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## G = C12.9S4order 288 = 25·32

### 9th non-split extension by C12 of S4 acting via S4/A4=C2

Aliases: C12.9S4, Q8.Dic9, C3.U2(𝔽3), Q8⋊C92C4, C4○D4.1D9, C6.3(A4⋊C4), C4.5(C3.S4), (C3×Q8).2Dic3, Q8.C18.2C2, C2.3(C6.S4), (C3×C4○D4).1S3, SmallGroup(288,70)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — Q8⋊C9 — C12.9S4
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — Q8.C18 — C12.9S4
 Lower central Q8⋊C9 — C12.9S4
 Upper central C1 — C4

Generators and relations for C12.9S4
G = < a,b,c,d,e | a12=1, b2=c2=a6, d3=a4, e2=a9, ab=ba, ac=ca, ad=da, eae-1=a5, cbc-1=a6b, dbd-1=a6bc, ebe-1=bc, dcd-1=b, ece-1=a6c, ede-1=a8d2 >

Character table of C12.9S4

 class 1 2A 2B 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 8A 8B 9A 9B 9C 12A 12B 12C 18A 18B 18C 36A 36B 36C 36D 36E 36F size 1 1 6 2 1 1 6 18 18 18 18 2 12 36 36 8 8 8 2 2 12 8 8 8 8 8 8 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 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 1 1 1 linear of order 2 ρ3 1 1 -1 1 -1 -1 1 i -i i -i 1 -1 i -i 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ4 1 1 -1 1 -1 -1 1 -i i -i i 1 -1 -i i 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ5 2 2 2 2 2 2 2 0 0 0 0 2 2 0 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 -1 2 2 2 0 0 0 0 -1 -1 0 0 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -1 -1 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 orthogonal lifted from D9 ρ7 2 2 2 -1 2 2 2 0 0 0 0 -1 -1 0 0 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -1 -1 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 orthogonal lifted from D9 ρ8 2 2 2 -1 2 2 2 0 0 0 0 -1 -1 0 0 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -1 -1 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 orthogonal lifted from D9 ρ9 2 2 -2 2 -2 -2 2 0 0 0 0 2 -2 0 0 -1 -1 -1 -2 -2 2 -1 -1 -1 1 1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ10 2 2 -2 -1 -2 -2 2 0 0 0 0 -1 1 0 0 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 1 1 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 symplectic lifted from Dic9, Schur index 2 ρ11 2 2 -2 -1 -2 -2 2 0 0 0 0 -1 1 0 0 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 1 1 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 symplectic lifted from Dic9, Schur index 2 ρ12 2 2 -2 -1 -2 -2 2 0 0 0 0 -1 1 0 0 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 1 1 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 symplectic lifted from Dic9, Schur index 2 ρ13 2 -2 0 2 2i -2i 0 -1-i -1+i 1+i 1-i -2 0 0 0 -1 -1 -1 2i -2i 0 1 1 1 -i i -i i -i i complex lifted from U2(𝔽3) ρ14 2 -2 