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## G = Q8.D9order 144 = 24·32

### The non-split extension by Q8 of D9 acting via D9/C3=S3

Aliases: Q8.D9, C6.1S4, C3.CSU2(𝔽3), Q8⋊C9.C2, (C3×Q8).1S3, C2.2(C3.S4), SmallGroup(144,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — Q8⋊C9 — Q8.D9
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — Q8.D9
 Lower central Q8⋊C9 — Q8.D9
 Upper central C1 — C2

Generators and relations for Q8.D9
G = < a,b,c,d | a4=c9=1, b2=d2=a2, bab-1=a-1, cac-1=b, dad-1=a-1b, cbc-1=ab, dbd-1=a2b, dcd-1=c-1 >

Character table of Q8.D9

 class 1 2 3 4A 4B 6 8A 8B 9A 9B 9C 12 18A 18B 18C size 1 1 2 6 36 2 18 18 8 8 8 12 8 8 8 ρ1 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 linear of order 2 ρ3 2 2 2 2 0 2 0 0 -1 -1 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ4 2 2 -1 2 0 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D9 ρ5 2 2 -1 2 0 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D9 ρ6 2 2 -1 2 0 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D9 ρ7 2 -2 2 0 0 -2 √2 -√2 -1 -1 -1 0 1 1 1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ8 2 -2 2 0 0 -2 -√2 √2 -1 -1 -1 0 1 1 1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ9 3 3 3 -1 -1 3 1 1 0 0 0 -1 0 0 0 orthogonal lifted from S4 ρ10 3 3 3 -1 1 3 -1 -1 0 0 0 -1 0 0 0 orthogonal lifted from S4 ρ11 4 -4 4 0 0 -4 0 0 1 1 1 0 -1 -1 -1 symplectic lifted from CSU2(𝔽3), Schur index 2 ρ12 4 -4 -2 0 0 2 0 0 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 0 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 symplectic faithful, Schur index 2 ρ13 4 -4 -2 0 0 2 0 0 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 0 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 symplectic faithful, Schur index 2 ρ14 4 -4 -2 0 0 2 0 0 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 0 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 symplectic faithful, Schur index 2 ρ15 6 6 -3 -2 0 -3 0 0 0 0 0 1 0 0 0 orthogonal lifted from C3.S4

