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## G = C12.6S4order 288 = 25·32

### 6th 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) — C12.6S4
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C6.5S4 — C12.6S4
 Lower central C3×SL2(𝔽3) — C12.6S4
 Upper central C1 — C2 — C4

Generators and relations for C12.6S4
G = < a,b,c,d,e | a12=d3=1, b2=c2=e2=a6, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc-1=a6b, dbd-1=a6bc, ebe-1=bc, dcd-1=b, ece-1=a6c, ede-1=d-1 >

Subgroups: 480 in 94 conjugacy classes, 21 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×C6, C3⋊C8, SL2(𝔽3), Dic6, C2×Dic3, C2×C12, C3×D4, C3×Q8, C8.C22, C3⋊Dic3, C3×C12, C4.Dic3, D4.S3, C3⋊Q16, CSU2(𝔽3), C4.A4, C2×Dic6, C3×C4○D4, C3×SL2(𝔽3), C324Q8, Q8.14D6, C4.S4, C6.5S4, C3×C4.A4, C12.6S4
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S4, C2×C3⋊S3, C2×S4, C3⋊S4, C4.S4, C2×C3⋊S4, C12.6S4

Character table of C12.6S4

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

Smallest permutation representation of C12.6S4
On 96 points
Generators in S96
```(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 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 28 7 34)(2 29 8 35)(3 30 9 36)(4 31 10 25)(5 32 11 26)(6 33 12 27)(13 78 19 84)(14 79 20 73)(15 80 21 74)(16 81 22 75)(17 82 23 76)(18 83 24 77)(37 89 43 95)(38 90 44 96)(39 91 45 85)(40 92 46 86)(41 93 47 87)(42 94 48 88)(49 70 55 64)(50 71 56 65)(51 72 57 66)(52 61 58 67)(53 62 59 68)(54 63 60 69)
(1 86 7 92)(2 87 8 93)(3 88 9 94)(4 89 10 95)(5 90 11 96)(6 91 12 85)(13 63 19 69)(14 64 20 70)(15 65 21 71)(16 66 22 72)(17 67 23 61)(18 68 24 62)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)(49 79 55 73)(50 80 56 74)(51 81 57 75)(52 82 58 76)(53 83 59 77)(54 84 60 78)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 52 65)(14 53 66)(15 54 67)(16 55 68)(17 56 69)(18 57 70)(19 58 71)(20 59 72)(21 60 61)(22 49 62)(23 50 63)(24 51 64)(25 87 45)(26 88 46)(27 89 47)(28 90 48)(29 91 37)(30 92 38)(31 93 39)(32 94 40)(33 95 41)(34 96 42)(35 85 43)(36 86 44)(73 77 81)(74 78 82)(75 79 83)(76 80 84)
(1 81 7 75)(2 80 8 74)(3 79 9 73)(4 78 10 84)(5 77 11 83)(6 76 12 82)(13 37 19 43)(14 48 20 42)(15 47 21 41)(16 46 22 40)(17 45 23 39)(18 44 24 38)(25 63 31 69)(26 62 32 68)(27 61 33 67)(28 72 34 66)(29 71 35 65)(30 70 36 64)(49 94 55 88)(50 93 56 87)(51 92 57 86)(52 91 58 85)(53 90 59 96)(54 89 60 95)```

`G:=sub<Sym(96)| (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)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,31,10,25)(5,32,11,26)(6,33,12,27)(13,78,19,84)(14,79,20,73)(15,80,21,74)(16,81,22,75)(17,82,23,76)(18,83,24,77)(37,89,43,95)(38,90,44,96)(39,91,45,85)(40,92,46,86)(41,93,47,87)(42,94,48,88)(49,70,55,64)(50,71,56,65)(51,72,57,66)(52,61,58,67)(53,62,59,68)(54,63,60,69), (1,86,7,92)(2,87,8,93)(3,88,9,94)(4,89,10,95)(5,90,11,96)(6,91,12,85)(13,63,19,69)(14,64,20,70)(15,65,21,71)(16,66,22,72)(17,67,23,61)(18,68,24,62)(25,43,31,37)(26,44,32,38)(27,45,33,39)(28,46,34,40)(29,47,35,41)(30,48,36,42)(49,79,55,73)(50,80,56,74)(51,81,57,75)(52,82,58,76)(53,83,59,77)(54,84,60,78), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,52,65)(14,53,66)(15,54,67)(16,55,68)(17,56,69)(18,57,70)(19,58,71)(20,59,72)(21,60,61)(22,49,62)(23,50,63)(24,51,64)(25,87,45)(26,88,46)(27,89,47)(28,90,48)(29,91,37)(30,92,38)(31,93,39)(32,94,40)(33,95,41)(34,96,42)(35,85,43)(36,86,44)(73,77,81)(74,78,82)(75,79,83)(76,80,84), (1,81,7,75)(2,80,8,74)(3,79,9,73)(4,78,10,84)(5,77,11,83)(6,76,12,82)(13,37,19,43)(14,48,20,42)(15,47,21,41)(16,46,22,40)(17,45,23,39)(18,44,24,38)(25,63,31,69)(26,62,32,68)(27,61,33,67)(28,72,34,66)(29,71,35,65)(30,70,36,64)(49,94,55,88)(50,93,56,87)(51,92,57,86)(52,91,58,85)(53,90,59,96)(54,89,60,95)>;`