0 2 -2i 2i 0 1-i 1+i -1+i -1-i -2 0 0 0 -1 -1 -1 -2i 2i 0 1 1 1 i -i i -i i -i complex lifted from U2(𝔽3) ρ15 2 -2 0 2 2i -2i 0 1+i 1-i -1-i -1+i -2 0 0 0 -1 -1 -1 2i -2i 0 1 1 1 -i i -i i -i i complex lifted from U2(𝔽3) ρ16 2 -2 0 2 -2i 2i 0 -1+i -1-i 1-i 1+i -2 0 0 0 -1 -1 -1 -2i 2i 0 1 1 1 i -i i -i i -i complex lifted from U2(𝔽3) ρ17 3 3 -1 3 3 3 -1 -1 -1 -1 -1 3 -1 1 1 0 0 0 3 3 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ18 3 3 -1 3 3 3 -1 1 1 1 1 3 -1 -1 -1 0 0 0 3 3 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ19 3 3 1 3 -3 -3 -1 i -i i -i 3 1 -i i 0 0 0 -3 -3 -1 0 0 0 0 0 0 0 0 0 complex lifted from A4⋊C4 ρ20 3 3 1 3 -3 -3 -1 -i i -i i 3 1 i -i 0 0 0 -3 -3 -1 0 0 0 0 0 0 0 0 0 complex lifted from A4⋊C4 ρ21 4 -4 0 4 4i -4i 0 0 0 0 0 -4 0 0 0 1 1 1 4i -4i 0 -1 -1 -1 i -i i -i i -i complex lifted from U2(𝔽3) ρ22 4 -4 0 4 -4i 4i 0 0 0 0 0 -4 0 0 0 1 1 1 -4i 4i 0 -1 -1 -1 -i i -i i -i i complex lifted from U2(𝔽3) ρ23 4 -4 0 -2 -4i 4i 0 0 0 0 0 2 0 0 0 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 2i -2i 0 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ4ζ95+ζ4ζ94 ζ43ζ95+ζ43ζ94 ζ4ζ98+ζ4ζ9 ζ43ζ98+ζ43ζ9 ζ4ζ97+ζ4ζ92 ζ43ζ97+ζ43ζ92 complex faithful ρ24 4 -4 0 -2 4i -4i 0 0 0 0 0 2 0 0 0 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 -2i 2i 0 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ43ζ98+ζ43ζ9 ζ4ζ98+ζ4ζ9 ζ43ζ97+ζ43ζ92 ζ4ζ97+ζ4ζ92 ζ43ζ95+ζ43ζ94 ζ4ζ95+ζ4ζ94 complex faithful ρ25 4 -4 0 -2 -4i 4i 0 0 0 0 0 2 0 0 0 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 2i -2i 0 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ4ζ98+ζ4ζ9 ζ43ζ98+ζ43ζ9 ζ4ζ97+ζ4ζ92 ζ43ζ97+ζ43ζ92 ζ4ζ95+ζ4ζ94 ζ43ζ95+ζ43ζ94 complex faithful ρ26 4 -4 0 -2 -4i 4i 0 0 0 0 0 2 0 0 0 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 2i -2i 0 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ4ζ97+ζ4ζ92 ζ43ζ97+ζ43ζ92 ζ4ζ95+ζ4ζ94 ζ43ζ95+ζ43ζ94 ζ4ζ98+ζ4ζ9 ζ43ζ98+ζ43ζ9 complex faithful ρ27 4 -4 0 -2 4i -4i 0 0 0 0 0 2 0 0 0 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 -2i 2i 0 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ43ζ95+ζ43ζ94 ζ4ζ95+ζ4ζ94 ζ43ζ98+ζ43ζ9 ζ4ζ98+ζ4ζ9 ζ43ζ97+ζ43ζ92 ζ4ζ97+ζ4ζ92 complex faithful ρ28 4 -4 0 -2 4i -4i 0 0 0 0 0 2 0 0 0 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 -2i 2i 0 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ43ζ97+ζ43ζ92 ζ4ζ97+ζ4ζ92 ζ43ζ95+ζ43ζ94 ζ4ζ95+ζ4ζ94 ζ43ζ98+ζ43ζ9 ζ4ζ98+ζ4ζ9 complex faithful ρ29 6 6 -2 -3 6 6 -2 0 0 0 0 -3 1 0 0 0 0 0 -3 -3 1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C3.S4 ρ30 6 6 2 -3 -6 -6 -2 0 0 0 0 -3 -1 0 0 0 0 0 3 3 1 0 0 0 0 0 0 0 0 0 symplectic lifted from C6.S4, Schur index 2

