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## G = D6.9D12order 288 = 25·32

### 6th non-split extension by D6 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — D6.9D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — D6.9D12
 Lower central C32 — C62 — D6.9D12
 Upper central C1 — C22 — C2×C4

Generators and relations for D6.9D12
G = < a,b,c,d | a6=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=a3c-1 >

Subgroups: 650 in 169 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×7], S3 [×3], C6 [×6], C6 [×6], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C32, Dic3 [×8], C12 [×6], D6 [×2], D6 [×5], C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3 [×3], C3×C6 [×3], C4×S3 [×2], C2×Dic3 [×2], C2×Dic3 [×8], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×3], C22×S3 [×2], C22×C6 [×2], C22.D4, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12, S3×C6 [×2], S3×C6 [×5], C62, Dic3⋊C4, C4⋊Dic3 [×4], D6⋊C4 [×2], D6⋊C4, C6.D4, C3×C22⋊C4 [×2], S3×C2×C4, C22×Dic3, C2×C3⋊D4 [×2], S3×Dic3 [×2], D6⋊S3 [×2], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6 [×2], C23.9D6, C23.21D6, D6⋊Dic3, Dic3⋊Dic3, C3×D6⋊C4 [×2], C12⋊Dic3, C2×S3×Dic3, C2×D6⋊S3, D6.9D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], D12 [×2], C22×S3 [×2], C22.D4, S32, C2×D12, C4○D12, S3×D4, D42S3 [×3], C2×S32, C23.9D6, C23.21D6, D125S3, S3×D12, D6.4D6, D6.9D12

Smallest permutation representation of D6.9D12
On 96 points
Generators in S96
```(1 28 5 32 9 36)(2 29 6 33 10 25)(3 30 7 34 11 26)(4 31 8 35 12 27)(13 95 21 91 17 87)(14 96 22 92 18 88)(15 85 23 93 19 89)(16 86 24 94 20 90)(37 71 41 63 45 67)(38 72 42 64 46 68)(39 61 43 65 47 69)(40 62 44 66 48 70)(49 77 57 73 53 81)(50 78 58 74 54 82)(51 79 59 75 55 83)(52 80 60 76 56 84)
(1 76)(2 53)(3 78)(4 55)(5 80)(6 57)(7 82)(8 59)(9 84)(10 49)(11 74)(12 51)(13 61)(14 48)(15 63)(16 38)(17 65)(18 40)(19 67)(20 42)(21 69)(22 44)(23 71)(24 46)(25 81)(26 58)(27 83)(28 60)(29 73)(30 50)(31 75)(32 52)(33 77)(34 54)(35 79)(36 56)(37 93)(39 95)(41 85)(43 87)(45 89)(47 91)(62 92)(64 94)(66 96)(68 86)(70 88)(72 90)
(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 20)(2 85)(3 18)(4 95)(5 16)(6 93)(7 14)(8 91)(9 24)(10 89)(11 22)(12 87)(13 27)(15 25)(17 35)(19 33)(21 31)(23 29)(26 92)(28 90)(30 88)(32 86)(34 96)(36 94)(37 81)(38 56)(39 79)(40 54)(41 77)(42 52)(43 75)(44 50)(45 73)(46 60)(47 83)(48 58)(49 71)(51 69)(53 67)(55 65)(57 63)(59 61)(62 82)(64 80)(66 78)(68 76)(70 74)(72 84)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A ··· 12H 12I 12J 12K 12L order 1 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 12 2 2 4 4 6 6 12 18 18 36 2 ··· 2 4 4 4 12 12 12 12 4 ··· 4 12 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 D12 C4○D12 S32 S3×D4 D4⋊2S3 C2×S32 D12⋊5S3 S3×D12 D6.4D6 kernel D6.9D12 D6⋊Dic3 Dic3⋊Dic3 C3×D6⋊C4 C12⋊Dic3 C2×S3×Dic3 C2×D6⋊S3 D6⋊C4 S3×C6 C2×Dic3 C2×C12 C22×S3 C3×C6 D6 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 2 1 1 1 2 2 2 2 2 4 4 4 1 1 3 1 2 2 2

Matrix representation of D6.9D12 in GL6(𝔽13)

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

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

D6.9D12 in GAP, Magma, Sage, TeX

`D_6._9D_{12}`
`% in TeX`

`G:=Group("D6.9D12");`
`// GroupNames label`

`G:=SmallGroup(288,539);`
`// by ID`

`G=gap.SmallGroup(288,539);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,142,1356,9414]);`
`// Polycyclic`

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

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