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## G = C12.26D6order 144 = 24·32

### 26th non-split extension by C12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12.26D6
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C4×C3⋊S3 — C12.26D6
 Lower central C32 — C3×C6 — C12.26D6
 Upper central C1 — C2 — Q8

Generators and relations for C12.26D6
G = < a,b,c | a12=1, b6=c2=a6, bab-1=a7, cac-1=a5, cbc-1=b5 >

Subgroups: 410 in 120 conjugacy classes, 47 normal (8 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C2×C4, D4, Q8, C32, Dic3, C12, D6, C4○D4, C3⋊S3, C3×C6, C4×S3, D12, C3×Q8, C3⋊Dic3, C3×C12, C2×C3⋊S3, Q83S3, C4×C3⋊S3, C12⋊S3, Q8×C32, C12.26D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C3⋊S3, C22×S3, C2×C3⋊S3, Q83S3, C22×C3⋊S3, C12.26D6

Character table of C12.26D6

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 4A 4B 4C 4D 4E 6A 6B 6C 6D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L size 1 1 18 18 18 2 2 2 2 2 2 2 9 9 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 -1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ6 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ7 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 -1 -1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ9 2 2 0 0 0 -1 -1 2 -1 -2 2 -2 0 0 -1 -1 -1 2 2 -1 1 1 1 -2 1 -2 1 1 -1 -1 orthogonal lifted from D6 ρ10 2 2 0 0 0 -1 2 -1 -1 2 2 2 0 0 -1 -1 2 -1 -1 -1 2 -1 2 -1 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ11 2 2 0 0 0 -1 -1 -1 2 2 -2 -2 0 0 2 -1 -1 -1 1 -2 -1 1 1 1 -2 -1 2 -1 1 1 orthogonal lifted from D6 ρ12 2 2 0 0 0 2 -1 -1 -1 2 2 2 0 0 -1 2 -1 -1 -1 -1 -1 2 -1 -1 -1 -1 -1 2 2 -1 orthogonal lifted from S3 ρ13 2 2 0 0 0 2 -1 -1 -1 2 -2 -2 0 0 -1 2 -1 -1 1 1 -1 -2 1 1 1 -1 -1 2 -2 1 orthogonal lifted from D6 ρ14 2 2 0 0 0 -1 -1 2 -1 -2 -2 2 0 0 -1 -1 -1 2 -2 1 1 -1 -1 2 -1 -2 1 1 1 1 orthogonal lifted from D6 ρ15 2 2 0 0 0 -1 2 -1 -1 -2 2 -2 0 0 -1 -1 2 -1 -1 -1 -2 1 -2 1 1 1 1 1 -1 2 orthogonal lifted from D6 ρ16 2 2 0 0 0 -1 -1 -1 2 2 2 2 0 0 2 -1 -1 -1 -1 2 -1 -1 -1 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ17 2 2 0 0 0 -1 -1 2 -1 2 -2 -2 0 0 -1 -1 -1 2 -2 1 -1 1 1 -2 1 2 -1 -1 1 1 orthogonal lifted from D6 ρ18 2 2 0 0 0 -1 -1 -1 2 -2 2 -2 0 0 2 -1 -1 -1 -1 2 1 1 1 1 -2 1 -2 1 -1 -1 orthogonal lifted from D6 ρ19 2 2 0 0 0 -1 2 -1 -1 -2 -2 2 0 0 -1 -1 2 -1 1 1 -2 -1 2 -1 -1 1 1 1 1 -2 orthogonal lifted from D6 ρ20 2 2 0 0 0 2 -1 -1 -1 -2 -2 2 0 0 -1 2 -1 -1 1 1 1 2 -1 -1 -1 1 1 -2 -2 1 orthogonal lifted from D6 ρ21 2 2 0 0 0 -1 -1 2 -1 2 2 2 0 0 -1 -1 -1 2 2 -1 -1 -1 -1 2 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ22 2 2 0 0 0 2 -1 -1 -1 -2 2 -2 0 0 -1 2 -1 -1 -1 -1 1 -2 1 1 1 1 1 -2 2 -1 orthogonal lifted from D6 ρ23 2 2 0 0 0 -1 -1 -1 2 -2 -2 2 0 0 2 -1 -1 -1 1 -2 1 -1 -1 -1 2 1 -2 1 1 1 orthogonal lifted from D6 ρ24 2 2 0 0 0 -1 2 -1 -1 2 -2 -2 0 0 -1 -1 2 -1 1 1 2 1 -2 1 1 -1 -1 -1 1 -2 orthogonal lifted from D6 ρ25 2 -2 0 0 0 2 2 2 2 0 0 0 -2i 2i -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 -2 0 0 0 2 2 2 2 0 0 0 2i -2i -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 4 -4 0 0 0 -2 4 -2 -2 0 0 0 0 0 2 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ28 4 -4 0 0 0 4 -2 -2 -2 0 0 0 0 0 2 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ29 4 -4 0 0 0 -2 -2 4 -2 0 0 0 0 0 2 2 2 -4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ30 4 -4 0 0 0 -2 -2 -2 4 0 0 0 0 0 -4 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2

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

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

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

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

Matrix representation of C12.26D6 in GL6(𝔽13)

 0 12 0 0 0 0 1 1 0 0 0 0 0 0 0 12 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 1 1 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 1 1 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 1 1 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5

`G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;`

C12.26D6 in GAP, Magma, Sage, TeX

`C_{12}._{26}D_6`
`% in TeX`

`G:=Group("C12.26D6");`
`// GroupNames label`

`G:=SmallGroup(144,175);`
`// by ID`

`G=gap.SmallGroup(144,175);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,116,50,964,3461]);`
`// Polycyclic`

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

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