metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊3D9, C8.8D18, D18.1D4, D4.1D18, C24.11D6, Dic36⋊4C2, C72.6C22, C36.3C23, Dic9.12D4, Dic18.1C22, (C8×D9)⋊2C2, (C9×D8)⋊3C2, C9⋊2(C4○D8), D4.D9⋊2C2, (C3×D8).4S3, (C3×D4).3D6, C6.91(S3×D4), C2.17(D4×D9), C9⋊C8.5C22, D4⋊2D9⋊2C2, C3.(D8⋊3S3), C18.29(C2×D4), C4.3(C22×D9), (C4×D9).8C22, (D4×C9).1C22, C12.42(C22×S3), SmallGroup(288,122)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊3D9
G = < a,b,c,d | a8=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 408 in 93 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C9, Dic3, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, D9, C18, C18, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C4○D8, Dic9, Dic9, C36, D18, C2×C18, S3×C8, Dic12, D4.S3, C3×D8, D4⋊2S3, C9⋊C8, C72, Dic18, C4×D9, C2×Dic9, C9⋊D4, D4×C9, D8⋊3S3, Dic36, C8×D9, D4.D9, C9×D8, D4⋊2D9, D8⋊3D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C22×S3, C4○D8, D18, S3×D4, C22×D9, D8⋊3S3, D4×D9, D8⋊3D9
►On 144 points
Generators in S144
(1 131 23 122 14 140 32 113)(2 132 24 123 15 141 33 114)(3 133 25 124 16 142 34 115)(4 134 26 125 17 143 35 116)(5 135 27 126 18 144 36 117)(6 127 19 118 10 136 28 109)(7 128 20 119 11 137 29 110)(8 129 21 120 12 138 30 111)(9 130 22 121 13 139 31 112)(37 100 64 82 46 91 55 73)(38 101 65 83 47 92 56 74)(39 102 66 84 48 93 57 75)(40 103 67 85 49 94 58 76)(41 104 68 86 50 95 59 77)(42 105 69 87 51 96 60 78)(43 106 70 88 52 97 61 79)(44 107 71 89 53 98 62 80)(45 108 72 90 54 99 63 81)
(1 77)(2 78)(3 79)(4 80)(5 81)(6 73)(7 74)(8 75)(9 76)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 35)(29 34)(30 33)(31 32)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(45 54)(55 71)(56 70)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(63 72)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(81 90)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(99 108)(109 116)(110 115)(111 114)(112 113)(118 125)(119 124)(120 123)(121 122)(127 134)(128 133)(129 132)(130 131)(136 143)(137 142)(138 141)(139 140)
G:=sub<Sym(144)| (1,131,23,122,14,140,32,113)(2,132,24,123,15,141,33,114)(3,133,25,124,16,142,34,115)(4,134,26,125,17,143,35,116)(5,135,27,126,18,144,36,117)(6,127,19,118,10,136,28,109)(7,128,20,119,11,137,29,110)(8,129,21,120,12,138,30,111)(9,130,22,121,13,139,31,112)(37,100,64,82,46,91,55,73)(38,101,65,83,47,92,56,74)(39,102,66,84,48,93,57,75)(40,103,67,85,49,94,58,76)(41,104,68,86,50,95,59,77)(42,105,69,87,51,96,60,78)(43,106,70,88,52,97,61,79)(44,107,71,89,53,98,62,80)(45,108,72,90,54,99,63,81), (1,77)(2,78)(3,79)(4,80)(5,81)(6,73)(7,74)(8,75)(9,76)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(45,54)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(81,90)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(99,108)(109,116)(110,115)(111,114)(112,113)(118,125)(119,124)(120,123)(121,122)(127,134)(128,133)(129,132)(130,131)(136,143)(137,142)(138,141)(139,140)>;
