Copied to
clipboard

## G = C62.Q8order 288 = 25·32

### 1st non-split extension by C62 of Q8 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C62.Q8
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C22×C3⋊S3 — C22×C32⋊C4 — C62.Q8
 Lower central C32 — C3⋊S3 — C62.Q8
 Upper central C1 — C22

Generators and relations for C62.Q8
G = < a,b,c,d | a6=b6=c4=1, d2=a3c2, ab=ba, cac-1=a3b2, dad-1=a-1b4, cbc-1=a4b3, dbd-1=a4b, dcd-1=a3b3c-1 >

Subgroups: 612 in 92 conjugacy classes, 31 normal (7 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22, C22 [×6], S3 [×4], C6 [×3], C2×C4 [×12], C23, C32, D6 [×6], C2×C6, C22×C4 [×3], C3⋊S3, C3⋊S3 [×3], C3×C6 [×3], C22×S3, C2.C42, C32⋊C4 [×6], C2×C3⋊S3 [×6], C62, C2×C32⋊C4 [×6], C2×C32⋊C4 [×6], C22×C3⋊S3, C22×C32⋊C4 [×3], C62.Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, PSU3(𝔽2), C2.PSU3(𝔽2) [×3], C62.Q8

Character table of C62.Q8

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C size 1 1 1 1 9 9 9 9 8 18 18 18 18 18 18 18 18 18 18 18 18 8 8 8 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 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 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 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 linear of order 2 ρ5 1 -1 -1 1 1 1 -1 -1 1 i i -i -i -1 -1 1 1 -i -i i i 1 -1 -1 linear of order 4 ρ6 1 -1 -1 1 1 1 -1 -1 1 -i -i i i -1 -1 1 1 i i -i -i 1 -1 -1 linear of order 4 ρ7 1 1 -1 -1 -1 1 1 -1 1 -i 1 1 -1 i -i -i i -i i i -1 -1 1 -1 linear of order 4 ρ8 1 1 -1 -1 -1 1 1 -1 1 i -1 -1 1 i -i -i i i -i -i 1 -1 1 -1 linear of order 4 ρ9 1 1 -1 -1 -1 1 1 -1 1 -i -1 -1 1 -i i i -i -i i i 1 -1 1 -1 linear of order 4 ρ10 1 -1 -1 1 1 1 -1 -1 1 i -i i i 1 1 -1 -1 -i -i i -i 1 -1 -1 linear of order 4 ρ11 1 -1 -1 1 1 1 -1 -1 1 -i i -i -i 1 1 -1 -1 i i -i i 1 -1 -1 linear of order 4 ρ12 1 1 -1 -1 -1 1 1 -1 1 i 1 1 -1 -i i i -i i -i -i -1 -1 1 -1 linear of order 4 ρ13 1 -1 1 -1 -1 1 -1 1 1 1 i -i i -i i -i i -1 1 -1 -i -1 -1 1 linear of order 4 ρ14 1 -1 1 -1 -1 1 -1 1 1 -1 -i i -i -i i -i i 1 -1 1 i -1 -1 1 linear of order 4 ρ15 1 -1 1 -1 -1 1 -1 1 1 1 -i i -i i -i i -i -1 1 -1 i -1 -1 1 linear of order 4 ρ16 1 -1 1 -1 -1 1 -1 1 1 -1 i -i i i -i i -i 1 -1 1 -i -1 -1 1 linear of order 4 ρ17 2 -2 2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 2 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 -2 orthogonal lifted from D4 ρ19 2 -2 -2 2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 orthogonal lifted from D4 ρ20 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 symplectic lifted from Q8, Schur index 2 ρ21 8 8 -8 -8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 1 orthogonal lifted from C2.PSU3(𝔽2) ρ22 8 -8 8 -8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 orthogonal lifted from C2.PSU3(𝔽2) ρ23 8 -8 -8 8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 orthogonal lifted from C2.PSU3(𝔽2) ρ24 8 8 8 8 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from PSU3(𝔽2)

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

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

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

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

Matrix representation of C62.Q8 in GL10(𝔽13)

 12 0 0 0 0 0 0 0 0 0 0 12 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 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 6 6 1 0 0 5 12 12 0 0 0 0 0 12 8 0 1 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 12 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 6 0 0 5 0 0 12 0 0 7 0 12 12 0 8 1 1
,
 0 1 0 0 0 0 0 0 0 0 1 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 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 6 6 1 1 5 5 11 12 0 0 0 0 0 0 0 0 12 1 0 0 1 1 8 8 0 0 8 0 0 0 1 1 8 8 12 0 8 0
,
 8 0 0 0 0 0 0 0 0 0 0 5 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 12 1 0 0 6 6 1 1 5 5 11 12 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 9 9 5 5 8 8 12 0 0 0 9 9 5 4 8 8 12 0

`G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,6,0,0,0,0,1,0,0,0,0,6,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,12,12,0,0,0,12,0,0,0,0,0,0,12,12,0,8,0,0,0,0,0,0,1,0,5,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,7,0,0,1,1,0,0,0,0,6,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,1,1,5,0,0,0,0,0,0,0,12,0,0,8,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,6,0,1,1,0,0,0,0,12,0,6,0,1,1,0,0,1,0,0,0,1,0,8,8,0,0,0,1,0,0,1,0,8,8,0,0,0,0,0,0,5,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,11,12,8,8,0,0,0,0,0,0,12,1,0,0],[8,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,12,9,9,0,0,0,0,0,6,12,0,9,9,0,0,0,0,0,1,0,0,5,5,0,0,0,0,0,1,0,0,5,4,0,0,1,0,0,5,0,0,8,8,0,0,0,1,0,5,0,0,8,8,0,0,0,0,12,11,0,0,12,12,0,0,0,0,1,12,0,0,0,0] >;`

C62.Q8 in GAP, Magma, Sage, TeX

`C_6^2.Q_8`
`% in TeX`

`G:=Group("C6^2.Q8");`
`// GroupNames label`

`G:=SmallGroup(288,395);`
`// by ID`

`G=gap.SmallGroup(288,395);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,309,92,9413,2028,691,12550,1581,2372]);`
`// Polycyclic`

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

Export

׿
×
𝔽