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## G = C12.27D12order 288 = 25·32

### 27th non-split extension by C12 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12.27D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — C12.27D12
 Lower central C32 — C62 — C12.27D12
 Upper central C1 — C22 — C2×C4

Generators and relations for C12.27D12
G = < a,b,c | a12=b12=1, c2=a6, bab-1=a5, cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 642 in 171 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6 [×2], C6 [×4], C6 [×5], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×10], C12 [×4], C12 [×4], D6 [×6], C2×C6 [×2], C2×C6 [×7], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×C6, C3×C6 [×2], Dic6 [×8], D12 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C22×S3 [×2], C22×C6 [×2], C4.4D4, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×6], C62, C4×Dic3, D6⋊C4 [×4], C6.D4 [×4], C4×C12, C2×Dic6 [×3], C2×D12, C6×D4, C3×D12 [×2], C6×Dic3 [×2], C324Q8 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6 [×2], C427S3, C23.12D6, D6⋊Dic3 [×4], Dic3×C12, C6×D12, C2×C324Q8, C12.27D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4.4D4, S32, C2×D12, C4○D12 [×2], D42S3 [×2], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C427S3, C23.12D6, D125S3 [×2], C2×C3⋊D12, C12.27D12

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 C4○D4 D12 C3⋊D4 C4○D12 S32 D4⋊2S3 C3⋊D12 C2×S32 D12⋊5S3 kernel C12.27D12 D6⋊Dic3 Dic3×C12 C6×D12 C2×C32⋊4Q8 C4×Dic3 C2×D12 C3×C12 C2×Dic3 C2×C12 C22×S3 C3×C6 C12 C12 C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 1 1 2 2 2 2 4 4 4 8 1 2 2 1 4

Matrix representation of C12.27D12 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 3 6 0 0 0 0 7 10
,
 5 2 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 9 11 0 0 0 0 2 11
,
 9 9 0 0 0 0 7 4 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 2 4 0 0 0 0 2 11

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,3,7,0,0,0,0,6,10],[5,0,0,0,0,0,2,8,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,9,2,0,0,0,0,11,11],[9,7,0,0,0,0,9,4,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,2,2,0,0,0,0,4,11] >;`

C12.27D12 in GAP, Magma, Sage, TeX

`C_{12}._{27}D_{12}`
`% in TeX`

`G:=Group("C12.27D12");`
`// GroupNames label`

`G:=SmallGroup(288,508);`
`// by ID`

`G=gap.SmallGroup(288,508);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,422,100,1356,9414]);`
`// Polycyclic`

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

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