G:=Group( (1,131,23,122,14,140,32,113)(2,132,24,123,15,141,33,114)(3,133,25,124,16,142,34,115)(4,134,26,125,17,143,35,116)(5,135,27,126,18,144,36,117)(6,127,19,118,10,136,28,109)(7,128,20,119,11,137,29,110)(8,129,21,120,12,138,30,111)(9,130,22,121,13,139,31,112)(37,100,64,82,46,91,55,73)(38,101,65,83,47,92,56,74)(39,102,66,84,48,93,57,75)(40,103,67,85,49,94,58,76)(41,104,68,86,50,95,59,77)(42,105,69,87,51,96,60,78)(43,106,70,88,52,97,61,79)(44,107,71,89,53,98,62,80)(45,108,72,90,54,99,63,81), (1,77)(2,78)(3,79)(4,80)(5,81)(6,73)(7,74)(8,75)(9,76)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,35)(29,34)(30,33)(31,32)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(45,54)(55,71)(56,70)(57,69)(58,68)(59,67)(60,66)(61,65)(62,64)(63,72)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(81,90)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(99,108)(109,116)(110,115)(111,114)(112,113)(118,125)(119,124)(120,123)(121,122)(127,134)(128,133)(129,132)(130,131)(136,143)(137,142)(138,141)(139,140) );
G=PermutationGroup([[(1,131,23,122,14,140,32,113),(2,132,24,123,15,141,33,114),(3,133,25,124,16,142,34,115),(4,134,26,125,17,143,35,116),(5,135,27,126,18,144,36,117),(6,127,19,118,10,136,28,109),(7,128,20,119,11,137,29,110),(8,129,21,120,12,138,30,111),(9,130,22,121,13,139,31,112),(37,100,64,82,46,91,55,73),(38,101,65,83,47,92,56,74),(39,102,66,84,48,93,57,75),(40,103,67,85,49,94,58,76),(41,104,68,86,50,95,59,77),(42,105,69,87,51,96,60,78),(43,106,70,88,52,97,61,79),(44,107,71,89,53,98,62,80),(45,108,72,90,54,99,63,81)], [(1,77),(2,78),(3,79),(4,80),(5,81),(6,73),(7,74),(8,75),(9,76),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23),(28,35),(29,34),(30,33),(31,32),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(45,54),(55,71),(56,70),(57,69),(58,68),(59,67),(60,66),(61,65),(62,64),(63,72),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(81,90),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100),(99,108),(109,116),(110,115),(111,114),(112,113),(118,125),(119,124),(120,123),(121,122),(127,134),(128,133),(129,132),(130,131),(136,143),(137,142),(138,141),(139,140)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12 | 18A | 18B | 18C | 18D | ··· | 18I | 24A | 24B | 36A | 36B | 36C | 72A | ··· | 72F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 18 | 18 | 18 | 18 | ··· | 18 | 24 | 24 | 36 | 36 | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 4 | 4 | 18 | 2 | 2 | 9 | 9 | 36 | 36 | 2 | 8 | 8 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 8 | ··· | 8 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D9 | C4○D8 | D18 | D18 | S3×D4 | D8⋊3S3 | D4×D9 | D8⋊3D9 |
| kernel | D8⋊3D9 | Dic36 | C8×D9 | D4.D9 | C9×D8 | D4⋊2D9 | C3×D8 | Dic9 | D18 | C24 | C3×D4 | D8 | C9 | C8 | D4 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 3 | 6 | 1 | 2 | 3 | 6 |
Matrix representation of D8⋊3D9 ►in GL4(𝔽73) generated by
| 22 | 0 | 0 | 0 |
| 23 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 52 | 30 | 0 | 0 |
| 34 | 21 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 70 | 45 |
| 0 | 0 | 28 | 42 |
| 1 | 0 | 0 | 0 |
| 16 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 72 | 1 |
G:=sub<GL(4,GF(73))| [22,23,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[52,34,0,0,30,21,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,70,28,0,0,45,42],[1,16,0,0,0,72,0,0,0,0,72,72,0,0,0,1] >;
D8⋊3D9 in GAP, Magma, Sage, TeX
D_8\rtimes_3D_9
% in TeX
G:=Group("D8:3D9"); // GroupNames label
G:=SmallGroup(288,122);
// by ID
G=gap.SmallGroup(288,122);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,346,185,80,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations