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## G = C12.7S4order 288 = 25·32

### 7th non-split extension by C12 of S4 acting via S4/A4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — C12.7S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C6.6S4 — C12.7S4
 Lower central C3×SL2(𝔽3) — C12.7S4
 Upper central C1 — C2 — C4

Generators and relations for C12.7S4
G = < a,b,c,d,e | a12=d3=e2=1, b2=c2=a6, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a6b, dbd-1=a6bc, ebe=bc, dcd-1=b, ece=a6c, ede=d-1 >

Subgroups: 832 in 110 conjugacy classes, 21 normal (13 characteristic)
C1, C2, C2 [×3], C3, C3 [×3], C4, C4, C22 [×5], S3 [×8], C6, C6 [×4], C8 [×2], C2×C4, D4 [×4], Q8, C23, C32, C12, C12 [×4], D6 [×10], C2×C6, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], SL2(𝔽3) [×3], D12 [×6], C2×C12, C3×D4, C3×Q8, C22×S3, C8⋊C22, C3×C12, C2×C3⋊S3 [×2], C4.Dic3, D4⋊S3 [×2], Q82S3 [×2], GL2(𝔽3) [×6], C4.A4 [×3], C2×D12, C3×C4○D4, C3×SL2(𝔽3), C12⋊S3, D4⋊D6, C4.3S4 [×3], C6.6S4 [×2], C3×C4.A4, C12.7S4
Quotients: C1, C2 [×3], C22, S3 [×4], D6 [×4], C3⋊S3, S4, C2×C3⋊S3, C2×S4, C3⋊S4, C4.3S4, C2×C3⋊S4, C12.7S4

Character table of C12.7S4

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 12I size 1 1 6 36 36 2 8 8 8 2 6 2 8 8 8 12 36 36 2 2 8 8 8 8 8 8 12 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 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 -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 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 -1 -1 1 linear of order 2 ρ5 2 2 -2 0 0 -1 -1 -1 2 -2 2 -1 2 -1 -1 1 0 0 1 1 1 -2 -2 1 1 1 -1 orthogonal lifted from D6 ρ6 2 2 -2 0 0 -1 2 -1 -1 -2 2 -1 -1 2 -1 1 0 0 1 1 -2 1 1 1 1 -2 -1 orthogonal lifted from D6 ρ7 2 2 -2 0 0 -1 -1 2 -1 -2 2 -1 -1 -1 2 1 0 0 1 1 1 1 1 -2 -2 1 -1 orthogonal lifted from D6 ρ8 2 2 2 0 0 -1 -1 -1 2 2 2 -1 2 -1 -1 -1 0 0 -1 -1 -1 2 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ9 2 2 -2 0 0 2 -1 -1 -1 -2 2 2 -1 -1 -1 -2 0 0 -2 -2 1 1 1 1 1 1 2 orthogonal lifted from D6 ρ10 2 2 2 0 0 -1 -1 2 -1 2 2 -1 -1 -1 2 -1 0 0 -1 -1 -1 -1 -1 2 2 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 0 0 -1 2 -1 -1 2 2 -1 -1 2 -1 -1 0 0 -1 -1 2 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ12 2 2 2 0 0 2 -1 -1 -1 2 2 2 -1 -1 -1 2 0 0 2 2 -1 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ13 3 3 -1 -1 -1 3 0 0 0 3 -1 3 0 0 0 -1 1 1 3 3 0 0 0 0 0 0 -1 orthogonal lifted from S4 ρ14 3 3 1 1 -1 3 0 0 0 -3 -1 3 0 0 0 1 -1 1 -3 -3 0 0 0 0 0 0 -1 orthogonal lifted from C2×S4 ρ15 3 3 -1 1 1 3 0 0 0 3 -1 3 0 0 0 -1 -1 -1 3 3 0 0 0 0 0 0 -1 orthogonal lifted from S4 ρ16 3 3 1 -1 1 3 0 0 0 -3 -1 3 0 0 0 1 1 -1 -3 -3 0 0 0 0 0 0 -1 orthogonal lifted from C2×S4 ρ17 4 -4 0 0 0 4 -2 -2 -2 0 0 -4 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.3S4 ρ18 4 -4 0 0 0 -2 -2 1 1 0 0 2 -1 2 -1 0 0 0 2√3 -2√3 0 -√3 √3 √3 -√3 0 0 orthogonal faithful ρ19 4 -4 0 0 0 4 1 1 1 0 0 -4 -1 -1 -1 0 0 0 0 0 -√3 √3 -√3 √3 -√3 √3 0 orthogonal lifted from C4.3S4 ρ20 4 -4 0 0 0 4 1 1 1 0 0 -4 -1 -1 -1 0 0 0 0 0 √3 -√3 √3 -√3 √3 -√3 0 orthogonal lifted from C4.3S4 ρ21 4 -4 0 0 0 -2 1 1 -2 0 0 2 2 -1 -1 0 0 0 -2√3 2√3 √3 0 0 √3 -√3 -√3 0 orthogonal faithful ρ22 4 -4 0 0 0 -2 -2 1 1 0 0 2 -1 2 -1 0 0 0 -2√3 2√3 0 √3 -√3 -√3 √3 0 0 orthogonal faithful ρ23 4 -4 0 0 0 -2 1 -2 1 0 0 2 -1 -1 2 0 0 0 -2√3 2√3 -√3 -√3 √3 0 0 √3 0 orthogonal faithful ρ24 4 -4 0 0 0 -2 1 1 -2 0 0 2 2 -1 -1 0 0 0 2√3 -2√3 -√3 0 0 -√3 √3 √3 0 orthogonal faithful ρ25 4 -4 0 0 0 -2 1 -2 1 0 0 2 -1 -1 2 0 0 0 2√3 -2√3 √3 √3 -√3 0 0 -√3 0 orthogonal faithful ρ26 6 6 2 0 0 -3 0 0 0 -6 -2 -3 0 0 0 -1 0 0 3 3 0 0 0 0 0 0 1 orthogonal lifted from C2×C3⋊S4 ρ27 6 6 -2 0 0 -3 0 0 0 6 -2 -3 0 0 0 1 0 0 -3 -3 0 0 0 0 0 0 1 orthogonal lifted from C3⋊S4

