] >;`

C4.10D8 in GAP, Magma, Sage, TeX

`C_4._{10}D_8`
`% in TeX`

`G:=Group("C4.10D8");`
`// GroupNames label`

`G:=SmallGroup(64,13);`
`// by ID`

`G=gap.SmallGroup(64,13);`
`# by ID`

`G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,332,158,681,165]);`
`// Polycyclic`

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

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