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## G = C6.11D12order 144 = 24·32

### 11st non-split extension by C6 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C6.11D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C6.11D12
 Lower central C32 — C3×C6 — C6.11D12
 Upper central C1 — C22 — C2×C4

Generators and relations for C6.11D12
G = < a,b,c | a6=b12=1, c2=a3, ab=ba, cac-1=a-1, cbc-1=a3b-1 >

Subgroups: 378 in 102 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C3⋊S3, C3×C6, C2×Dic3, C2×C12, C22×S3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, D6⋊C4, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C6.11D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C3⋊S3, C4×S3, D12, C3⋊D4, C2×C3⋊S3, D6⋊C4, C4×C3⋊S3, C12⋊S3, C327D4, C6.11D12

Smallest permutation representation of C6.11D12
On 72 points
Generators in S72
```(1 19 67 53 40 25)(2 20 68 54 41 26)(3 21 69 55 42 27)(4 22 70 56 43 28)(5 23 71 57 44 29)(6 24 72 58 45 30)(7 13 61 59 46 31)(8 14 62 60 47 32)(9 15 63 49 48 33)(10 16 64 50 37 34)(11 17 65 51 38 35)(12 18 66 52 39 36)
(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 52 53 12)(2 11 54 51)(3 50 55 10)(4 9 56 49)(5 60 57 8)(6 7 58 59)(13 72 46 30)(14 29 47 71)(15 70 48 28)(16 27 37 69)(17 68 38 26)(18 25 39 67)(19 66 40 36)(20 35 41 65)(21 64 42 34)(22 33 43 63)(23 62 44 32)(24 31 45 61)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C4 S3 D4 D6 C4×S3 D12 C3⋊D4 kernel C6.11D12 C2×C3⋊Dic3 C6×C12 C22×C3⋊S3 C2×C3⋊S3 C2×C12 C3×C6 C2×C6 C6 C6 C6 # reps 1 1 1 1 4 4 2 4 8 8 8

Matrix representation of C6.11D12 in GL4(𝔽13) generated by

 0 1 0 0 12 1 0 0 0 0 12 0 0 0 0 12
,
 2 9 0 0 4 11 0 0 0 0 0 8 0 0 5 8
,
 4 11 0 0 2 9 0 0 0 0 5 8 0 0 0 8
`G:=sub<GL(4,GF(13))| [0,12,0,0,1,1,0,0,0,0,12,0,0,0,0,12],[2,4,0,0,9,11,0,0,0,0,0,5,0,0,8,8],[4,2,0,0,11,9,0,0,0,0,5,0,0,0,8,8] >;`

C6.11D12 in GAP, Magma, Sage, TeX

`C_6._{11}D_{12}`
`% in TeX`

`G:=Group("C6.11D12");`
`// GroupNames label`

`G:=SmallGroup(144,95);`
`// by ID`

`G=gap.SmallGroup(144,95);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,964,3461]);`
`// Polycyclic`

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

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