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## G = Q8.15D18order 288 = 25·32

### 1st non-split extension by Q8 of D18 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — Q8.15D18
 Chief series C1 — C3 — C9 — C18 — D18 — C4×D9 — Q8×D9 — Q8.15D18
 Lower central C9 — C18 — Q8.15D18
 Upper central C1 — C2 — C2×Q8

Generators and relations for Q8.15D18
G = < a,b,c,d | a4=c18=1, b2=d2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c-1 >

Subgroups: 824 in 219 conjugacy classes, 102 normal (13 characteristic)
C1, C2, C2 [×5], C3, C4 [×6], C4 [×4], C22, C22 [×4], S3 [×4], C6, C6, C2×C4 [×3], C2×C4 [×12], D4 [×10], Q8 [×4], Q8 [×6], C9, Dic3 [×4], C12 [×6], D6 [×4], C2×C6, C2×Q8, C2×Q8 [×4], C4○D4 [×10], D9 [×4], C18, C18, Dic6 [×6], C4×S3 [×12], D12 [×6], C3⋊D4 [×4], C2×C12 [×3], C3×Q8 [×4], 2- 1+4, Dic9 [×4], C36 [×6], D18 [×4], C2×C18, C4○D12 [×6], S3×Q8 [×4], Q83S3 [×4], C6×Q8, Dic18 [×6], C4×D9 [×12], D36 [×6], C9⋊D4 [×4], C2×C36 [×3], Q8×C9 [×4], Q8.15D6, D365C2 [×6], Q8×D9 [×4], Q83D9 [×4], Q8×C18, Q8.15D18
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, D9, C22×S3 [×7], 2- 1+4, D18 [×7], S3×C23, C22×D9 [×7], Q8.15D6, C23×D9, Q8.15D18

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

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A ··· 4F 4G 4H 4I 4J 6A 6B 6C 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 2 2 2 2 3 4 ··· 4 4 4 4 4 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 2 18 18 18 18 2 2 ··· 2 18 18 18 18 2 2 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 S3 D6 D6 D9 D18 D18 2- 1+4 Q8.15D6 Q8.15D18 kernel Q8.15D18 D36⋊5C2 Q8×D9 Q8⋊3D9 Q8×C18 C6×Q8 C2×C12 C3×Q8 C2×Q8 C2×C4 Q8 C9 C3 C1 # reps 1 6 4 4 1 1 3 4 3 9 12 1 2 6

Matrix representation of Q8.15D18 in GL6(𝔽37)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 1 0 0 0 1 2 36 35 0 0 0 2 36 35 0 0 1 1 0 36
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 6 34 31 0 0 0 26 3 0 0 0 0 21 11 0 0 0 29 29 0 5
,
 11 20 0 0 0 0 17 31 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 2 2 36 35 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 5 31 3 6 0 0 0 26 3 0 0 0 0 21 11 0 0 0 8 2 32 32

`G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,36,2,2,1,0,0,1,36,36,0,0,0,0,35,35,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,29,0,0,6,26,21,29,0,0,34,3,11,0,0,0,31,0,0,5],[11,17,0,0,0,0,20,31,0,0,0,0,0,0,0,1,2,0,0,0,1,0,2,0,0,0,0,0,36,0,0,0,0,0,35,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,8,0,0,31,26,21,2,0,0,3,3,11,32,0,0,6,0,0,32] >;`

Q8.15D18 in GAP, Magma, Sage, TeX

`Q_8._{15}D_{18}`
`% in TeX`

`G:=Group("Q8.15D18");`
`// GroupNames label`

`G:=SmallGroup(288,361);`
`// by ID`

`G=gap.SmallGroup(288,361);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,6725,292,9414]);`
`// Polycyclic`

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

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