`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)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,28,7,34)(2,29,8,35)(3,30,9,36)(4,31,10,25)(5,32,11,26)(6,33,12,27)(13,78,19,84)(14,79,20,73)(15,80,21,74)(16,81,22,75)(17,82,23,76)(18,83,24,77)(37,89,43,95)(38,90,44,96)(39,91,45,85)(40,92,46,86)(41,93,47,87)(42,94,48,88)(49,70,55,64)(50,71,56,65)(51,72,57,66)(52,61,58,67)(53,62,59,68)(54,63,60,69), (1,86,7,92)(2,87,8,93)(3,88,9,94)(4,89,10,95)(5,90,11,96)(6,91,12,85)(13,63,19,69)(14,64,20,70)(15,65,21,71)(16,66,22,72)(17,67,23,61)(18,68,24,62)(25,43,31,37)(26,44,32,38)(27,45,33,39)(28,46,34,40)(29,47,35,41)(30,48,36,42)(49,79,55,73)(50,80,56,74)(51,81,57,75)(52,82,58,76)(53,83,59,77)(54,84,60,78), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,52,65)(14,53,66)(15,54,67)(16,55,68)(17,56,69)(18,57,70)(19,58,71)(20,59,72)(21,60,61)(22,49,62)(23,50,63)(24,51,64)(25,87,45)(26,88,46)(27,89,47)(28,90,48)(29,91,37)(30,92,38)(31,93,39)(32,94,40)(33,95,41)(34,96,42)(35,85,43)(36,86,44)(73,77,81)(74,78,82)(75,79,83)(76,80,84), (1,81,7,75)(2,80,8,74)(3,79,9,73)(4,78,10,84)(5,77,11,83)(6,76,12,82)(13,37,19,43)(14,48,20,42)(15,47,21,41)(16,46,22,40)(17,45,23,39)(18,44,24,38)(25,63,31,69)(26,62,32,68)(27,61,33,67)(28,72,34,66)(29,71,35,65)(30,70,36,64)(49,94,55,88)(50,93,56,87)(51,92,57,86)(52,91,58,85)(53,90,59,96)(54,89,60,95) );`

`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),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,28,7,34),(2,29,8,35),(3,30,9,36),(4,31,10,25),(5,32,11,26),(6,33,12,27),(13,78,19,84),(14,79,20,73),(15,80,21,74),(16,81,22,75),(17,82,23,76),(18,83,24,77),(37,89,43,95),(38,90,44,96),(39,91,45,85),(40,92,46,86),(41,93,47,87),(42,94,48,88),(49,70,55,64),(50,71,56,65),(51,72,57,66),(52,61,58,67),(53,62,59,68),(54,63,60,69)], [(1,86,7,92),(2,87,8,93),(3,88,9,94),(4,89,10,95),(5,90,11,96),(6,91,12,85),(13,63,19,69),(14,64,20,70),(15,65,21,71),(16,66,22,72),(17,67,23,61),(18,68,24,62),(25,43,31,37),(26,44,32,38),(27,45,33,39),(28,46,34,40),(29,47,35,41),(30,48,36,42),(49,79,55,73),(50,80,56,74),(51,81,57,75),(52,82,58,76),(53,83,59,77),(54,84,60,78)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,52,65),(14,53,66),(15,54,67),(16,55,68),(17,56,69),(18,57,70),(19,58,71),(20,59,72),(21,60,61),(22,49,62),(23,50,63),(24,51,64),(25,87,45),(26,88,46),(27,89,47),(28,90,48),(29,91,37),(30,92,38),(31,93,39),(32,94,40),(33,95,41),(34,96,42),(35,85,43),(36,86,44),(73,77,81),(74,78,82),(75,79,83),(76,80,84)], [(1,81,7,75),(2,80,8,74),(3,79,9,73),(4,78,10,84),(5,77,11,83),(6,76,12,82),(13,37,19,43),(14,48,20,42),(15,47,21,41),(16,46,22,40),(17,45,23,39),(18,44,24,38),(25,63,31,69),(26,62,32,68),(27,61,33,67),(28,72,34,66),(29,71,35,65),(30,70,36,64),(49,94,55,88),(50,93,56,87),(51,92,57,86),(52,91,58,85),(53,90,59,96),(54,89,60,95)]])`

Matrix representation of C12.6S4 in GL4(𝔽73) generated by

 66 0 7 7 7 66 0 7 0 7 66 7 66 66 66 52
,
 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
,
 72 0 72 72 0 0 0 1 1 1 0 1 0 72 0 72
,
 22 0 12 12 0 12 22 12 12 22 0 12 51 51 51 39
`G:=sub<GL(4,GF(73))| [66,7,0,66,0,66,7,66,7,0,66,66,7,7,7,52],[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],[72,0,1,0,0,0,1,72,72,0,0,0,72,1,1,72],[22,0,12,51,0,12,22,51,12,22,0,51,12,12,12,39] >;`

C12.6S4 in GAP, Magma, Sage, TeX

`C_{12}._6S_4`
`% in TeX`

`G:=Group("C12.6S4");`
`// GroupNames label`

`G:=SmallGroup(288,913);`
`// by ID`

`G=gap.SmallGroup(288,913);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1008,2045,1016,170,675,2524,1908,172,1517,1153,285,124]);`
`// Polycyclic`

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

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