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## G = C62.2Q8order 288 = 25·32

### 2nd non-split extension by C62 of Q8 acting faithfully

Aliases: C62.2Q8, C22.PSU3(𝔽2), C3⋊Dic3.8D4, C322C8.3C4, C322(C8.C4), C62.C4.1C2, C2.5(C2.PSU3(𝔽2)), (C3×C6).10(C4⋊C4), C3⋊Dic3.16(C2×C4), (C2×C322C8).7C2, (C2×C3⋊Dic3).6C22, SmallGroup(288,396)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — C62.2Q8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×C32⋊2C8 — C62.2Q8
 Lower central C32 — C3×C6 — C3⋊Dic3 — C62.2Q8
 Upper central C1 — C2 — C22

Generators and relations for C62.2Q8
G = < a,b,c,d | a6=b6=1, c4=b3, d2=a3c2, ab=ba, cac-1=a3b-1, dad-1=a-1b4, cbc-1=a4b3, dbd-1=a4b, dcd-1=a3c3 >

Character table of C62.2Q8

 class 1 2A 2B 3 4A 4B 4C 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 8 9 9 18 8 8 8 18 18 18 18 36 36 36 36 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 -1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 i -i -i i linear of order 4 ρ6 1 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 i -i i -i linear of order 4 ρ7 1 1 -1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 -i i i -i linear of order 4 ρ8 1 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 -i i -i i linear of order 4 ρ9 2 2 -2 2 2 2 -2 -2 2 -2 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 symplectic lifted from Q8, Schur index 2 ρ11 2 -2 0 2 -2i 2i 0 0 -2 0 -√2 √2 -√-2 √-2 0 0 0 0 complex lifted from C8.C4 ρ12 2 -2 0 2 -2i 2i 0 0 -2 0 √2 -√2 √-2 -√-2 0 0 0 0 complex lifted from C8.C4 ρ13 2 -2 0 2 2i -2i 0 0 -2 0 √2 -√2 -√-2 √-2 0 0 0 0 complex lifted from C8.C4 ρ14 2 -2 0 2 2i -2i 0 0 -2 0 -√2 √2 √-2 -√-2 0 0 0 0 complex lifted from C8.C4 ρ15 8 8 -8 -1 0 0 0 1 -1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2.PSU3(𝔽2) ρ16 8 8 8 -1 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from PSU3(𝔽2) ρ17 8 -8 0 -1 0 0 0 3 1 -3 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ18 8 -8 0 -1 0 0 0 -3 1 3 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C62.2Q8
On 48 points
Generators in S48
```(1 39 29)(2 6)(3 31 33)(4 8)(5 35 25)(7 27 37)(9 17 48)(10 22 41 14 18 45)(11 42 19)(12 47 20 16 43 24)(13 21 44)(15 46 23)(26 30)(28 32)(34 38)(36 40)
(1 5)(2 36 30 6 40 26)(3 7)(4 28 34 8 32 38)(9 44 17 13 48 21)(10 22 41 14 18 45)(11 23 42 15 19 46)(12 47 20 16 43 24)(25 29)(27 31)(33 37)(35 39)
(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)
(1 13 3 15 5 9 7 11)(2 12 8 10 6 16 4 14)(17 27 19 29 21 31 23 25)(18 26 24 32 22 30 20 28)(33 46 35 48 37 42 39 44)(34 45 40 43 38 41 36 47)```

`G:=sub<Sym(48)| (1,39,29)(2,6)(3,31,33)(4,8)(5,35,25)(7,27,37)(9,17,48)(10,22,41,14,18,45)(11,42,19)(12,47,20,16,43,24)(13,21,44)(15,46,23)(26,30)(28,32)(34,38)(36,40), (1,5)(2,36,30,6,40,26)(3,7)(4,28,34,8,32,38)(9,44,17,13,48,21)(10,22,41,14,18,45)(11,23,42,15,19,46)(12,47,20,16,43,24)(25,29)(27,31)(33,37)(35,39), (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), (1,13,3,15,5,9,7,11)(2,12,8,10,6,16,4,14)(17,27,19,29,21,31,23,25)(18,26,24,32,22,30,20,28)(33,46,35,48,37,42,39,44)(34,45,40,43,38,41,36,47)>;`

`G:=Group( (1,39,29)(2,6)(3,31,33)(4,8)(5,35,25)(7,27,37)(9,17,48)(10,22,41,14,18,45)(11,42,19)(12,47,20,16,43,24)(13,21,44)(15,46,23)(26,30)(28,32)(34,38)(36,40), (1,5)(2,36,30,6,40,26)(3,7)(4,28,34,8,32,38)(9,44,17,13,48,21)(10,22,41,14,18,45)(11,23,42,15,19,46)(12,47,20,16,43,24)(25,29)(27,31)(33,37)(35,39), (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), (1,13,3,15,5,9,7,11)(2,12,8,10,6,16,4,14)(17,27,19,29,21,31,23,25)(18,26,24,32,22,30,20,28)(33,46,35,48,37,42,39,44)(34,45,40,43,38,41,36,47) );`

`G=PermutationGroup([(1,39,29),(2,6),(3,31,33),(4,8),(5,35,25),(7,27,37),(9,17,48),(10,22,41,14,18,45),(11,42,19),(12,47,20,16,43,24),(13,21,44),(15,46,23),(26,30),(28,32),(34,38),(36,40)], [(1,5),(2,36,30,6,40,26),(3,7),(4,28,34,8,32,38),(9,44,17,13,48,21),(10,22,41,14,18,45),(11,23,42,15,19,46),(12,47,20,16,43,24),(25,29),(27,31),(33,37),(35,39)], [(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)], [(1,13,3,15,5,9,7,11),(2,12,8,10,6,16,4,14),(17,27,19,29,21,31,23,25),(18,26,24,32,22,30,20,28),(33,46,35,48,37,42,39,44),(34,45,40,43,38,41,36,47)])`

Matrix representation of C62.2Q8 in GL10(𝔽73)

 1 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 1 0
,
 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 1 0
,
 0 1 0 0 0 0 0 0 0 0 46 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 13 11 0 0 0 0 0 0 0 0 71 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 11 0 0 0 0 0 0 0 0 71 60 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0
,
 51 0 0 0 0 0 0 0 0 0 0 63 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 13 11 0 0 0 0 0 0 0 0 71 60 0 0 0 0 0 0 0 0 0 0 13 11 0 0 0 0 0 0 0 0 71 60 0 0 0 0

`G:=sub<GL(10,GF(73))| [1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0],[72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0],[0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,13,71,0,0,0,0,0,0,0,0,11,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,13,71,0,0,0,0,0,0,0,0,11,60,0,0],[51,0,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,0,0,0,0,0,13,71,0,0,0,0,0,0,0,0,11,60,0,0,0,0,0,0,0,0,0,0,13,71,0,0,0,0,0,0,0,0,11,60,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C62.2Q8 in GAP, Magma, Sage, TeX

`C_6^2._2Q_8`
`% in TeX`

`G:=Group("C6^2.2Q8");`
`// GroupNames label`

`G:=SmallGroup(288,396);`
`// by ID`

`G=gap.SmallGroup(288,396);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,85,92,219,100,346,9413,2028,691,12550,1581,2372]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^4=b^3,d^2=a^3*c^2,a*b=b*a,c*a*c^-1=a^3*b^-1,d*a*d^-1=a^-1*b^4,c*b*c^-1=a^4*b^3,d*b*d^-1=a^4*b,d*c*d^-1=a^3*c^3>;`
`// generators/relations`

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