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## G = C12.S4order 288 = 25·32

### 8th non-split extension by C12 of S4 acting via S4/A4=C2

Aliases: C12.8S4, C23.Dic9, C3.A4⋊C8, C22⋊(C9⋊C8), C3.(A4⋊C8), C6.1(A4⋊C4), C4.4(C3.S4), (C22×C4).1D9, (C22×C12).5S3, C2.1(C6.S4), (C22×C6).2Dic3, (C2×C6).(C3⋊C8), (C2×C3.A4).C4, (C4×C3.A4).2C2, SmallGroup(288,68)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3.A4 — C12.S4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C2×C3.A4 — C4×C3.A4 — C12.S4
 Lower central C3.A4 — C12.S4
 Upper central C1 — C4

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

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 8A ··· 8H 9A 9B 9C 12A 12B 12C 12D 18A 18B 18C 36A ··· 36F order 1 2 2 2 3 4 4 4 4 6 6 6 8 ··· 8 9 9 9 12 12 12 12 18 18 18 36 ··· 36 size 1 1 3 3 2 1 1 3 3 2 6 6 18 ··· 18 8 8 8 2 2 6 6 8 8 8 8 ··· 8

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 3 3 6 6 6 type + + + - + - + + - image C1 C2 C4 C8 S3 Dic3 D9 C3⋊C8 Dic9 C9⋊C8 S4 A4⋊C4 A4⋊C8 C3.S4 C6.S4 C12.S4 kernel C12.S4 C4×C3.A4 C2×C3.A4 C3.A4 C22×C12 C22×C6 C22×C4 C2×C6 C23 C22 C12 C6 C3 C4 C2 C1 # reps 1 1 2 4 1 1 3 2 3 6 2 2 4 1 1 2

Matrix representation of C12.S4 in GL5(𝔽73)

 46 19 0 0 0 4 54 0 0 0 0 0 46 0 0 0 0 0 46 0 0 0 0 0 46
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 72
,
 34 6 0 0 0 32 25 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 47 11 0 0 0 29 26 0 0 0 0 0 0 0 51 0 0 0 51 0 0 0 51 0 0

`G:=sub<GL(5,GF(73))| [46,4,0,0,0,19,54,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,46],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72],[34,32,0,0,0,6,25,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[47,29,0,0,0,11,26,0,0,0,0,0,0,0,51,0,0,0,51,0,0,0,51,0,0] >;`

C12.S4 in GAP, Magma, Sage, TeX

`C_{12}.S_4`
`% in TeX`

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

`G:=SmallGroup(288,68);`
`// by ID`

`G=gap.SmallGroup(288,68);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,14,36,1123,192,1684,6053,782,3534,1350]);`
`// Polycyclic`

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

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