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## G = Q8.D18order 288 = 25·32

### 2nd non-split extension by Q8 of D18 acting via D18/C6=S3

Aliases: Q8.2D18, Q8⋊D91C2, (C2×C6).9S4, (C2×Q8)⋊2D9, C6.21(C2×S4), Q8.D91C2, (C6×Q8).4S3, (C3×Q8).10D6, Q8⋊C9.2C22, C3.(Q8.D6), C22.2(C3.S4), (C2×Q8⋊C9)⋊3C2, C2.7(C2×C3.S4), SmallGroup(288,337)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — Q8⋊C9 — Q8.D18
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — Q8⋊D9 — Q8.D18
 Lower central Q8⋊C9 — Q8.D18
 Upper central C1 — C2 — C22

Generators and relations for Q8.D18
G = < a,b,c,d | a4=c18=1, b2=d2=a2, bab-1=a-1, cac-1=b, dad-1=a-1b, cbc-1=ab, dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 359 in 65 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C9, Dic3, C12, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, D9, C18, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, C3×Q8, C8.C22, Dic9, D18, C2×C18, C4.Dic3, Q82S3, C3⋊Q16, C4○D12, C6×Q8, Q8⋊C9, C9⋊D4, Q8.11D6, Q8.D9, Q8⋊D9, C2×Q8⋊C9, Q8.D18
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, C2×S4, C3.S4, Q8.D6, C2×C3.S4, Q8.D18

Character table of Q8.D18

 class 1 2A 2B 2C 3 4A 4B 4C 6A 6B 6C 8A 8B 9A 9B 9C 12A 12B 18A 18B 18C 18D 18E 18F 18G 18H 18I size 1 1 2 36 2 6 6 36 2 2 2 36 36 8 8 8 12 12 8 8 8 8 8 8 8 8 8 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 2 0 2 2 2 0 2 2 2 0 0 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 -2 0 2 -2 2 0 -2 -2 2 0 0 -1 -1 -1 -2 2 1 1 1 1 1 -1 -1 -1 1 orthogonal lifted from D6 ρ7 2 2 -2 0 -1 -2 2 0 1 1 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 1 -1 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ98-ζ9 orthogonal lifted from D18 ρ8 2 2 -2 0 -1 -2 2 0 1 1 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 1 -1 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ95-ζ94 orthogonal lifted from D18 ρ9 2 2 -2 0 -1 -2 2 0 1 1 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 1 -1 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ97-ζ92 orthogonal lifted from D18 ρ10 2 2 2 0 -1 2 2 0 -1 -1 -1 0 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95+ζ94 orthogonal lifted from D9 ρ11 2 2 2 0 -1 2 2 0 -1 -1 -1 0 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97+ζ92 orthogonal lifted from D9 ρ12 2 2 2 0 -1 2 2 0 -1 -1 -1 0 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ9 orthogonal lifted from D9 ρ13 3 3 3 1 3 -1 -1 1 3 3 3 -1 -1 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ14 3 3 3 -1 3 -1 -1 -1 3 3 3 1 1 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from S4 ρ15 3 3 -3 -1 3 1 -1 1 -3 -3 3 -1 1 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ16 3 3 -3 1 3 1 -1 -1 -3 -3 3 1 -1 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×S4 ρ17 4 -4 0 0 4 0 0 0 0 0 -4 0 0 -2 -2 -2 0 0 0 0 0 0 0 2 2 2 0 symplectic lifted from Q8.D6, Schur index 2 ρ18 4 -4 0 0 4 0 0 0 0 0 -4 0 0 1 1 1 0 0 -√-3 -√-3 √-3 -√-3 √-3 -1 -1 -1 √-3 complex lifted from Q8.D6 ρ19 4 -4 0 0 4 0 0 0 0 0 -4 0 0 1 1 1 0 0 √-3 √-3 -√-3 √-3 -√-3 -1 -1 -1 -√-3 complex lifted from Q8.D6 ρ20 4 -4 0 0 -2 0 0 0 2√-3 -2√-3 2 0 0 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 0 0 -ζ98+ζ9 ζ97-ζ92 -ζ97+ζ92 -ζ95+ζ94 ζ95-ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98-ζ9 complex faithful ρ21 4 -4 0 0 -2 0 0 0 -2√-3 2√-3 2 0 0 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 0 0 ζ95-ζ94 ζ98-ζ9 -ζ98+ζ9 -ζ97+ζ92 ζ97-ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 -ζ95+ζ94 complex faithful ρ22 4 -4 0 0 -2 0 0 0 -2√-3 2√-3 2 0 0 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 0 0 -ζ97+ζ92 ζ95-ζ94 -ζ95+ζ94 ζ98-ζ9 -ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ97-ζ92 complex faithful ρ23 4 -4 0 0 -2 0 0 0 -2√-3 2√-3 2 0 0 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 0 0 ζ98-ζ9 -ζ97+ζ92 ζ97-ζ92 ζ95-ζ94 -ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 -ζ98+ζ9 complex faithful ρ24 4 -4 0 0 -2 0 0 0 2√-3 -2√-3 2 0 0 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 0 0 ζ97-ζ92 -ζ95+ζ94 ζ95-ζ94 -ζ98+ζ9 ζ98-ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 -ζ97+ζ92 complex faithful ρ25 4 -4 0 0 -2 0 0 0 2√-3 -2√-3 2 0 0 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 0 0 -ζ95+ζ94 -ζ98+ζ9 ζ98-ζ9 ζ97-ζ92 -ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ95-ζ94 complex faithful ρ26 6 6 -6 0 -3 2 -2 0 3 3 -3 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C3.S4 ρ27 6 6 6 0 -3 -2 -2 0 -3 -3 -3 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 orthogonal lifted from C3.S4

