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## G = D12.19D4order 192 = 26·3

### 2nd non-split extension by D12 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.19D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4×D12 — D12.19D4
 Lower central C3 — C6 — C2×C12 — D12.19D4
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for D12.19D4
G = < a,b,c,d | a12=b2=c4=1, d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a6c-1 >

Subgroups: 456 in 124 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C24⋊C2, D24, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, D4.2D4, C2.Dic12, C2.D24, C3×C4⋊C8, C4×D12, C427S3, C2×C24⋊C2, C2×D24, D12.19D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C4○D8, C8⋊C22, C2×D12, S3×D4, Q83S3, D4.2D4, C12⋊D4, C4○D24, C8⋊D6, D12.19D4

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 D12 C4○D8 C4○D24 C8⋊C22 S3×D4 Q8⋊3S3 C8⋊D6 kernel D12.19D4 C2.Dic12 C2.D24 C3×C4⋊C8 C4×D12 C42⋊7S3 C2×C24⋊C2 C2×D24 C4⋊C8 D12 C2×C12 C42 C2×C8 C12 C2×C4 C6 C2 C6 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 2 2 4 4 8 1 1 1 2

Matrix representation of D12.19D4 in GL4(𝔽73) generated by

 66 7 0 0 66 59 0 0 0 0 1 0 0 0 0 1
,
 18 23 0 0 5 55 0 0 0 0 1 0 0 0 0 1
,
 27 0 0 0 0 27 0 0 0 0 0 2 0 0 36 0
,
 18 68 0 0 5 23 0 0 0 0 72 0 0 0 0 1
`G:=sub<GL(4,GF(73))| [66,66,0,0,7,59,0,0,0,0,1,0,0,0,0,1],[18,5,0,0,23,55,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,0,36,0,0,2,0],[18,5,0,0,68,23,0,0,0,0,72,0,0,0,0,1] >;`

D12.19D4 in GAP, Magma, Sage, TeX

`D_{12}._{19}D_4`
`% in TeX`

`G:=Group("D12.19D4");`
`// GroupNames label`

`G:=SmallGroup(192,403);`
`// by ID`

`G=gap.SmallGroup(192,403);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,344,254,219,58,1123,136,6278]);`
`// Polycyclic`

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

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