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## G = D4.1D12order 192 = 26·3

### 1st non-split extension by D4 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D4.1D12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C42⋊7S3 — D4.1D12
 Lower central C3 — C6 — C2×C12 — D4.1D12
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for D4.1D12
G = < a,b,c,d | a4=b2=c12=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 376 in 124 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×7], S3, C6 [×3], C6 [×2], C8 [×2], C2×C4 [×3], C2×C4 [×4], D4 [×2], D4 [×3], Q8 [×2], C23 [×2], Dic3, C12 [×2], C12 [×3], D6 [×3], C2×C6, C2×C6 [×4], C42, C22⋊C4 [×3], C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, C3⋊C8 [×2], Dic6 [×2], D12 [×2], C2×Dic3, C2×C12 [×3], C2×C12 [×3], C3×D4 [×2], C3×D4, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C2×C3⋊C8 [×2], D6⋊C4 [×2], D4⋊S3 [×2], D4.S3 [×2], C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×D12, C22×C12, C6×D4, D4.2D4, C12⋊C8, C6.D8, C6.SD16, C427S3, C2×D4⋊S3, C2×D4.S3, D4×C12, D4.1D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C4○D8, C8⋊C22, C2×D12, C4○D12, C2×C3⋊D4, D4.2D4, C127D4, D126C22, Q8.13D6, D4.1D12

Smallest permutation representation of D4.1D12
On 96 points
Generators in S96
```(1 13 70 32)(2 14 71 33)(3 15 72 34)(4 16 61 35)(5 17 62 36)(6 18 63 25)(7 19 64 26)(8 20 65 27)(9 21 66 28)(10 22 67 29)(11 23 68 30)(12 24 69 31)(37 74 55 90)(38 75 56 91)(39 76 57 92)(40 77 58 93)(41 78 59 94)(42 79 60 95)(43 80 49 96)(44 81 50 85)(45 82 51 86)(46 83 52 87)(47 84 53 88)(48 73 54 89)
(1 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 85)(24 86)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 73)(35 74)(36 75)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)
(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 69 70 12)(2 11 71 68)(3 67 72 10)(4 9 61 66)(5 65 62 8)(6 7 63 64)(13 24 32 31)(14 30 33 23)(15 22 34 29)(16 28 35 21)(17 20 36 27)(18 26 25 19)(37 95 55 79)(38 78 56 94)(39 93 57 77)(40 76 58 92)(41 91 59 75)(42 74 60 90)(43 89 49 73)(44 84 50 88)(45 87 51 83)(46 82 52 86)(47 85 53 81)(48 80 54 96)```

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

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

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

39 conjugacy classes

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

39 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 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 D12 C4○D8 C4○D12 C8⋊C22 D12⋊6C22 Q8.13D6 kernel D4.1D12 C12⋊C8 C6.D8 C6.SD16 C42⋊7S3 C2×D4⋊S3 C2×D4.S3 D4×C12 C4×D4 C2×C12 C3×D4 C42 C4⋊C4 C2×D4 C12 C2×C4 D4 C6 C4 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 2 4 4 4 4 1 2 2

Matrix representation of D4.1D12 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 71 0 0 1 1
,
 30 13 0 0 60 43 0 0 0 0 32 32 0 0 57 41
,
 59 7 0 0 66 66 0 0 0 0 46 0 0 0 0 46
,
 66 66 0 0 59 7 0 0 0 0 46 0 0 0 27 27
`G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,1,0,0,71,1],[30,60,0,0,13,43,0,0,0,0,32,57,0,0,32,41],[59,66,0,0,7,66,0,0,0,0,46,0,0,0,0,46],[66,59,0,0,66,7,0,0,0,0,46,27,0,0,0,27] >;`

D4.1D12 in GAP, Magma, Sage, TeX

`D_4._1D_{12}`
`% in TeX`

`G:=Group("D4.1D12");`
`// GroupNames label`

`G:=SmallGroup(192,575);`
`// by ID`

`G=gap.SmallGroup(192,575);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,1123,297,136,6278]);`
`// Polycyclic`

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

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