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## G = C12.19D12order 288 = 25·32

### 19th non-split extension by C12 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12.19D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C2×C12⋊S3 — C12.19D12
 Lower central C32 — C3×C6 — C62 — C12.19D12
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C12.19D12
G = < a,b,c | a12=1, b12=a6, c2=a9, bab-1=a7, cac-1=a5, cbc-1=a3b11 >

Subgroups: 748 in 138 conjugacy classes, 47 normal (17 characteristic)
C1, C2, C2 [×3], C3 [×4], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×4], C6 [×4], C8 [×2], C2×C4, D4 [×2], C23 [×2], C32, C12 [×8], D6 [×16], C2×C6 [×4], M4(2), M4(2), C2×D4, C3⋊S3 [×2], C3×C6, C3×C6, C3⋊C8 [×4], C24 [×4], D12 [×8], C2×C12 [×4], C22×S3 [×8], C4.D4, C3×C12 [×2], C2×C3⋊S3 [×4], C62, C4.Dic3 [×4], C3×M4(2) [×4], C2×D12 [×4], C324C8, C3×C24, C12⋊S3 [×2], C6×C12, C22×C3⋊S3 [×2], C12.46D4 [×4], C12.58D6, C32×M4(2), C2×C12⋊S3, C12.19D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], C4.D4, C2×C3⋊S3, D6⋊C4 [×4], C4×C3⋊S3, C12⋊S3, C327D4, C12.46D4 [×4], C6.11D12, C12.19D12

Smallest permutation representation of C12.19D12
On 72 points
Generators in S72
```(1 49 30 7 55 36 13 61 42 19 67 48)(2 62 31 20 56 25 14 50 43 8 68 37)(3 51 32 9 57 38 15 63 44 21 69 26)(4 64 33 22 58 27 16 52 45 10 70 39)(5 53 34 11 59 40 17 65 46 23 71 28)(6 66 35 24 60 29 18 54 47 12 72 41)
(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 12 19 18 13 24 7 6)(2 5 8 23 14 17 20 11)(3 10 21 16 15 22 9 4)(25 53 31 71 37 65 43 59)(26 58 44 64 38 70 32 52)(27 51 33 69 39 63 45 57)(28 56 46 62 40 68 34 50)(29 49 35 67 41 61 47 55)(30 54 48 60 42 66 36 72)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 8C 8D 12A ··· 12H 12I 12J 12K 12L 24A ··· 24P order 1 2 2 2 2 3 3 3 3 4 4 6 6 6 6 6 6 6 6 8 8 8 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 2 36 36 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 36 36 2 ··· 2 4 4 4 4 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D6 D12 C3⋊D4 C4×S3 C4.D4 C12.46D4 kernel C12.19D12 C12.58D6 C32×M4(2) C2×C12⋊S3 C22×C3⋊S3 C3×M4(2) C3×C12 C2×C12 C12 C12 C2×C6 C32 C3 # reps 1 1 1 1 4 4 2 4 8 8 8 1 8

Matrix representation of C12.19D12 in GL6(𝔽73)

 36 47 0 0 0 0 26 36 0 0 0 0 0 0 14 66 0 0 0 0 7 7 0 0 0 0 56 25 59 7 0 0 22 40 66 66
,
 26 37 0 0 0 0 36 26 0 0 0 0 0 0 31 4 72 2 0 0 22 41 71 1 0 0 18 44 41 59 0 0 65 57 40 33
,
 37 47 0 0 0 0 47 36 0 0 0 0 0 0 18 35 52 0 0 0 6 23 52 21 0 0 64 19 20 35 0 0 69 5 49 12

`G:=sub<GL(6,GF(73))| [36,26,0,0,0,0,47,36,0,0,0,0,0,0,14,7,56,22,0,0,66,7,25,40,0,0,0,0,59,66,0,0,0,0,7,66],[26,36,0,0,0,0,37,26,0,0,0,0,0,0,31,22,18,65,0,0,4,41,44,57,0,0,72,71,41,40,0,0,2,1,59,33],[37,47,0,0,0,0,47,36,0,0,0,0,0,0,18,6,64,69,0,0,35,23,19,5,0,0,52,52,20,49,0,0,0,21,35,12] >;`

C12.19D12 in GAP, Magma, Sage, TeX

`C_{12}._{19}D_{12}`
`% in TeX`

`G:=Group("C12.19D12");`
`// GroupNames label`

`G:=SmallGroup(288,298);`
`// by ID`

`G=gap.SmallGroup(288,298);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,100,346,2693,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;`
`// generators/relations`

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