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## G = D8.9D6order 192 = 26·3

### 4th non-split extension by D8 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — D8.9D6
 Chief series C1 — C3 — C6 — C12 — C24 — Dic12 — C2×Dic12 — D8.9D6
 Lower central C3 — C6 — C12 — C24 — D8.9D6
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4○D8

Generators and relations for D8.9D6
G = < a,b,c,d | a8=b2=c6=1, d2=a4, bab=dad-1=a-1, ac=ca, cbc-1=a4b, dbd-1=ab, dcd-1=c-1 >

Subgroups: 232 in 82 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, C2×C6, C2×C6, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, M5(2), SD32, Q32, C2×Q16, C4○D8, C3⋊C16, Dic12, Dic12, C2×C24, C3×D8, C3×SD16, C3×Q16, C2×Dic6, C3×C4○D4, Q32⋊C2, C12.C8, D8.S3, C3⋊Q32, C2×Dic12, C3×C4○D8, D8.9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C2×D8, D4⋊S3, C2×C3⋊D4, Q32⋊C2, C2×D4⋊S3, D8.9D6

Character table of D8.9D6

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 8A 8B 8C 12A 12B 12C 12D 12E 16A 16B 16C 16D 24A 24B 24C 24D size 1 1 2 8 2 2 2 8 24 24 2 4 8 8 2 2 4 2 2 4 8 8 12 12 12 12 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 1 -1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ5 1 1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ6 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ7 1 1 1 -1 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ8 1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 linear of order 2 ρ9 2 2 2 -2 -1 2 2 -2 0 0 -1 -1 1 1 2 2 2 -1 -1 -1 1 1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from D6 ρ10 2 2 -2 -2 -1 -2 2 2 0 0 -1 1 1 1 2 2 -2 1 1 -1 -1 -1 0 0 0 0 1 -1 1 -1 orthogonal lifted from D6 ρ11 2 2 2 0 2 2 2 0 0 0 2 2 0 0 -2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 orthogonal lifted from D4 ρ12 2 2 2 2 -1 2 2 2 0 0 -1 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 -2 2 -1 -2 2 -2 0 0 -1 1 -1 -1 2 2 -2 1 1 -1 1 1 0 0 0 0 1 -1 1 -1 orthogonal lifted from D6 ρ14 2 2 -2 0 2 -2 2 0 0 0 2 -2 0 0 -2 -2 2 -2 -2 2 0 0 0 0 0 0 2 -2 2 -2 orthogonal lifted from D4 ρ15 2 2 2 0 2 -2 -2 0 0 0 2 2 0 0 0 0 0 -2 -2 -2 0 0 √2 √2 -√2 -√2 0 0 0 0 orthogonal lifted from D8 ρ16 2 2 -2 0 2 2 -2 0 0 0 2 -2 0 0 0 0 0 2 2 -2 0 0 √2 -√2 √2 -√2 0 0 0 0 orthogonal lifted from D8 ρ17 2 2 2 0 2 -2 -2 0 0 0 2 2 0 0 0 0 0 -2 -2 -2 0 0 -√2 -√2 √2 √2 0 0 0 0 orthogonal lifted from D8 ρ18 2 2 -2 0 2 2 -2 0 0 0 2 -2 0 0 0 0 0 2 2 -2 0 0 -√2 √2 -√2 √2 0 0 0 0 orthogonal lifted from D8 ρ19 2 2 -2 0 -1 -2 2 0 0 0 -1 1 -√-3 √-3 -2 -2 2 1 1 -1 -√-3 √-3 0 0 0 0 -1 1 -1 1 complex lifted from C3⋊D4 ρ20 2 2 -2 0 -1 -2 2 0 0 0 -1 1 √-3 -√-3 -2 -2 2 1 1 -1 √-3 -√-3 0 0 0 0 -1 1 -1 1 complex lifted from C3⋊D4 ρ21 2 2 2 0 -1 2 2 0 0 0 -1 -1 -√-3 √-3 -2 -2 -2 -1 -1 -1 √-3 -√-3 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ22 2 2 2 0 -1 2 2 0 0 0 -1 -1 √-3 -√-3 -2 -2 -2 -1 -1 -1 -√-3 √-3 0 0 0 0 1 1 1 1 complex lifted from C3⋊D4 ρ23 4 4 -4 0 -2 4 -4 0 0 0 -2 2 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ24 4 4 4 0 -2 -4 -4 0 0 0 -2 -2 0 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ25 4 -4 0 0 4 0 0 0 0 0 -4 0 0 0 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 symplectic lifted from Q32⋊C2, Schur index 2 ρ26 4 -4 0 0 4 0 0 0 0 0 -4 0 0 0 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 symplectic lifted from Q32⋊C2, Schur index 2 ρ27 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 2√2 -2√2 0 -2√3 2√3 0 0 0 0 0 0 0 √6 √2 -√6 -√2 symplectic faithful, Schur index 2 ρ28 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 2√2 -2√2 0 2√3 -2√3 0 0 0 0 0 0 0 -√6 √2 √6 -√2 symplectic faithful, Schur index 2 ρ29 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 -2√2 2√2 0 2√3 -2√3 0 0 0 0 0 0 0 √6 -√2 -√6 √2 symplectic faithful, Schur index 2 ρ30 4 -4 0 0 -2 0 0 0 0 0 2 0 0 0 -2√2 2√2 0 -2√3 2√3 0 0 0 0 0 0 0 -√6 -√2 √6 √2 symplectic faithful, Schur index 2

