Copied to
clipboard

## G = D12.Q8order 192 = 26·3

### 1st non-split extension by D12 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — Dic3⋊5D4 — D12.Q8
 Lower central C3 — C6 — C2×C12 — D12.Q8
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for D12.Q8
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a9c2, bab=a-1, ac=ca, dad-1=a7, cbc-1=a3b, bd=db, dcd-1=c3 >

Subgroups: 320 in 102 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, D4.Q8, C6.Q16, C6.D8, Dic3⋊C8, C2.D24, C3×C4.Q8, C4.Dic6, Dic35D4, D12.Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C22×S3, C22⋊Q8, C4○D8, C8⋊C22, C4○D12, S3×D4, S3×Q8, D4.Q8, D6⋊Q8, Q83D6, Q8.7D6, D12.Q8

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

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

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

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

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 12B 12C 12D 12E 12F 24A 24B 24C 24D 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 24 24 24 24 size 1 1 1 1 12 12 2 2 2 4 4 6 6 8 12 24 2 2 2 4 4 12 12 4 4 8 8 8 8 4 4 4 4

33 irreducible representations

 dim 1 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 C2 S3 Q8 D4 D6 D6 C4○D4 C4○D8 C4○D12 C8⋊C22 S3×Q8 S3×D4 Q8⋊3D6 Q8.7D6 kernel D12.Q8 C6.Q16 C6.D8 Dic3⋊C8 C2.D24 C3×C4.Q8 C4.Dic6 Dic3⋊5D4 C4.Q8 D12 C2×Dic3 C4⋊C4 C2×C8 C12 C6 C4 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 2 1 2 4 4 1 1 1 2 2

Matrix representation of D12.Q8 in GL6(𝔽73)

 0 72 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 72 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 63 1
,
 67 67 0 0 0 0 6 67 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 58 3 0 0 0 0 47 15
,
 46 0 0 0 0 0 0 27 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 0 0 0 0 0 0 46

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,72,63,0,0,0,0,0,1],[67,6,0,0,0,0,67,67,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,58,47,0,0,0,0,3,15],[46,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46] >;`

D12.Q8 in GAP, Magma, Sage, TeX

`D_{12}.Q_8`
`% in TeX`

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

`G:=SmallGroup(192,430);`
`// by ID`

`G=gap.SmallGroup(192,430);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,555,268,1684,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^9*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

׿
×
𝔽