direct product, metabelian, supersoluble, monomial, A-group
Aliases: D9×C3⋊C8, C36.37D6, C12.37D18, D18.2Dic3, Dic9.2Dic3, (C3×D9)⋊C8, C3⋊3(C8×D9), C12.59S32, (C6×D9).1C4, (C4×D9).3S3, C4.22(S3×D9), C6.16(C4×D9), (C12×D9).3C2, C32.2(S3×C8), C6.1(S3×Dic3), C2.1(Dic3×D9), C36.S3⋊7C2, (C3×C12).155D6, (C3×Dic9).1C4, C18.1(C2×Dic3), (C3×C36).36C22, C9⋊1(C2×C3⋊C8), (C9×C3⋊C8)⋊4C2, (C3×C9)⋊1(C2×C8), C3.1(S3×C3⋊C8), (C3×C3⋊C8).7S3, (C3×C6).33(C4×S3), (C3×C18).1(C2×C4), SmallGroup(432,58)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — D9×C3⋊C8 |
Generators and relations for D9×C3⋊C8
G = < a,b,c,d | a9=b2=c3=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 300 in 74 conjugacy classes, 33 normal (29 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C2×C12, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, S3×C8, C2×C3⋊C8, C3×D9, C3×C18, C9⋊C8, C72, C4×D9, C3×C3⋊C8, C32⋊4C8, S3×C12, C3×Dic9, C3×C36, C6×D9, C8×D9, S3×C3⋊C8, C9×C3⋊C8, C36.S3, C12×D9, D9×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, D9, C3⋊C8, C4×S3, C2×Dic3, D18, S32, S3×C8, C2×C3⋊C8, C4×D9, S3×Dic3, S3×D9, C8×D9, S3×C3⋊C8, Dic3×D9, D9×C3⋊C8
►On 144 points
Generators in S144
(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 75)(2 74)(3 73)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 90)(18 89)(19 97)(20 96)(21 95)(22 94)(23 93)(24 92)(25 91)(26 99)(27 98)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 108)(36 107)(37 115)(38 114)(39 113)(40 112)(41 111)(42 110)(43 109)(44 117)(45 116)(46 124)(47 123)(48 122)(49 121)(50 120)(51 119)(52 118)(53 126)(54 125)(55 133)(56 132)(57 131)(58 130)(59 129)(60 128)(61 127)(62 135)(63 134)(64 142)(65 141)(66 140)(67 139)(68 138)(69 137)(70 136)(71 144)(72 143)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)(136 142 139)(137 143 140)(138 144 141)
(1 68 32 50 14 59 23 41)(2 69 33 51 15 60 24 42)(3 70 34 52 16 61 25 43)(4 71 35 53 17 62 26 44)(5 72 36 54 18 63 27 45)(6 64 28 46 10 55 19 37)(7 65 29 47 11 56 20 38)(8 66 30 48 12 57 21 39)(9 67 31 49 13 58 22 40)(73 136 100 118 82 127 91 109)(74 137 101 119 83 128 92 110)(75 138 102 120 84 129 93 111)(76 139 103 121 85 130 94 112)(77 140 104 122 86 131 95 113)(78 141 105 123 87 132 96 114)(79 142 106 124 88 133 97 115)(80 143 107 125 89 134 98 116)(81 144 108 126 90 135 99 117)
G:=sub<Sym(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), (1,75)(2,74)(3,73)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,90)(18,89)(19,97)(20,96)(21,95)(22,94)(23,93)(24,92)(25,91)(26,99)(27,98)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,108)(36,107)(37,115)(38,114)(39,113)(40,112)(41,111)(42,110)(43,109)(44,117)(45,116)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,126)(54,125)(55,133)(56,132)(57,131)(58,130)(59,129)(60,128)(61,127)(62,135)(63,134)(64,142)(65,141)(66,140)(67,139)(68,138)(69,137)(70,136)(71,144)(72,143), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,115,112)(110,116,113)(111,117,114)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132)(136,142,139)(137,143,140)(138,144,141), (1,68,32,50,14,59,23,41)(2,69,33,51,15,60,24,42)(3,70,34,52,16,61,25,43)(4,71,35,53,17,62,26,44)(5,72,36,54,18,63,27,45)(6,64,28,46,10,55,19,37)(7,65,29,47,11,56,20,38)(8,66,30,48,12,57,21,39)(9,67,31,49,13,58,22,40)(73,136,100,118,82,127,91,109)(74,137,101,119,83,128,92,110)(75,138,102,120,84,129,93,111)(76,139,103,121,85,130,94,112)(77,140,104,122,86,131,95,113)(78,141,105,123,87,132,96,114)(79,142,106,124,88,133,97,115)(80,143,107,125,89,134,98,116)(81,144,108,126,90,135,99,117)>;
