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## G = C12.59D6order 144 = 24·32

### 20th non-split extension by C12 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12.59D6
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C12.59D6
 Lower central C32 — C3×C6 — C12.59D6
 Upper central C1 — C4 — C2×C4

Generators and relations for C12.59D6
G = < a,b,c | a12=b6=1, c2=a6, ab=ba, cac-1=a5, cbc-1=a6b-1 >

Subgroups: 378 in 120 conjugacy classes, 47 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C4○D4, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C4○D12, C324Q8, C4×C3⋊S3, C12⋊S3, C327D4, C6×C12, C12.59D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, C2×C3⋊S3, C4○D12, C22×C3⋊S3, C12.59D6

Smallest permutation representation of C12.59D6
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 29 58 48 68 18)(2 30 59 37 69 19)(3 31 60 38 70 20)(4 32 49 39 71 21)(5 33 50 40 72 22)(6 34 51 41 61 23)(7 35 52 42 62 24)(8 36 53 43 63 13)(9 25 54 44 64 14)(10 26 55 45 65 15)(11 27 56 46 66 16)(12 28 57 47 67 17)
(1 49 7 55)(2 54 8 60)(3 59 9 53)(4 52 10 58)(5 57 11 51)(6 50 12 56)(13 44 19 38)(14 37 20 43)(15 42 21 48)(16 47 22 41)(17 40 23 46)(18 45 24 39)(25 30 31 36)(26 35 32 29)(27 28 33 34)(61 72 67 66)(62 65 68 71)(63 70 69 64)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 4A 4B 4C 4D 4E 6A ··· 6L 12A ··· 12P order 1 2 2 2 2 3 3 3 3 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 2 18 18 2 2 2 2 1 1 2 18 18 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 C4○D12 kernel C12.59D6 C32⋊4Q8 C4×C3⋊S3 C12⋊S3 C32⋊7D4 C6×C12 C2×C12 C12 C2×C6 C32 C3 # reps 1 1 2 1 2 1 4 8 4 2 16

Matrix representation of C12.59D6 in GL4(𝔽13) generated by

 0 1 0 0 12 12 0 0 0 0 8 0 0 0 0 8
,
 12 0 0 0 0 12 0 0 0 0 4 11 0 0 2 2
,
 1 0 0 0 12 12 0 0 0 0 5 8 0 0 0 8
`G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,8,0,0,0,0,8],[12,0,0,0,0,12,0,0,0,0,4,2,0,0,11,2],[1,12,0,0,0,12,0,0,0,0,5,0,0,0,8,8] >;`

C12.59D6 in GAP, Magma, Sage, TeX

`C_{12}._{59}D_6`
`% in TeX`

`G:=Group("C12.59D6");`
`// GroupNames label`

`G:=SmallGroup(144,171);`
`// by ID`

`G=gap.SmallGroup(144,171);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,964,3461]);`
`// Polycyclic`

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

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