Smallest permutation representation of Q8.D9
Regular action on 144 points
Generators in S144
```(1 102 74 129)(2 125 75 115)(3 86 76 97)(4 105 77 132)(5 119 78 109)(6 89 79 91)(7 108 80 135)(8 122 81 112)(9 83 73 94)(10 55 37 53)(11 66 38 142)(12 36 39 26)(13 58 40 47)(14 69 41 136)(15 30 42 20)(16 61 43 50)(17 72 44 139)(18 33 45 23)(19 59 29 48)(21 138 31 71)(22 62 32 51)(24 141 34 65)(25 56 35 54)(27 144 28 68)(46 67 57 143)(49 70 60 137)(52 64 63 140)(82 127 93 100)(84 124 95 114)(85 130 96 103)(87 118 98 117)(88 133 99 106)(90 121 92 111)(101 113 128 123)(104 116 131 126)(107 110 134 120)
(1 124 74 114)(2 85 75 96)(3 104 76 131)(4 118 77 117)(5 88 78 99)(6 107 79 134)(7 121 80 111)(8 82 81 93)(9 101 73 128)(10 65 37 141)(11 35 38 25)(12 57 39 46)(13 68 40 144)(14 29 41 19)(15 60 42 49)(16 71 43 138)(17 32 44 22)(18 63 45 52)(20 137 30 70)(21 61 31 50)(23 140 33 64)(24 55 34 53)(26 143 36 67)(27 58 28 47)(48 69 59 136)(51 72 62 139)(54 66 56 142)(83 123 94 113)(84 129 95 102)(86 126 97 116)(87 132 98 105)(89 120 91 110)(90 135 92 108)(100 112 127 122)(103 115 130 125)(106 109 133 119)
(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 97 98 99)(100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135)(136 137 138 139 140 141 142 143 144)
(1 70 74 137)(2 69 75 136)(3 68 76 144)(4 67 77 143)(5 66 78 142)(6 65 79 141)(7 64 80 140)(8 72 81 139)(9 71 73 138)(10 134 37 107)(11 133 38 106)(12 132 39 105)(13 131 40 104)(14 130 41 103)(15 129 42 102)(16 128 43 101)(17 127 44 100)(18 135 45 108)(19 125 29 115)(20 124 30 114)(21 123 31 113)(22 122 32 112)(23 121 33 111)(24 120 34 110)(25 119 35 109)(26 118 36 117)(27 126 28 116)(46 98 57 87)(47 97 58 86)(48 96 59 85)(49 95 60 84)(50 94 61 83)(51 93 62 82)(52 92 63 90)(53 91 55 89)(54 99 56 88)```

`G:=sub<Sym(144)| (1,102,74,129)(2,125,75,115)(3,86,76,97)(4,105,77,132)(5,119,78,109)(6,89,79,91)(7,108,80,135)(8,122,81,112)(9,83,73,94)(10,55,37,53)(11,66,38,142)(12,36,39,26)(13,58,40,47)(14,69,41,136)(15,30,42,20)(16,61,43,50)(17,72,44,139)(18,33,45,23)(19,59,29,48)(21,138,31,71)(22,62,32,51)(24,141,34,65)(25,56,35,54)(27,144,28,68)(46,67,57,143)(49,70,60,137)(52,64,63,140)(82,127,93,100)(84,124,95,114)(85,130,96,103)(87,118,98,117)(88,133,99,106)(90,121,92,111)(101,113,128,123)(104,116,131,126)(107,110,134,120), (1,124,74,114)(2,85,75,96)(3,104,76,131)(4,118,77,117)(5,88,78,99)(6,107,79,134)(7,121,80,111)(8,82,81,93)(9,101,73,128)(10,65,37,141)(11,35,38,25)(12,57,39,46)(13,68,40,144)(14,29,41,19)(15,60,42,49)(16,71,43,138)(17,32,44,22)(18,63,45,52)(20,137,30,70)(21,61,31,50)(23,140,33,64)(24,55,34,53)(26,143,36,67)(27,58,28,47)(48,69,59,136)(51,72,62,139)(54,66,56,142)(83,123,94,113)(84,129,95,102)(86,126,97,116)(87,132,98,105)(89,120,91,110)(90,135,92,108)(100,112,127,122)(103,115,130,125)(106,109,133,119), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,70,74,137)(2,69,75,136)(3,68,76,144)(4,67,77,143)(5,66,78,142)(6,65,79,141)(7,64,80,140)(8,72,81,139)(9,71,73,138)(10,134,37,107)(11,133,38,106)(12,132,39,105)(13,131,40,104)(14,130,41,103)(15,129,42,102)(16,128,43,101)(17,127,44,100)(18,135,45,108)(19,125,29,115)(20,124,30,114)(21,123,31,113)(22,122,32,112)(23,121,33,111)(24,120,34,110)(25,119,35,109)(26,118,36,117)(27,126,28,116)(46,98,57,87)(47,97,58,86)(48,96,59,85)(49,95,60,84)(50,94,61,83)(51,93,62,82)(52,92,63,90)(53,91,55,89)(54,99,56,88)>;`

