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## G = D12.30D4order 192 = 26·3

### 13rd non-split extension by D12 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D12.30D4
 Chief series C1 — C3 — C6 — C12 — C4×S3 — C4○D12 — Q8.15D6 — D12.30D4
 Lower central C3 — C6 — C12 — D12.30D4
 Upper central C1 — C2 — C2×C4 — C2×Q16

Generators and relations for D12.30D4
G = < a,b,c,d | a12=b2=1, c4=d2=a6, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=a6c3 >

Subgroups: 616 in 248 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, C2×C8, M4(2), D8, SD16, Q16, Q16, C2×Q8, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, D12, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×Q8, C3×Q8, C8○D4, C2×Q16, C2×Q16, C4○D8, C8.C22, 2- 1+4, S3×C8, C8⋊S3, C24⋊C2, D24, Dic12, C4.Dic3, Q82S3, C3⋊Q16, C2×C24, C3×Q16, C4○D12, C4○D12, C4○D12, S3×Q8, S3×Q8, Q83S3, Q83S3, C6×Q8, Q8○D8, C8○D12, C4○D24, S3×Q16, Q16⋊S3, D24⋊C2, Q8.11D6, C6×Q16, Q8.15D6, D12.30D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, S3×D4, S3×C23, Q8○D8, C2×S3×D4, D12.30D4

Smallest permutation representation of D12.30D4
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 38)(2 37)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 84)(14 83)(15 82)(16 81)(17 80)(18 79)(19 78)(20 77)(21 76)(22 75)(23 74)(24 73)(25 89)(26 88)(27 87)(28 86)(29 85)(30 96)(31 95)(32 94)(33 93)(34 92)(35 91)(36 90)(49 66)(50 65)(51 64)(52 63)(53 62)(54 61)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)
(1 71 92 19 7 65 86 13)(2 72 93 20 8 66 87 14)(3 61 94 21 9 67 88 15)(4 62 95 22 10 68 89 16)(5 63 96 23 11 69 90 17)(6 64 85 24 12 70 91 18)(25 81 47 53 31 75 41 59)(26 82 48 54 32 76 42 60)(27 83 37 55 33 77 43 49)(28 84 38 56 34 78 44 50)(29 73 39 57 35 79 45 51)(30 74 40 58 36 80 46 52)
(1 26 7 32)(2 33 8 27)(3 28 9 34)(4 35 10 29)(5 30 11 36)(6 25 12 31)(13 82 19 76)(14 77 20 83)(15 84 21 78)(16 79 22 73)(17 74 23 80)(18 81 24 75)(37 87 43 93)(38 94 44 88)(39 89 45 95)(40 96 46 90)(41 91 47 85)(42 86 48 92)(49 72 55 66)(50 67 56 61)(51 62 57 68)(52 69 58 63)(53 64 59 70)(54 71 60 65)```

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D6 S3×D4 S3×D4 Q8○D8 D12.30D4 kernel D12.30D4 C8○D12 C4○D24 S3×Q16 Q16⋊S3 D24⋊C2 Q8.11D6 C6×Q16 Q8.15D6 C2×Q16 Dic6 D12 C3⋊D4 C2×C8 Q16 C2×Q8 C4 C22 C3 C1 # reps 1 1 1 2 4 2 2 1 2 1 1 1 2 1 4 2 1 1 2 4

Matrix representation of D12.30D4 in GL4(𝔽7) generated by

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

D12.30D4 in GAP, Magma, Sage, TeX

`D_{12}._{30}D_4`
`% in TeX`

`G:=Group("D12.30D4");`
`// GroupNames label`

`G:=SmallGroup(192,1325);`
`// by ID`

`G=gap.SmallGroup(192,1325);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,184,185,136,438,235,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^4=d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=a^6*c^3>;`
`// generators/relations`

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