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G = C8.5Q8order 64 = 26

4th non-split extension by C8 of Q8 acting via Q8/C4=C2

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

Aliases: C8.5Q8, C42.84C22, (C4×C8).9C2, C4.6(C2×Q8), (C2×C4).59D4, C2.7(C4⋊Q8), C2.D8.6C2, C4.Q8.7C2, C2.19(C4○D8), C4⋊C4.23C22, (C2×C8).94C22, C42.C2.4C2, (C2×C4).124C23, C22.120(C2×D4), SmallGroup(64,180)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C8.5Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C8.5Q8
 Lower central C1 — C2 — C2×C4 — C8.5Q8
 Upper central C1 — C22 — C42 — C8.5Q8
 Jennings C1 — C2 — C2 — C2×C4 — C8.5Q8

Generators and relations for C8.5Q8
G = < a,b,c | a8=b4=1, c2=a4b2, ab=ba, cac-1=a3, cbc-1=a4b-1 >

Character table of C8.5Q8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 2 2 2 2 8 8 8 8 2 2 2 2 2 2 2 2 ρ1 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 2 2 2 2 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 -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 0 2 0 0 0 0 0 0 2 0 0 -2 -2 0 2 0 symplectic lifted from Q8, Schur index 2 ρ12 2 -2 2 -2 0 2 0 -2 0 0 0 0 0 0 0 2 -2 0 0 -2 0 2 symplectic lifted from Q8, Schur index 2 ρ13 2 -2 2 -2 0 2 0 -2 0 0 0 0 0 0 0 -2 2 0 0 2 0 -2 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 2 -2 0 -2 0 2 0 0 0 0 0 0 -2 0 0 2 2 0 -2 0 symplectic lifted from Q8, Schur index 2 ρ15 2 -2 -2 2 2i 0 0 0 -2i 0 0 0 0 0 -√2 -√-2 -√-2 √2 -√2 √-2 √2 √-2 complex lifted from C4○D8 ρ16 2 2 -2 -2 0 0 -2i 0 0 2i 0 0 0 0 -√2 -√-2 √-2 -√2 √2 -√-2 √2 √-2 complex lifted from C4○D8 ρ17 2 -2 -2 2 -2i 0 0 0 2i 0 0 0 0 0 √2 -√-2 -√-2 -√2 √2 √-2 -√2 √-2 complex lifted from C4○D8 ρ18 2 2 -2 -2 0 0 2i 0 0 -2i 0 0 0 0 √2 -√-2 √-2 √2 -√2 -√-2 -√2 √-2 complex lifted from C4○D8 ρ19 2 2 -2 -2 0 0 2i 0 0 -2i 0 0 0 0 -√2 √-2 -√-2 -√2 √2 √-2 √2 -√-2 complex lifted from C4○D8 ρ20 2 2 -2 -2 0 0 -2i 0 0 2i 0 0 0 0 √2 √-2 -√-2 √2 -√2 √-2 -√2 -√-2 complex lifted from C4○D8 ρ21 2 -2 -2 2 2i 0 0 0 -2i 0 0 0 0 0 √2 √-2 √-2 -√2 √2 -√-2 -√2 -√-2 complex lifted from C4○D8 ρ22 2 -2 -2 2 -2i 0 0 0 2i 0 0 0 0 0 -√2 √-2 √-2 √2 -√2 -√-2 √2 -√-2 complex lifted from C4○D8

Smallest permutation representation of C8.5Q8
Regular action on 64 points
Generators in S64
```(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 33 19 26)(2 34 20 27)(3 35 21 28)(4 36 22 29)(5 37 23 30)(6 38 24 31)(7 39 17 32)(8 40 18 25)(9 46 58 55)(10 47 59 56)(11 48 60 49)(12 41 61 50)(13 42 62 51)(14 43 63 52)(15 44 64 53)(16 45 57 54)
(1 51 23 46)(2 54 24 41)(3 49 17 44)(4 52 18 47)(5 55 19 42)(6 50 20 45)(7 53 21 48)(8 56 22 43)(9 37 62 26)(10 40 63 29)(11 35 64 32)(12 38 57 27)(13 33 58 30)(14 36 59 25)(15 39 60 28)(16 34 61 31)```

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

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

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

Matrix representation of C8.5Q8 in GL4(𝔽17) generated by

 1 0 0 0 0 1 0 0 0 0 15 0 0 0 0 9
,
 1 13 0 0 9 16 0 0 0 0 4 0 0 0 0 4
,
 13 16 0 0 0 4 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,15,0,0,0,0,9],[1,9,0,0,13,16,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,16,4,0,0,0,0,0,1,0,0,1,0] >;`

C8.5Q8 in GAP, Magma, Sage, TeX

`C_8._5Q_8`
`% in TeX`

`G:=Group("C8.5Q8");`
`// GroupNames label`

`G:=SmallGroup(64,180);`
`// by ID`

`G=gap.SmallGroup(64,180);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,247,362,86,1444,88]);`
`// Polycyclic`

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

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