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## G = C12.12S4order 288 = 25·32

### 12nd non-split extension by C12 of S4 acting via S4/A4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C12.12S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — C12×A4 — C12.12S4
 Lower central C3×A4 — C12.12S4
 Upper central C1 — C4

Generators and relations for C12.12S4
G = < a,b,c,d,e | a12=b2=c2=d3=1, e2=a9, ab=ba, ac=ca, ad=da, eae-1=a5, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >

Subgroups: 248 in 68 conjugacy classes, 25 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C23, C32, C12, C12, A4, C2×C6, C2×C6, C2×C8, C22×C4, C3×C6, C3⋊C8, C2×C12, C2×A4, C22×C6, C22⋊C8, C3×C12, C3×A4, C2×C3⋊C8, C4×A4, C22×C12, C324C8, C6×A4, C12.55D4, A4⋊C8, C12×A4, C12.12S4
Quotients: C1, C2, C4, S3, C8, Dic3, C3⋊S3, C3⋊C8, S4, C3⋊Dic3, A4⋊C4, C324C8, C3⋊S4, A4⋊C8, C6.7S4, C12.12S4

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 8A ··· 8H 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 3 3 3 3 4 4 4 4 6 6 6 6 6 6 8 ··· 8 12 12 12 12 12 ··· 12 size 1 1 3 3 2 8 8 8 1 1 3 3 2 6 6 8 8 8 18 ··· 18 2 2 6 6 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 S3 Dic3 Dic3 C3⋊C8 C3⋊C8 S4 A4⋊C4 A4⋊C8 C3⋊S4 C6.7S4 C12.12S4 kernel C12.12S4 C12×A4 C6×A4 C3×A4 C4×A4 C22×C12 C2×A4 C22×C6 A4 C2×C6 C12 C6 C3 C4 C2 C1 # reps 1 1 2 4 3 1 3 1 6 2 2 2 4 1 1 2

Matrix representation of C12.12S4 in GL5(𝔽73)

 49 0 0 0 0 7 70 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
,
 64 0 0 0 0 30 8 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 19 57 0 0 0 30 54 0 0 0 0 0 0 0 51 0 0 0 51 0 0 0 51 0 0

`G:=sub<GL(5,GF(73))| [49,7,0,0,0,0,70,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,0,0,46],[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],[64,30,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[19,30,0,0,0,57,54,0,0,0,0,0,0,0,51,0,0,0,51,0,0,0,51,0,0] >;`

C12.12S4 in GAP, Magma, Sage, TeX

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

`G:=Group("C12.12S4");`
`// GroupNames label`

`G:=SmallGroup(288,402);`
`// by ID`

`G=gap.SmallGroup(288,402);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,14,36,451,1684,6053,782,3534,1350]);`
`// Polycyclic`

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

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