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## G = D12⋊5Q8order 192 = 26·3

### 3rd semidirect product of D12 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12⋊5Q8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4×D12 — D12⋊5Q8
 Lower central C3 — C6 — C2×C12 — D12⋊5Q8
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for D125Q8
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, dbd-1=a6b, dcd-1=c-1 >

Subgroups: 320 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], S3 [×2], C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×2], C23, Dic3, C12 [×2], C12 [×2], C12 [×3], D6 [×4], C2×C6, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C2×D4, C2×Q8, C3⋊C8 [×2], C4×S3 [×2], D12 [×2], D12, C2×Dic3, C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C22×S3, D4⋊C4 [×2], C4⋊C8, C4.Q8 [×2], C4×D4, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C2×D12, C6×Q8, D42Q8, C12⋊C8, C12.Q8 [×2], C6.D8 [×2], C4×D12, C3×C4⋊Q8, D125Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], SD16 [×2], C2×D4, C2×Q8, C4○D4, C3⋊D4 [×2], C22×S3, C22⋊Q8, C2×SD16, C8⋊C22, Q82S3 [×2], S3×Q8, Q83S3, C2×C3⋊D4, D42Q8, D126C22, C2×Q82S3, D63Q8, D125Q8

Smallest permutation representation of D125Q8
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 25)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 89)(14 88)(15 87)(16 86)(17 85)(18 96)(19 95)(20 94)(21 93)(22 92)(23 91)(24 90)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 60)(48 59)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 84)(70 83)(71 82)(72 81)
(1 53 32 40)(2 60 33 47)(3 55 34 42)(4 50 35 37)(5 57 36 44)(6 52 25 39)(7 59 26 46)(8 54 27 41)(9 49 28 48)(10 56 29 43)(11 51 30 38)(12 58 31 45)(13 69 89 81)(14 64 90 76)(15 71 91 83)(16 66 92 78)(17 61 93 73)(18 68 94 80)(19 63 95 75)(20 70 96 82)(21 65 85 77)(22 72 86 84)(23 67 87 79)(24 62 88 74)
(1 83 32 71)(2 84 33 72)(3 73 34 61)(4 74 35 62)(5 75 36 63)(6 76 25 64)(7 77 26 65)(8 78 27 66)(9 79 28 67)(10 80 29 68)(11 81 30 69)(12 82 31 70)(13 51 89 38)(14 52 90 39)(15 53 91 40)(16 54 92 41)(17 55 93 42)(18 56 94 43)(19 57 95 44)(20 58 96 45)(21 59 85 46)(22 60 86 47)(23 49 87 48)(24 50 88 37)```

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

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

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

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 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + - + + + + + - + image C1 C2 C2 C2 C2 C2 S3 Q8 D4 D6 D6 SD16 C4○D4 C3⋊D4 C8⋊C22 Q8⋊2S3 S3×Q8 Q8⋊3S3 D12⋊6C22 kernel D12⋊5Q8 C12⋊C8 C12.Q8 C6.D8 C4×D12 C3×C4⋊Q8 C4⋊Q8 D12 C2×C12 C42 C4⋊C4 C12 C12 C2×C4 C6 C4 C4 C4 C2 # reps 1 1 2 2 1 1 1 2 2 1 2 4 2 4 1 2 1 1 2

Matrix representation of D125Q8 in GL6(𝔽73)

 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0 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 72 72 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 62 72
,
 6 6 0 0 0 0 6 67 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 62 71 0 0 0 0 61 11
,
 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 46 0 0 0 0 0 5 27

`G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,1,62,0,0,0,0,0,72],[6,6,0,0,0,0,6,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,62,61,0,0,0,0,71,11],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,5,0,0,0,0,0,27] >;`

D125Q8 in GAP, Magma, Sage, TeX

`D_{12}\rtimes_5Q_8`
`% in TeX`

`G:=Group("D12:5Q8");`
`// GroupNames label`

`G:=SmallGroup(192,643);`
`// by ID`

`G=gap.SmallGroup(192,643);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,268,1123,297,136,6278]);`
`// Polycyclic`

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

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