`G:=Group( (1,102,74,129)(2,125,75,115)(3,86,76,97)(4,105,77,132)(5,119,78,109)(6,89,79,91)(7,108,80,135)(8,122,81,112)(9,83,73,94)(10,55,37,53)(11,66,38,142)(12,36,39,26)(13,58,40,47)(14,69,41,136)(15,30,42,20)(16,61,43,50)(17,72,44,139)(18,33,45,23)(19,59,29,48)(21,138,31,71)(22,62,32,51)(24,141,34,65)(25,56,35,54)(27,144,28,68)(46,67,57,143)(49,70,60,137)(52,64,63,140)(82,127,93,100)(84,124,95,114)(85,130,96,103)(87,118,98,117)(88,133,99,106)(90,121,92,111)(101,113,128,123)(104,116,131,126)(107,110,134,120), (1,124,74,114)(2,85,75,96)(3,104,76,131)(4,118,77,117)(5,88,78,99)(6,107,79,134)(7,121,80,111)(8,82,81,93)(9,101,73,128)(10,65,37,141)(11,35,38,25)(12,57,39,46)(13,68,40,144)(14,29,41,19)(15,60,42,49)(16,71,43,138)(17,32,44,22)(18,63,45,52)(20,137,30,70)(21,61,31,50)(23,140,33,64)(24,55,34,53)(26,143,36,67)(27,58,28,47)(48,69,59,136)(51,72,62,139)(54,66,56,142)(83,123,94,113)(84,129,95,102)(86,126,97,116)(87,132,98,105)(89,120,91,110)(90,135,92,108)(100,112,127,122)(103,115,130,125)(106,109,133,119), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135)(136,137,138,139,140,141,142,143,144), (1,70,74,137)(2,69,75,136)(3,68,76,144)(4,67,77,143)(5,66,78,142)(6,65,79,141)(7,64,80,140)(8,72,81,139)(9,71,73,138)(10,134,37,107)(11,133,38,106)(12,132,39,105)(13,131,40,104)(14,130,41,103)(15,129,42,102)(16,128,43,101)(17,127,44,100)(18,135,45,108)(19,125,29,115)(20,124,30,114)(21,123,31,113)(22,122,32,112)(23,121,33,111)(24,120,34,110)(25,119,35,109)(26,118,36,117)(27,126,28,116)(46,98,57,87)(47,97,58,86)(48,96,59,85)(49,95,60,84)(50,94,61,83)(51,93,62,82)(52,92,63,90)(53,91,55,89)(54,99,56,88) );`

`G=PermutationGroup([[(1,102,74,129),(2,125,75,115),(3,86,76,97),(4,105,77,132),(5,119,78,109),(6,89,79,91),(7,108,80,135),(8,122,81,112),(9,83,73,94),(10,55,37,53),(11,66,38,142),(12,36,39,26),(13,58,40,47),(14,69,41,136),(15,30,42,20),(16,61,43,50),(17,72,44,139),(18,33,45,23),(19,59,29,48),(21,138,31,71),(22,62,32,51),(24,141,34,65),(25,56,35,54),(27,144,28,68),(46,67,57,143),(49,70,60,137),(52,64,63,140),(82,127,93,100),(84,124,95,114),(85,130,96,103),(87,118,98,117),(88,133,99,106),(90,121,92,111),(101,113,128,123),(104,116,131,126),(107,110,134,120)], [(1,124,74,114),(2,85,75,96),(3,104,76,131),(4,118,77,117),(5,88,78,99),(6,107,79,134),(7,121,80,111),(8,82,81,93),(9,101,73,128),(10,65,37,141),(11,35,38,25),(12,57,39,46),(13,68,40,144),(14,29,41,19),(15,60,42,49),(16,71,43,138),(17,32,44,22),(18,63,45,52),(20,137,30,70),(21,61,31,50),(23,140,33,64),(24,55,34,53),(26,143,36,67),(27,58,28,47),(48,69,59,136),(51,72,62,139),(54,66,56,142),(83,123,94,113),(84,129,95,102),(86,126,97,116),(87,132,98,105),(89,120,91,110),(90,135,92,108),(100,112,127,122),(103,115,130,125),(106,109,133,119)], [(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),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135),(136,137,138,139,140,141,142,143,144)], [(1,70,74,137),(2,69,75,136),(3,68,76,144),(4,67,77,143),(5,66,78,142),(6,65,79,141),(7,64,80,140),(8,72,81,139),(9,71,73,138),(10,134,37,107),(11,133,38,106),(12,132,39,105),(13,131,40,104),(14,130,41,103),(15,129,42,102),(16,128,43,101),(17,127,44,100),(18,135,45,108),(19,125,29,115),(20,124,30,114),(21,123,31,113),(22,122,32,112),(23,121,33,111),(24,120,34,110),(25,119,35,109),(26,118,36,117),(27,126,28,116),(46,98,57,87),(47,97,58,86),(48,96,59,85),(49,95,60,84),(50,94,61,83),(51,93,62,82),(52,92,63,90),(53,91,55,89),(54,99,56,88)]])`

Q8.D9 is a maximal subgroup of   Q8.D18  C12.3S4  C12.11S4  C32.CSU2(𝔽3)  C18.5S4  C32.3CSU2(𝔽3)
Q8.D9 is a maximal quotient of   Q8⋊Dic9  Q8.D27  C32.3CSU2(𝔽3)

Matrix representation of Q8.D9 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 32 22 0 0 43 41
,
 31 70 0 0 3 28 0 0 0 0 29 33 0 0 9 43
,
 1 0 0 0 1 72 0 0 0 0 45 43 0 0 14 28
`G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,71,72],[1,0,0,0,0,1,0,0,0,0,32,43,0,0,22,41],[31,3,0,0,70,28,0,0,0,0,29,9,0,0,33,43],[1,1,0,0,0,72,0,0,0,0,45,14,0,0,43,28] >;`

Q8.D9 in GAP, Magma, Sage, TeX

`Q_8.D_9`
`% in TeX`

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

`G:=SmallGroup(144,31);`
`// by ID`

`G=gap.SmallGroup(144,31);`
`# by ID`

`G:=PCGroup([6,-2,-3,-3,-2,2,-2,432,121,79,218,867,1305,117,544,820,202,88]);`
`// Polycyclic`

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

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