Smallest permutation representation of D8.9D6
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 8)(2 7)(3 6)(4 5)(9 15)(10 14)(11 13)(18 24)(19 23)(20 22)(25 30)(26 29)(27 28)(31 32)(33 36)(34 35)(37 40)(38 39)(41 46)(42 45)(43 44)(47 48)(49 52)(50 51)(53 56)(54 55)(57 62)(58 61)(59 60)(63 64)(66 72)(67 71)(68 70)(73 75)(76 80)(77 79)(81 85)(82 84)(86 88)(89 93)(90 92)(94 96)
(1 46 39 62 32 49)(2 47 40 63 25 50)(3 48 33 64 26 51)(4 41 34 57 27 52)(5 42 35 58 28 53)(6 43 36 59 29 54)(7 44 37 60 30 55)(8 45 38 61 31 56)(9 73 96 82 22 72)(10 74 89 83 23 65)(11 75 90 84 24 66)(12 76 91 85 17 67)(13 77 92 86 18 68)(14 78 93 87 19 69)(15 79 94 88 20 70)(16 80 95 81 21 71)
(1 94 5 90)(2 93 6 89)(3 92 7 96)(4 91 8 95)(9 33 13 37)(10 40 14 36)(11 39 15 35)(12 38 16 34)(17 31 21 27)(18 30 22 26)(19 29 23 25)(20 28 24 32)(41 76 45 80)(42 75 46 79)(43 74 47 78)(44 73 48 77)(49 88 53 84)(50 87 54 83)(51 86 55 82)(52 85 56 81)(57 67 61 71)(58 66 62 70)(59 65 63 69)(60 72 64 68)```

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

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

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

Matrix representation of D8.9D6 in GL4(𝔽97) generated by

 7 0 0 90 46 14 94 46 53 65 0 0 7 0 0 7
,
 7 0 0 7 51 0 3 51 53 65 0 0 7 0 0 90
,
 52 29 43 81 90 39 11 22 10 69 6 0 68 29 0 0
,
 73 61 86 89 74 16 48 94 21 75 82 37 74 42 68 23
`G:=sub<GL(4,GF(97))| [7,46,53,7,0,14,65,0,0,94,0,0,90,46,0,7],[7,51,53,7,0,0,65,0,0,3,0,0,7,51,0,90],[52,90,10,68,29,39,69,29,43,11,6,0,81,22,0,0],[73,74,21,74,61,16,75,42,86,48,82,68,89,94,37,23] >;`

D8.9D6 in GAP, Magma, Sage, TeX

`D_8._9D_6`
`% in TeX`

`G:=Group("D8.9D6");`
`// GroupNames label`

`G:=SmallGroup(192,754);`
`// by ID`

`G=gap.SmallGroup(192,754);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,387,675,185,192,1684,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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