Smallest permutation representation of C12.7S4
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 29 7 35)(2 30 8 36)(3 31 9 25)(4 32 10 26)(5 33 11 27)(6 34 12 28)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)
(1 44 7 38)(2 45 8 39)(3 46 9 40)(4 47 10 41)(5 48 11 42)(6 37 12 43)(13 34 19 28)(14 35 20 29)(15 36 21 30)(16 25 22 31)(17 26 23 32)(18 27 24 33)
(13 34 37)(14 35 38)(15 36 39)(16 25 40)(17 26 41)(18 27 42)(19 28 43)(20 29 44)(21 30 45)(22 31 46)(23 32 47)(24 33 48)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(37 45)(38 44)(39 43)(40 42)(46 48)```

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

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

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

Matrix representation of C12.7S4 in GL4(𝔽73) generated by

 59 0 66 66 66 59 0 66 0 66 59 66 7 7 7 0
,
 0 0 1 0 72 72 72 71 72 0 0 0 1 1 0 1
,
 1 1 1 2 0 0 1 0 0 72 0 0 72 0 72 72
,
 1 0 0 0 72 72 72 71 0 72 0 0 0 1 1 1
,
 72 0 72 72 0 72 72 72 72 72 0 72 1 1 1 2
`G:=sub<GL(4,GF(73))| [59,66,0,7,0,59,66,7,66,0,59,7,66,66,66,0],[0,72,72,1,0,72,0,1,1,72,0,0,0,71,0,1],[1,0,0,72,1,0,72,0,1,1,0,72,2,0,0,72],[1,72,0,0,0,72,72,1,0,72,0,1,0,71,0,1],[72,0,72,1,0,72,72,1,72,72,0,1,72,72,72,2] >;`

C12.7S4 in GAP, Magma, Sage, TeX

`C_{12}._7S_4`
`% in TeX`

`G:=Group("C12.7S4");`
`// GroupNames label`

`G:=SmallGroup(288,915);`
`// by ID`

`G=gap.SmallGroup(288,915);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,1016,170,675,2524,1908,172,1517,1153,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=d^3=e^2=1,b^2=c^2=a^6,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=a^6*b,d*b*d^-1=a^6*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^6*c,e*d*e=d^-1>;`
`// generators/relations`

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