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## G = C62⋊4Q8order 288 = 25·32

### 2nd semidirect product of C62 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62⋊4Q8
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C32⋊2Q8 — C62⋊4Q8
 Lower central C32 — C62 — C62⋊4Q8
 Upper central C1 — C22 — C23

Generators and relations for C624Q8
G = < a,b,c,d | a6=b6=c4=1, d2=c2, ab=ba, cac-1=a-1b3, dad-1=ab3, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 594 in 175 conjugacy classes, 54 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×2], C6 [×6], C6 [×11], C2×C4 [×8], Q8 [×2], C23, C32, Dic3 [×14], C12 [×4], C2×C6 [×6], C2×C6 [×11], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3×C6 [×3], C3×C6 [×2], Dic6 [×4], C2×Dic3 [×4], C2×Dic3 [×14], C2×C12 [×4], C22×C6 [×2], C22×C6, C22⋊Q8, C3×Dic3 [×4], C3⋊Dic3 [×2], C3⋊Dic3, C62, C62 [×2], C62 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C6.D4 [×2], C3×C22⋊C4 [×2], C2×Dic6 [×2], C22×Dic3 [×3], C322Q8 [×2], C6×Dic3 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], C2×C62, Dic3.D4 [×2], Dic3⋊Dic3 [×2], C62.C22, C3×C6.D4 [×2], C2×C322Q8, C22×C3⋊Dic3, C624Q8
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, Dic6 [×4], C22×S3 [×2], C22⋊Q8, S32, C2×Dic6 [×2], S3×D4 [×2], D42S3 [×2], C322Q8 [×2], C2×S32, Dic3.D4 [×2], D6.4D6, C2×C322Q8, Dic3⋊D6, C624Q8

Smallest permutation representation of C624Q8
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 13 3 15 5 17)(2 14 4 16 6 18)(7 46 9 48 11 44)(8 47 10 43 12 45)(19 25 23 29 21 27)(20 26 24 30 22 28)(31 41 35 39 33 37)(32 42 36 40 34 38)
(1 23 4 30)(2 26 5 19)(3 21 6 28)(7 39 10 34)(8 36 11 41)(9 37 12 32)(13 29 16 22)(14 24 17 25)(15 27 18 20)(31 45 42 48)(33 43 38 46)(35 47 40 44)
(1 31 4 42)(2 40 5 35)(3 33 6 38)(7 20 10 27)(8 25 11 24)(9 22 12 29)(13 37 16 32)(14 36 17 41)(15 39 18 34)(19 44 26 47)(21 46 28 43)(23 48 30 45)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6Q 12A ··· 12H order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 4 12 12 12 12 18 18 18 18 2 ··· 2 4 ··· 4 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + - + + - + + - - + - + image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 C4○D4 Dic6 S32 S3×D4 D4⋊2S3 C32⋊2Q8 C2×S32 D6.4D6 Dic3⋊D6 kernel C62⋊4Q8 Dic3⋊Dic3 C62.C22 C3×C6.D4 C2×C32⋊2Q8 C22×C3⋊Dic3 C6.D4 C3⋊Dic3 C62 C2×Dic3 C22×C6 C3×C6 C2×C6 C23 C6 C6 C22 C22 C2 C2 # reps 1 2 1 2 1 1 2 2 2 4 2 2 8 1 2 2 2 1 2 2

Matrix representation of C624Q8 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12
,
 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 9 0 0 0 0 0 0 9 10 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 12

`G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,9,0,0,0,0,0,0,9,10,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12] >;`

C624Q8 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_4Q_8`
`% in TeX`

`G:=Group("C6^2:4Q8");`
`// GroupNames label`

`G:=SmallGroup(288,630);`
`// by ID`

`G=gap.SmallGroup(288,630);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,64,422,219,1356,9414]);`
`// Polycyclic`

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

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