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## G = D4.2D12order 192 = 26·3

### 2nd non-split extension by D4 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D4.2D12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12⋊2Q8 — D4.2D12
 Lower central C3 — C6 — C2×C12 — D4.2D12
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for D4.2D12
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=c-1 >

Subgroups: 312 in 120 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×4], C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8 [×2], C3⋊C8 [×2], Dic6 [×4], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×3], C3×D4 [×2], C3×D4, C22×C6, Q8⋊C4 [×2], C4⋊C8, C4×D4, C4⋊Q8, C2×SD16 [×2], C2×C3⋊C8 [×2], C4⋊Dic3 [×2], D4.S3 [×4], C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6 [×2], C22×C12, C6×D4, D4.D4, C12⋊C8, C6.SD16 [×2], C122Q8, C2×D4.S3 [×2], D4×C12, D4.2D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], SD16 [×2], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C2×SD16, C8.C22, D4.S3 [×2], C2×D12, C4○D12, C2×C3⋊D4, D4.D4, C127D4, C2×D4.S3, Q8.14D6, D4.2D12

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

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12L order 1 2 2 2 2 2 3 4 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 2 2 2 2 2 4 4 4 24 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 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 SD16 C4○D4 C3⋊D4 D12 C4○D12 C8.C22 D4.S3 Q8.14D6 kernel D4.2D12 C12⋊C8 C6.SD16 C12⋊2Q8 C2×D4.S3 D4×C12 C4×D4 C2×C12 C3×D4 C42 C4⋊C4 C2×D4 C12 C12 C2×C4 D4 C4 C6 C4 C2 # reps 1 1 2 1 2 1 1 2 2 1 1 1 4 2 4 4 4 1 2 2

Matrix representation of D4.2D12 in GL6(𝔽73)

 0 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 0 1 0 0 0 0 1 0 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 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 65 0 0 0 0 0 2 9 0 0 0 0 0 0 46 0 0 0 0 0 0 27
,
 67 6 0 0 0 0 6 6 0 0 0 0 0 0 70 11 0 0 0 0 59 3 0 0 0 0 0 0 0 27 0 0 0 0 46 0

`G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,65,2,0,0,0,0,0,9,0,0,0,0,0,0,46,0,0,0,0,0,0,27],[67,6,0,0,0,0,6,6,0,0,0,0,0,0,70,59,0,0,0,0,11,3,0,0,0,0,0,0,0,46,0,0,0,0,27,0] >;`

D4.2D12 in GAP, Magma, Sage, TeX

`D_4._2D_{12}`
`% in TeX`

`G:=Group("D4.2D12");`
`// GroupNames label`

`G:=SmallGroup(192,578);`
`// by ID`

`G=gap.SmallGroup(192,578);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,1123,297,136,6278]);`
`// 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=c^-1>;`
`// generators/relations`

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