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## G = D12.24D6order 288 = 25·32

### 9th non-split extension by D12 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.24D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D12⋊5S3 — D12.24D6
 Lower central C32 — C3×C6 — C3×C12 — D12.24D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D12.24D6
G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, cac-1=dad-1=a7, cbc-1=a3b, dbd-1=a9b, dcd-1=c5 >

Subgroups: 466 in 129 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×Q8, C8.C22, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C8⋊S3, Dic12, C4.Dic3, D4.S3, Q82S3, Q82S3, C3⋊Q16, C3×SD16, C4○D12, D42S3, S3×Q8, C6×Q8, C3×C3⋊C8, C324C8, S3×Dic3, D6⋊S3, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C3×D12, C324Q8, Q8×C32, D4.D6, Q8.11D6, D6.Dic3, Dic6⋊S3, C323Q16, C3×Q82S3, C327Q16, D125S3, C3×S3×Q8, D12.24D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8.C22, S32, S3×D4, C2×C3⋊D4, C2×S32, D4.D6, Q8.11D6, S3×C3⋊D4, D12.24D6

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

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 24A 24B order 1 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 6 8 8 12 12 12 12 12 12 12 12 12 12 12 24 24 size 1 1 6 12 2 2 4 2 4 6 12 36 2 2 4 6 6 24 12 36 4 4 4 4 8 8 8 8 12 12 12 12 12

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D6 C3⋊D4 C3⋊D4 C8.C22 S32 S3×D4 C2×S32 D4.D6 Q8.11D6 S3×C3⋊D4 D12.24D6 kernel D12.24D6 D6.Dic3 Dic6⋊S3 C32⋊3Q16 C3×Q8⋊2S3 C32⋊7Q16 D12⋊5S3 C3×S3×Q8 Q8⋊2S3 S3×Q8 C3×Dic3 S3×C6 C3⋊C8 Dic6 C4×S3 D12 C3×Q8 Dic3 D6 C32 Q8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 2 2 2 1

Matrix representation of D12.24D6 in GL8(𝔽73)

 0 72 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 13 29 72 63 0 0 0 0 25 13 44 1
,
 0 0 1 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 19 67 60 8 0 0 0 0 36 35 0 8 0 0 0 0 30 66 17 71 0 0 0 0 49 20 22 2
,
 43 30 0 0 0 0 0 0 43 13 0 0 0 0 0 0 0 0 30 43 0 0 0 0 0 0 30 60 0 0 0 0 0 0 0 0 1 0 66 0 0 0 0 0 18 56 66 3 0 0 0 0 42 0 72 0 0 0 0 0 48 25 0 17
,
 0 0 34 26 0 0 0 0 0 0 65 39 0 0 0 0 39 47 0 0 0 0 0 0 8 34 0 0 0 0 0 0 0 0 0 0 38 57 20 54 0 0 0 0 53 67 53 0 0 0 0 0 41 58 59 19 0 0 0 0 40 61 69 55

`G:=sub<GL(8,GF(73))| [0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,13,25,0,0,0,0,1,0,29,13,0,0,0,0,0,0,72,44,0,0,0,0,0,0,63,1],[0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,19,36,30,49,0,0,0,0,67,35,66,20,0,0,0,0,60,0,17,22,0,0,0,0,8,8,71,2],[43,43,0,0,0,0,0,0,30,13,0,0,0,0,0,0,0,0,30,30,0,0,0,0,0,0,43,60,0,0,0,0,0,0,0,0,1,18,42,48,0,0,0,0,0,56,0,25,0,0,0,0,66,66,72,0,0,0,0,0,0,3,0,17],[0,0,39,8,0,0,0,0,0,0,47,34,0,0,0,0,34,65,0,0,0,0,0,0,26,39,0,0,0,0,0,0,0,0,0,0,38,53,41,40,0,0,0,0,57,67,58,61,0,0,0,0,20,53,59,69,0,0,0,0,54,0,19,55] >;`

D12.24D6 in GAP, Magma, Sage, TeX

`D_{12}._{24}D_6`
`% in TeX`

`G:=Group("D12.24D6");`
`// GroupNames label`

`G:=SmallGroup(288,594);`
`// by ID`

`G=gap.SmallGroup(288,594);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,135,100,346,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^7,c*b*c^-1=a^3*b,d*b*d^-1=a^9*b,d*c*d^-1=c^5>;`
`// generators/relations`

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