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G = Q8.D18order 288 = 25·32

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

non-abelian, soluble

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

C1C2Q8Q8⋊C9 — Q8.D18
C1C2Q8C3×Q8Q8⋊C9Q8⋊D9 — Q8.D18
Q8⋊C9 — Q8.D18
C1C2C22

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 12A2B2C34A4B4C6A6B6C8A8B9A9B9C12A12B18A18B18C18D18E18F18G18H18I
 size 112362663622236368881212888888888
ρ1111111111111111111111111111    trivial
ρ2111-1111-1111-1-111111111111111    linear of order 2
ρ311-1-11-111-1-111-1111-11-1-1-1-1-1111-1    linear of order 2
ρ411-111-11-1-1-11-11111-11-1-1-1-1-1111-1    linear of order 2
ρ52220222022200-1-1-122-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ622-202-220-2-2200-1-1-1-2211111-1-1-11    orthogonal lifted from D6
ρ722-20-1-22011-100ζ9594ζ989ζ97921-19899792979295949594ζ9594ζ9792ζ989989    orthogonal lifted from D18
ρ822-20-1-22011-100ζ9792ζ9594ζ9891-1959498998997929792ζ9792ζ989ζ95949594    orthogonal lifted from D18
ρ922-20-1-22011-100ζ989ζ9792ζ95941-1979295949594989989ζ989ζ9594ζ97929792    orthogonal lifted from D18
ρ102220-1220-1-1-100ζ9792ζ9594ζ989-1-1ζ9594ζ989ζ989ζ9792ζ9792ζ9792ζ989ζ9594ζ9594    orthogonal lifted from D9
ρ112220-1220-1-1-100ζ989ζ9792ζ9594-1-1ζ9792ζ9594ζ9594ζ989ζ989ζ989ζ9594ζ9792ζ9792    orthogonal lifted from D9
ρ122220-1220-1-1-100ζ9594ζ989ζ9792-1-1ζ989ζ9792ζ9792ζ9594ζ9594ζ9594ζ9792ζ989ζ989    orthogonal lifted from D9
ρ1333313-1-11333-1-1000-1-1000000000    orthogonal lifted from S4
ρ14333-13-1-1-133311000-1-1000000000    orthogonal lifted from S4
ρ1533-3-131-11-3-33-110001-1000000000    orthogonal lifted from C2×S4
ρ1633-3131-1-1-3-331-10001-1000000000    orthogonal lifted from C2×S4
ρ174-400400000-400-2-2-200000002220    symplectic lifted from Q8.D6, Schur index 2
ρ184-400400000-40011100--3--3-3--3-3-1-1-1-3    complex lifted from Q8.D6
ρ194-400400000-40011100-3-3--3-3--3-1-1-1--3    complex lifted from Q8.D6
ρ204-400-20002-3-2-32009594989979200989ζ979297929594ζ9594ζ9594ζ9792ζ989ζ989    complex faithful
ρ214-400-2000-2-32-32009792959498900ζ9594ζ9899899792ζ9792ζ9792ζ989ζ95949594    complex faithful
ρ224-400-2000-2-32-320098997929594009792ζ95949594ζ989989ζ989ζ9594ζ9792ζ9792    complex faithful
ρ234-400-2000-2-32-32009594989979200ζ9899792ζ9792ζ95949594ζ9594ζ9792ζ989989    complex faithful
ρ244-400-20002-3-2-32009899792959400ζ97929594ζ9594989ζ989ζ989ζ9594ζ97929792    complex faithful
ρ254-400-20002-3-2-320097929594989009594989ζ989ζ97929792ζ9792ζ989ζ9594ζ9594    complex faithful
ρ2666-60-32-2033-300000-11000000000    orthogonal lifted from C2×C3.S4
ρ276660-3-2-20-3-3-30000011000000000    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

0100
72000
00072
0010
,
00720
00072
1000
0100
,
2542154
21547119
54525471
1925452
,
71195219
1921192
5219254
1925452
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

Export

Character table of Q8.D18 in TeX

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