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## G = C12.81D12order 288 = 25·32

### 12nd non-split extension by C12 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12.81D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — Dic3×C12 — C12.81D12
 Lower central C32 — C3×C6 — C12.81D12
 Upper central C1 — C2×C4

Generators and relations for C12.81D12
G = < a,b,c | a12=b12=1, c2=a9, bab-1=cac-1=a5, cbc-1=b-1 >

Subgroups: 210 in 87 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×3], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×2], Dic3, C12 [×4], C12 [×5], C2×C6 [×2], C2×C6, C42, C2×C8 [×2], C3×C6 [×3], C3⋊C8 [×5], C24, C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C4⋊C8, C3×Dic3 [×2], C3×Dic3, C3×C12 [×2], C62, C2×C3⋊C8, C2×C3⋊C8 [×3], C4×Dic3, C4×C12, C2×C24, C3×C3⋊C8, C324C8, C6×Dic3 [×2], C6×C12, C12⋊C8, Dic3⋊C8, C6×C3⋊C8, Dic3×C12, C2×C324C8, C12.81D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, D4, Q8, Dic3 [×2], D6 [×2], C4⋊C4, C2×C8, M4(2), C3⋊C8 [×2], Dic6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4, C4⋊C8, S32, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C3⋊D12, C322Q8, C12⋊C8, Dic3⋊C8, S3×C3⋊C8, D6.Dic3, Dic3⋊Dic3, C12.81D12

Smallest permutation representation of C12.81D12
On 96 points
Generators in S96
```(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)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 42 91 74 5 38 95 82 9 46 87 78)(2 47 92 79 6 43 96 75 10 39 88 83)(3 40 93 84 7 48 85 80 11 44 89 76)(4 45 94 77 8 41 86 73 12 37 90 81)(13 72 36 49 17 68 28 57 21 64 32 53)(14 65 25 54 18 61 29 50 22 69 33 58)(15 70 26 59 19 66 30 55 23 62 34 51)(16 63 27 52 20 71 31 60 24 67 35 56)
(1 19 10 16 7 13 4 22)(2 24 11 21 8 18 5 15)(3 17 12 14 9 23 6 20)(25 95 34 92 31 89 28 86)(26 88 35 85 32 94 29 91)(27 93 36 90 33 87 30 96)(37 58 46 55 43 52 40 49)(38 51 47 60 44 57 41 54)(39 56 48 53 45 50 42 59)(61 74 70 83 67 80 64 77)(62 79 71 76 68 73 65 82)(63 84 72 81 69 78 66 75)```

`G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,42,91,74,5,38,95,82,9,46,87,78)(2,47,92,79,6,43,96,75,10,39,88,83)(3,40,93,84,7,48,85,80,11,44,89,76)(4,45,94,77,8,41,86,73,12,37,90,81)(13,72,36,49,17,68,28,57,21,64,32,53)(14,65,25,54,18,61,29,50,22,69,33,58)(15,70,26,59,19,66,30,55,23,62,34,51)(16,63,27,52,20,71,31,60,24,67,35,56), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,95,34,92,31,89,28,86)(26,88,35,85,32,94,29,91)(27,93,36,90,33,87,30,96)(37,58,46,55,43,52,40,49)(38,51,47,60,44,57,41,54)(39,56,48,53,45,50,42,59)(61,74,70,83,67,80,64,77)(62,79,71,76,68,73,65,82)(63,84,72,81,69,78,66,75)>;`

`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)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,42,91,74,5,38,95,82,9,46,87,78)(2,47,92,79,6,43,96,75,10,39,88,83)(3,40,93,84,7,48,85,80,11,44,89,76)(4,45,94,77,8,41,86,73,12,37,90,81)(13,72,36,49,17,68,28,57,21,64,32,53)(14,65,25,54,18,61,29,50,22,69,33,58)(15,70,26,59,19,66,30,55,23,62,34,51)(16,63,27,52,20,71,31,60,24,67,35,56), (1,19,10,16,7,13,4,22)(2,24,11,21,8,18,5,15)(3,17,12,14,9,23,6,20)(25,95,34,92,31,89,28,86)(26,88,35,85,32,94,29,91)(27,93,36,90,33,87,30,96)(37,58,46,55,43,52,40,49)(38,51,47,60,44,57,41,54)(39,56,48,53,45,50,42,59)(61,74,70,83,67,80,64,77)(62,79,71,76,68,73,65,82)(63,84,72,81,69,78,66,75) );`

`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),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,42,91,74,5,38,95,82,9,46,87,78),(2,47,92,79,6,43,96,75,10,39,88,83),(3,40,93,84,7,48,85,80,11,44,89,76),(4,45,94,77,8,41,86,73,12,37,90,81),(13,72,36,49,17,68,28,57,21,64,32,53),(14,65,25,54,18,61,29,50,22,69,33,58),(15,70,26,59,19,66,30,55,23,62,34,51),(16,63,27,52,20,71,31,60,24,67,35,56)], [(1,19,10,16,7,13,4,22),(2,24,11,21,8,18,5,15),(3,17,12,14,9,23,6,20),(25,95,34,92,31,89,28,86),(26,88,35,85,32,94,29,91),(27,93,36,90,33,87,30,96),(37,58,46,55,43,52,40,49),(38,51,47,60,44,57,41,54),(39,56,48,53,45,50,42,59),(61,74,70,83,67,80,64,77),(62,79,71,76,68,73,65,82),(63,84,72,81,69,78,66,75)])`

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G 6H 6I 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12H 12I 12J 12K 12L 12M ··· 12T 24A ··· 24H order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 8 8 8 8 8 8 8 8 12 ··· 12 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 4 1 1 1 1 6 6 6 6 2 ··· 2 4 4 4 6 6 6 6 18 18 18 18 2 ··· 2 4 4 4 4 6 ··· 6 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + - - + - + + + - - image C1 C2 C2 C2 C4 C8 S3 S3 D4 Q8 Dic3 D6 M4(2) C3⋊C8 Dic6 D12 C3⋊D4 C4×S3 S3×C8 C8⋊S3 C4.Dic3 S32 C3⋊D12 C32⋊2Q8 S3×Dic3 S3×C3⋊C8 D6.Dic3 kernel C12.81D12 C6×C3⋊C8 Dic3×C12 C2×C32⋊4C8 C6×Dic3 C3×Dic3 C2×C3⋊C8 C4×Dic3 C3×C12 C3×C12 C2×Dic3 C2×C12 C3×C6 Dic3 C12 C12 C12 C2×C6 C6 C6 C6 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 8 1 1 1 1 2 2 2 4 4 2 2 2 4 4 4 1 1 1 1 2 2

Matrix representation of C12.81D12 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 46 0 0 0 0 27 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 22 48 0 0 0 0 34 51 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 22 18 0 0 0 0 34 51 0 0 0 0 0 0 63 0 0 0 0 0 10 10 0 0 0 0 0 0 72 0 0 0 0 0 72 1

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,27,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,34,0,0,0,0,48,51,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[22,34,0,0,0,0,18,51,0,0,0,0,0,0,63,10,0,0,0,0,0,10,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;`

C12.81D12 in GAP, Magma, Sage, TeX

`C_{12}._{81}D_{12}`
`% in TeX`

`G:=Group("C12.81D12");`
`// GroupNames label`

`G:=SmallGroup(288,219);`
`// by ID`

`G=gap.SmallGroup(288,219);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,1356,9414]);`
`// Polycyclic`

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

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