Smallest permutation representation of Q8.D18
On 144 points
Generators in S144
```(1 133 41 97)(2 114 42 76)(3 32 43 64)(4 136 44 100)(5 117 45 79)(6 35 46 67)(7 139 47 103)(8 120 48 82)(9 20 49 70)(10 142 50 106)(11 123 51 85)(12 23 52 55)(13 127 53 91)(14 126 54 88)(15 26 37 58)(16 130 38 94)(17 111 39 73)(18 29 40 61)(19 104 69 140)(21 122 71 84)(22 107 72 143)(24 125 56 87)(25 92 57 128)(27 110 59 90)(28 95 60 131)(30 113 62 75)(31 98 63 134)(33 116 65 78)(34 101 66 137)(36 119 68 81)(74 96 112 132)(77 99 115 135)(80 102 118 138)(83 105 121 141)(86 108 124 144)(89 93 109 129)
(1 113 41 75)(2 31 42 63)(3 135 43 99)(4 116 44 78)(5 34 45 66)(6 138 46 102)(7 119 47 81)(8 19 48 69)(9 141 49 105)(10 122 50 84)(11 22 51 72)(12 144 52 108)(13 125 53 87)(14 25 54 57)(15 129 37 93)(16 110 38 90)(17 28 39 60)(18 132 40 96)(20 121 70 83)(21 106 71 142)(23 124 55 86)(24 91 56 127)(26 109 58 89)(27 94 59 130)(29 112 61 74)(30 97 62 133)(32 115 64 77)(33 100 65 136)(35 118 67 80)(36 103 68 139)(73 95 111 131)(76 98 114 134)(79 101 117 137)(82 104 120 140)(85 107 123 143)(88 92 126 128)
(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 18 41 40)(2 39 42 17)(3 16 43 38)(4 37 44 15)(5 14 45 54)(6 53 46 13)(7 12 47 52)(8 51 48 11)(9 10 49 50)(19 22 69 72)(20 71 70 21)(23 36 55 68)(24 67 56 35)(25 34 57 66)(26 65 58 33)(27 32 59 64)(28 63 60 31)(29 30 61 62)(73 134 111 98)(74 97 112 133)(75 132 113 96)(76 95 114 131)(77 130 115 94)(78 93 116 129)(79 128 117 92)(80 91 118 127)(81 144 119 108)(82 107 120 143)(83 142 121 106)(84 105 122 141)(85 140 123 104)(86 103 124 139)(87 138 125 102)(88 101 126 137)(89 136 109 100)(90 99 110 135)```

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

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

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

Matrix representation of Q8.D18 in GL4(𝔽73) generated by

 0 1 0 0 72 0 0 0 0 0 0 72 0 0 1 0
,
 0 0 72 0 0 0 0 72 1 0 0 0 0 1 0 0
,
 2 54 21 54 21 54 71 19 54 52 54 71 19 2 54 52
,
 71 19 52 19 19 21 19 2 52 19 2 54 19 2 54 52
`G:=sub<GL(4,GF(73))| [0,72,0,0,1,0,0,0,0,0,0,1,0,0,72,0],[0,0,1,0,0,0,0,1,72,0,0,0,0,72,0,0],[2,21,54,19,54,54,52,2,21,71,54,54,54,19,71,52],[71,19,52,19,19,21,19,2,52,19,2,54,19,2,54,52] >;`

Q8.D18 in GAP, Magma, Sage, TeX

`Q_8.D_{18}`
`% in TeX`

`G:=Group("Q8.D18");`
`// GroupNames label`

`G:=SmallGroup(288,337);`
`// by ID`

`G=gap.SmallGroup(288,337);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,422,142,675,2524,1908,172,1517,1153,285,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^18=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=a^2*c^-1>;`
`// generators/relations`

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