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G = Q8⋊2D9order 144 = 24·32

The semidirect product of Q8 and D9 acting via D9/C9=C2

Aliases: Q82D9, C93SD16, C4.4D18, C12.4D6, D36.2C2, C18.10D4, C36.4C22, C9⋊C83C2, (Q8×C9)⋊1C2, (C3×Q8).6S3, C2.7(C9⋊D4), C3.(Q82S3), C6.17(C3⋊D4), SmallGroup(144,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — Q8⋊2D9
 Chief series C1 — C3 — C9 — C18 — C36 — D36 — Q8⋊2D9
 Lower central C9 — C18 — C36 — Q8⋊2D9
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Q82D9
G = < a,b,c,d | a4=c9=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

Character table of Q82D9

 class 1 2A 2B 3 4A 4B 6 8A 8B 9A 9B 9C 12A 12B 12C 18A 18B 18C 36A 36B 36C 36D 36E 36F 36G 36H 36I size 1 1 36 2 2 4 2 18 18 2 2 2 4 4 4 2 2 2 4 4 4 4 4 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 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 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 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 linear of order 2 ρ5 2 2 0 2 2 2 2 0 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 0 2 -2 0 2 0 0 2 2 2 -2 0 0 2 2 2 0 0 0 0 0 0 -2 -2 -2 orthogonal lifted from D4 ρ7 2 2 0 2 2 -2 2 0 0 -1 -1 -1 2 -2 -2 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 orthogonal lifted from D6 ρ8 2 2 0 -1 2 -2 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -1 1 1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D18 ρ9 2 2 0 -1 2 2 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ10 2 2 0 -1 2 -2 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -1 1 1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D18 ρ11 2 2 0 -1 2 -2 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -1 1 1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D18 ρ12 2 2 0 -1 2 2 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ13 2 2 0 -1 2 2 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ14 2 -2 0 2 0 0 -2 √-2 -√-2 2 2 2 0 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ15 2 2 0 2 -2 0 2 0 0 -1 -1 -1 -2 0 0 -1 -1 -1 -√-3 -√-3 -√-3 √-3 √-3 √-3 1 1 1 complex lifted from C3⋊D4 ρ16 2 2 0 -1 -2 0 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 1 √-3 -√-3 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98-ζ9 -ζ97+ζ92 ζ95-ζ94 ζ97-ζ92 -ζ98+ζ9 -ζ95+ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 complex lifted from C9⋊D4 ρ17 2 2 0 -1 -2 0 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 1 √-3 -√-3 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95-ζ94 ζ98-ζ9 -ζ97+ζ92 -ζ98+ζ9 -ζ95+ζ94 ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 complex lifted from C9⋊D4 ρ18 2 2 0 -1 -2 0 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 1 -√-3 √-3 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97-ζ92 -ζ95+ζ94 -ζ98+ζ9 ζ95-ζ94 -ζ97+ζ92 ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 complex lifted from C9⋊D4 ρ19 2 -2 0 2 0 0 -2 -√-2 √-2 2 2 2 0 0 0 -2 -2 -2 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ20 2 2 0 2 -2 0 2 0 0 -1 -1 -1 -2 0 0 -1 -1 -1 √-3 √-3 √-3 -√-3 -√-3 -√-3 1 1 1 complex lifted from C3⋊D4 ρ21 2 2 0 -1 -2 0 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 1 -√-3 √-3 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ95+ζ94 -ζ98+ζ9 ζ97-ζ92 ζ98-ζ9 ζ95-ζ94 -ζ97+ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 complex lifted from C9⋊D4 ρ22 2 2 0 -1 -2 0 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 1 √-3 -√-3 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ97+ζ92 ζ95-ζ94 ζ98-ζ9 -ζ95+ζ94 ζ97-ζ92 -ζ98+ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 complex lifted from C9⋊D4 ρ23 2 2 0 -1 -2 0 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 1 -√-3 √-3 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ98+ζ9 ζ97-ζ92 -ζ95+ζ94 -ζ97+ζ92 ζ98-ζ9 ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 complex lifted from C9⋊D4 ρ24 4 -4 0 4 0 0 -4 0 0 -2 -2 -2 0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ25 4 -4 0 -2 0 0 2 0 0 2ζ95+2ζ94 2ζ98+2ζ9 2ζ97+2ζ92 0 0 0 -2ζ98-2ζ9 -2ζ97-2ζ92 -2ζ95-2ζ94 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ26 4 -4 0 -2 0 0 2 0 0 2ζ98+2ζ9 2ζ97+2ζ92 2ζ95+2ζ94 0 0 0 -2ζ97-2ζ92 -2ζ95-2ζ94 -2ζ98-2ζ9 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ27 4 -4 0 -2 0 0 2 0 0 2ζ97+2ζ92 2ζ95+2ζ94 2ζ98+2ζ9 0 0 0 -2ζ95-2ζ94 -2ζ98-2ζ9 -2ζ97-2ζ92 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of Q82D9
On 72 points
Generators in S72
```(1 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)
(1 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)
(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)
(1 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(27 36)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)```

`G:=sub<Sym(72)| (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72)>;`

`G:=Group( (1,32,14,23)(2,33,15,24)(3,34,16,25)(4,35,17,26)(5,36,18,27)(6,28,10,19)(7,29,11,20)(8,30,12,21)(9,31,13,22)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(27,36)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,63)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,72) );`

`G=PermutationGroup([[(1,32,14,23),(2,33,15,24),(3,34,16,25),(4,35,17,26),(5,36,18,27),(6,28,10,19),(7,29,11,20),(8,30,12,21),(9,31,13,22),(37,55,46,64),(38,56,47,65),(39,57,48,66),(40,58,49,67),(41,59,50,68),(42,60,51,69),(43,61,52,70),(44,62,53,71),(45,63,54,72)], [(1,50,14,41),(2,51,15,42),(3,52,16,43),(4,53,17,44),(5,54,18,45),(6,46,10,37),(7,47,11,38),(8,48,12,39),(9,49,13,40),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(27,36),(37,62),(38,61),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,63),(46,71),(47,70),(48,69),(49,68),(50,67),(51,66),(52,65),(53,64),(54,72)]])`

Q82D9 is a maximal subgroup of
SD16×D9  D72⋊C2  Q16⋊D9  D725C2  C36.C23  D4⋊D18  D4.9D18  Q82D27  Q8⋊D27  Dic6⋊D9  C18.D12  D36.C6  C36.20D6  C18.6S4
Q82D9 is a maximal quotient of
C4.Dic18  C18.D8  Q82Dic9  Q82D27  Dic6⋊D9  C18.D12  C36.20D6

Matrix representation of Q82D9 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 1 71 0 0 1 72
,
 1 0 0 0 0 1 0 0 0 0 61 12 0 0 67 12
,
 70 45 0 0 28 42 0 0 0 0 1 0 0 0 0 1
,
 28 42 0 0 70 45 0 0 0 0 1 0 0 0 1 72
`G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,71,72],[1,0,0,0,0,1,0,0,0,0,61,67,0,0,12,12],[70,28,0,0,45,42,0,0,0,0,1,0,0,0,0,1],[28,70,0,0,42,45,0,0,0,0,1,1,0,0,0,72] >;`

Q82D9 in GAP, Magma, Sage, TeX

`Q_8\rtimes_2D_9`
`% in TeX`

`G:=Group("Q8:2D9");`
`// GroupNames label`

`G:=SmallGroup(144,18);`
`// by ID`

`G=gap.SmallGroup(144,18);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,55,218,116,50,2404,208,3461]);`
`// Polycyclic`

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

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