Smallest permutation representation of C12.9S4
On 72 points
Generators in S72
```(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)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 72 43 66)(38 61 44 67)(39 62 45 68)(40 63 46 69)(41 64 47 70)(42 65 48 71)(49 58 55 52)(50 59 56 53)(51 60 57 54)
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 50 31 56)(26 51 32 57)(27 52 33 58)(28 53 34 59)(29 54 35 60)(30 55 36 49)(37 63 43 69)(38 64 44 70)(39 65 45 71)(40 66 46 72)(41 67 47 61)(42 68 48 62)
(1 67 33 5 71 25 9 63 29)(2 68 34 6 72 26 10 64 30)(3 69 35 7 61 27 11 65 31)(4 70 36 8 62 28 12 66 32)(13 41 49 17 45 53 21 37 57)(14 42 50 18 46 54 22 38 58)(15 43 51 19 47 55 23 39 59)(16 44 52 20 48 56 24 40 60)
(1 19 10 16 7 13 4 22)(2 24 11 21 8 18 5 15)(3 17 12 14 9 23 6 20)(25 39 34 48 31 45 28 42)(26 44 35 41 32 38 29 47)(27 37 36 46 33 43 30 40)(49 66 58 63 55 72 52 69)(50 71 59 68 56 65 53 62)(51 64 60 61 57 70 54 67)```

`G:=sub<Sym(72)| (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), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,72,43,66)(38,61,44,67)(39,62,45,68)(40,63,46,69)(41,64,47,70)(42,65,48,71)(49,58,55,52)(50,59,56,53)(51,60,57,54), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,50,31,56)(26,51,32,57)(27,52,33,58)(28,53,34,59)(29,54,35,60)(30,55,36,49)(37,63,43,69)(38,64,44,70)(39,65,45,71)(40,66,46,72)(41,67,47,61)(42,68,48,62), (1,67,33,5,71,25,9,63,29)(2,68,34,6,72,26,10,64,30)(3,69,35,7,61,27,11,65,31)(4,70,36,8,62,28,12,66,32)(13,41,49,17,45,53,21,37,57)(14,42,50,18,46,54,22,38,58)(15,43,51,19,47,55,23,39,59)(16,44,52,20,48,56,24,40,60), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,39,34,48,31,45,28,42)(26,44,35,41,32,38,29,47)(27,37,36,46,33,43,30,40)(49,66,58,63,55,72,52,69)(50,71,59,68,56,65,53,62)(51,64,60,61,57,70,54,67)>;`

`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), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,72,43,66)(38,61,44,67)(39,62,45,68)(40,63,46,69)(41,64,47,70)(42,65,48,71)(49,58,55,52)(50,59,56,53)(51,60,57,54), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,50,31,56)(26,51,32,57)(27,52,33,58)(28,53,34,59)(29,54,35,60)(30,55,36,49)(37,63,43,69)(38,64,44,70)(39,65,45,71)(40,66,46,72)(41,67,47,61)(42,68,48,62), (1,67,33,5,71,25,9,63,29)(2,68,34,6,72,26,10,64,30)(3,69,35,7,61,27,11,65,31)(4,70,36,8,62,28,12,66,32)(13,41,49,17,45,53,21,37,57)(14,42,50,18,46,54,22,38,58)(15,43,51,19,47,55,23,39,59)(16,44,52,20,48,56,24,40,60), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,39,34,48,31,45,28,42)(26,44,35,41,32,38,29,47)(27,37,36,46,33,43,30,40)(49,66,58,63,55,72,52,69)(50,71,59,68,56,65,53,62)(51,64,60,61,57,70,54,67) );`

`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)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,72,43,66),(38,61,44,67),(39,62,45,68),(40,63,46,69),(41,64,47,70),(42,65,48,71),(49,58,55,52),(50,59,56,53),(51,60,57,54)], [(1,4,7,10),(2,5,8,11),(3,6,9,12),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,50,31,56),(26,51,32,57),(27,52,33,58),(28,53,34,59),(29,54,35,60),(30,55,36,49),(37,63,43,69),(38,64,44,70),(39,65,45,71),(40,66,46,72),(41,67,47,61),(42,68,48,62)], [(1,67,33,5,71,25,9,63,29),(2,68,34,6,72,26,10,64,30),(3,69,35,7,61,27,11,65,31),(4,70,36,8,62,28,12,66,32),(13,41,49,17,45,53,21,37,57),(14,42,50,18,46,54,22,38,58),(15,43,51,19,47,55,23,39,59),(16,44,52,20,48,56,24,40,60)], [(1,19,10,16,7,13,4,22),(2,24,11,21,8,18,5,15),(3,17,12,14,9,23,6,20),(25,39,34,48,31,45,28,42),(26,44,35,41,32,38,29,47),(27,37,36,46,33,43,30,40),(49,66,58,63,55,72,52,69),(50,71,59,68,56,65,53,62),(51,64,60,61,57,70,54,67)]])`

Matrix representation of C12.9S4 in GL4(𝔽73) generated by

 27 0 0 0 0 27 0 0 0 0 1 2 0 0 35 71
,
 0 27 0 0 27 0 0 0 0 0 1 0 0 0 0 1
,
 27 0 0 0 0 46 0 0 0 0 1 0 0 0 0 1
,
 59 59 0 0 60 13 0 0 0 0 34 6 0 0 32 25
,
 0 27 0 0 1 0 0 0 0 0 1 0 0 0 35 72
`G:=sub<GL(4,GF(73))| [27,0,0,0,0,27,0,0,0,0,1,35,0,0,2,71],[0,27,0,0,27,0,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1],[59,60,0,0,59,13,0,0,0,0,34,32,0,0,6,25],[0,1,0,0,27,0,0,0,0,0,1,35,0,0,0,72] >;`

C12.9S4 in GAP, Magma, Sage, TeX

`C_{12}._9S_4`
`% in TeX`

`G:=Group("C12.9S4");`
`// GroupNames label`

`G:=SmallGroup(288,70);`
`// by ID`

`G=gap.SmallGroup(288,70);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,14,1016,422,142,675,2524,1908,172,1517,1153,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=1,b^2=c^2=a^6,d^3=a^4,e^2=a^9,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^5,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=a^8*d^2>;`
`// generators/relations`

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