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G = D12.4Q8order 192 = 26·3

2nd non-split extension by D12 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.4Q8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4×D12 — D12.4Q8
 Lower central C3 — C6 — C2×C12 — D12.4Q8
 Upper central C1 — C22 — C42 — C42.C2

Generators and relations for D12.4Q8
G = < a,b,c,d | a12=b2=c4=1, d2=a6c2, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 304 in 102 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], S3 [×2], C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×3], C23, Dic3, C12 [×2], C12 [×4], D6 [×4], C2×C6, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×D4, C3⋊C8 [×2], C4×S3 [×2], D12 [×2], D12, C2×Dic3, C2×C12 [×3], C2×C12 [×2], C22×S3, D4⋊C4 [×2], C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], S3×C2×C4, C2×D12, D4.Q8, C12⋊C8, C6.Q16, C12.Q8, C6.D8 [×2], C4×D12, C3×C42.C2, D12.4Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, C3⋊D4 [×2], C22×S3, C22⋊Q8, C4○D8, C8⋊C22, S3×Q8, Q83S3, C2×C3⋊D4, D4.Q8, D63Q8, D4⋊D6, Q8.13D6, D12.4Q8

Smallest permutation representation of D12.4Q8
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 65)(2 64)(3 63)(4 62)(5 61)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 48)(25 83)(26 82)(27 81)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 84)(49 85)(50 96)(51 95)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)
(1 27 72 79)(2 34 61 74)(3 29 62 81)(4 36 63 76)(5 31 64 83)(6 26 65 78)(7 33 66 73)(8 28 67 80)(9 35 68 75)(10 30 69 82)(11 25 70 77)(12 32 71 84)(13 52 39 95)(14 59 40 90)(15 54 41 85)(16 49 42 92)(17 56 43 87)(18 51 44 94)(19 58 45 89)(20 53 46 96)(21 60 47 91)(22 55 48 86)(23 50 37 93)(24 57 38 88)
(1 92 66 55)(2 93 67 56)(3 94 68 57)(4 95 69 58)(5 96 70 59)(6 85 71 60)(7 86 72 49)(8 87 61 50)(9 88 62 51)(10 89 63 52)(11 90 64 53)(12 91 65 54)(13 30 45 76)(14 31 46 77)(15 32 47 78)(16 33 48 79)(17 34 37 80)(18 35 38 81)(19 36 39 82)(20 25 40 83)(21 26 41 84)(22 27 42 73)(23 28 43 74)(24 29 44 75)```

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + - + + + + - + + image C1 C2 C2 C2 C2 C2 C2 S3 Q8 D4 D6 D6 C4○D4 C3⋊D4 C4○D8 C8⋊C22 S3×Q8 Q8⋊3S3 D4⋊D6 Q8.13D6 kernel D12.4Q8 C12⋊C8 C6.Q16 C12.Q8 C6.D8 C4×D12 C3×C42.C2 C42.C2 D12 C2×C12 C42 C4⋊C4 C12 C2×C4 C6 C6 C4 C4 C2 C2 # reps 1 1 1 1 2 1 1 1 2 2 1 2 2 4 4 1 1 1 2 2

Matrix representation of D12.4Q8 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 72 71 0 0 0 0 1 1
,
 1 0 0 0 0 0 24 72 0 0 0 0 0 0 1 72 0 0 0 0 0 72 0 0 0 0 0 0 1 2 0 0 0 0 0 72
,
 37 3 0 0 0 0 30 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 32 0 0 0 0 16 0
,
 27 0 0 0 0 0 64 46 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 46 0 0 0 0 0 0 46

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,71,1],[1,24,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,2,72],[37,30,0,0,0,0,3,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,32,0],[27,64,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46] >;`

D12.4Q8 in GAP, Magma, Sage, TeX

`D_{12}._4Q_8`
`% in TeX`

`G:=Group("D12.4Q8");`
`// GroupNames label`

`G:=SmallGroup(192,625);`
`// by ID`

`G=gap.SmallGroup(192,625);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,344,254,219,100,1123,297,136,6278]);`
`// Polycyclic`

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

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