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G = D8.D9order 288 = 25·32

The non-split extension by D8 of D9 acting via D9/C9=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.D9, C92SD32, C24.7D6, C8.5D18, C36.4D4, C18.9D8, Dic363C2, C72.3C22, C9⋊C162C2, (C9×D8).1C2, (C3×D8).2S3, C3.(D8.S3), C2.5(D4⋊D9), C4.2(C9⋊D4), C6.16(D4⋊S3), C12.2(C3⋊D4), SmallGroup(288,34)

Series: Derived Chief Lower central Upper central

C1C72 — D8.D9
C1C3C9C18C36C72Dic36 — D8.D9
C9C18C36C72 — D8.D9
C1C2C4C8D8

Generators and relations for D8.D9
 G = < a,b,c,d | a8=b2=c9=1, d2=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c-1 >

8C2
4C22
36C4
8C6
2D4
18Q8
4C2×C6
12Dic3
8C18
9Q16
9C16
2C3×D4
6Dic6
4Dic9
4C2×C18
9SD32
3C3⋊C16
3Dic12
2Dic18
2D4×C9
3D8.S3

Smallest permutation representation of D8.D9
On 144 points
Generators in S144
(1 59 23 50 14 68 32 41)(2 60 24 51 15 69 33 42)(3 61 25 52 16 70 34 43)(4 62 26 53 17 71 35 44)(5 63 27 54 18 72 36 45)(6 55 19 46 10 64 28 37)(7 56 20 47 11 65 29 38)(8 57 21 48 12 66 30 39)(9 58 22 49 13 67 31 40)(73 109 100 136 82 118 91 127)(74 110 101 137 83 119 92 128)(75 111 102 138 84 120 93 129)(76 112 103 139 85 121 94 130)(77 113 104 140 86 122 95 131)(78 114 105 141 87 123 96 132)(79 115 106 142 88 124 97 133)(80 116 107 143 89 125 98 134)(81 117 108 144 90 126 99 135)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 37)(7 38)(8 39)(9 40)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(109 118)(110 119)(111 120)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)
(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 84 14 75)(2 83 15 74)(3 82 16 73)(4 90 17 81)(5 89 18 80)(6 88 10 79)(7 87 11 78)(8 86 12 77)(9 85 13 76)(19 106 28 97)(20 105 29 96)(21 104 30 95)(22 103 31 94)(23 102 32 93)(24 101 33 92)(25 100 34 91)(26 108 35 99)(27 107 36 98)(37 124 46 115)(38 123 47 114)(39 122 48 113)(40 121 49 112)(41 120 50 111)(42 119 51 110)(43 118 52 109)(44 126 53 117)(45 125 54 116)(55 142 64 133)(56 141 65 132)(57 140 66 131)(58 139 67 130)(59 138 68 129)(60 137 69 128)(61 136 70 127)(62 144 71 135)(63 143 72 134)

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

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

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

39 conjugacy classes

class 1 2A2B 3 4A4B6A6B6C8A8B9A9B9C 12 16A16B16C16D18A18B18C18D···18I24A24B36A36B36C72A···72F
order12234466688999121616161618181818···18242436363672···72
size1182272288222224181818182228···8444444···4

39 irreducible representations

dim11112222222224444
type+++++++++++-+-
imageC1C2C2C2S3D4D6D8D9C3⋊D4SD32D18C9⋊D4D4⋊S3D8.S3D4⋊D9D8.D9
kernelD8.D9C9⋊C16Dic36C9×D8C3×D8C36C24C18D8C12C9C8C4C6C3C2C1
# reps11111112324361236

Matrix representation of D8.D9 in GL4(𝔽433) generated by

432000
043200
000213
00124206
,
432000
0100
000213
003090
,
296000
025600
0010
0001
,
025600
296000
00187159
00194246
G:=sub<GL(4,GF(433))| [432,0,0,0,0,432,0,0,0,0,0,124,0,0,213,206],[432,0,0,0,0,1,0,0,0,0,0,309,0,0,213,0],[296,0,0,0,0,256,0,0,0,0,1,0,0,0,0,1],[0,296,0,0,256,0,0,0,0,0,187,194,0,0,159,246] >;

D8.D9 in GAP, Magma, Sage, TeX

D_8.D_9
% in TeX

G:=Group("D8.D9");
// GroupNames label

G:=SmallGroup(288,34);
// by ID

G=gap.SmallGroup(288,34);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,85,254,135,142,675,346,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^9=1,d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of D8.D9 in TeX

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