G:=Group( (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), (1,75)(2,74)(3,73)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,90)(18,89)(19,97)(20,96)(21,95)(22,94)(23,93)(24,92)(25,91)(26,99)(27,98)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,108)(36,107)(37,115)(38,114)(39,113)(40,112)(41,111)(42,110)(43,109)(44,117)(45,116)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,126)(54,125)(55,133)(56,132)(57,131)(58,130)(59,129)(60,128)(61,127)(62,135)(63,134)(64,142)(65,141)(66,140)(67,139)(68,138)(69,137)(70,136)(71,144)(72,143), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54)(55,58,61)(56,59,62)(57,60,63)(64,67,70)(65,68,71)(66,69,72)(73,76,79)(74,77,80)(75,78,81)(82,85,88)(83,86,89)(84,87,90)(91,94,97)(92,95,98)(93,96,99)(100,103,106)(101,104,107)(102,105,108)(109,115,112)(110,116,113)(111,117,114)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132)(136,142,139)(137,143,140)(138,144,141), (1,68,32,50,14,59,23,41)(2,69,33,51,15,60,24,42)(3,70,34,52,16,61,25,43)(4,71,35,53,17,62,26,44)(5,72,36,54,18,63,27,45)(6,64,28,46,10,55,19,37)(7,65,29,47,11,56,20,38)(8,66,30,48,12,57,21,39)(9,67,31,49,13,58,22,40)(73,136,100,118,82,127,91,109)(74,137,101,119,83,128,92,110)(75,138,102,120,84,129,93,111)(76,139,103,121,85,130,94,112)(77,140,104,122,86,131,95,113)(78,141,105,123,87,132,96,114)(79,142,106,124,88,133,97,115)(80,143,107,125,89,134,98,116)(81,144,108,126,90,135,99,117) );
G=PermutationGroup([[(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)], [(1,75),(2,74),(3,73),(4,81),(5,80),(6,79),(7,78),(8,77),(9,76),(10,88),(11,87),(12,86),(13,85),(14,84),(15,83),(16,82),(17,90),(18,89),(19,97),(20,96),(21,95),(22,94),(23,93),(24,92),(25,91),(26,99),(27,98),(28,106),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,108),(36,107),(37,115),(38,114),(39,113),(40,112),(41,111),(42,110),(43,109),(44,117),(45,116),(46,124),(47,123),(48,122),(49,121),(50,120),(51,119),(52,118),(53,126),(54,125),(55,133),(56,132),(57,131),(58,130),(59,129),(60,128),(61,127),(62,135),(63,134),(64,142),(65,141),(66,140),(67,139),(68,138),(69,137),(70,136),(71,144),(72,143)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54),(55,58,61),(56,59,62),(57,60,63),(64,67,70),(65,68,71),(66,69,72),(73,76,79),(74,77,80),(75,78,81),(82,85,88),(83,86,89),(84,87,90),(91,94,97),(92,95,98),(93,96,99),(100,103,106),(101,104,107),(102,105,108),(109,115,112),(110,116,113),(111,117,114),(118,124,121),(119,125,122),(120,126,123),(127,133,130),(128,134,131),(129,135,132),(136,142,139),(137,143,140),(138,144,141)], [(1,68,32,50,14,59,23,41),(2,69,33,51,15,60,24,42),(3,70,34,52,16,61,25,43),(4,71,35,53,17,62,26,44),(5,72,36,54,18,63,27,45),(6,64,28,46,10,55,19,37),(7,65,29,47,11,56,20,38),(8,66,30,48,12,57,21,39),(9,67,31,49,13,58,22,40),(73,136,100,118,82,127,91,109),(74,137,101,119,83,128,92,110),(75,138,102,120,84,129,93,111),(76,139,103,121,85,130,94,112),(77,140,104,122,86,131,95,113),(78,141,105,123,87,132,96,114),(79,142,106,124,88,133,97,115),(80,143,107,125,89,134,98,116),(81,144,108,126,90,135,99,117)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | 18B | 18C | 18D | 18E | 18F | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36L | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 1 | 1 | 9 | 9 | 2 | 2 | 4 | 18 | 18 | 3 | 3 | 3 | 3 | 27 | 27 | 27 | 27 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | + | + | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | S3 | Dic3 | D6 | Dic3 | D6 | D9 | C3⋊C8 | C4×S3 | D18 | S3×C8 | C4×D9 | C8×D9 | S32 | S3×Dic3 | S3×D9 | S3×C3⋊C8 | Dic3×D9 | D9×C3⋊C8 |
| kernel | D9×C3⋊C8 | C9×C3⋊C8 | C36.S3 | C12×D9 | C3×Dic9 | C6×D9 | C3×D9 | C4×D9 | C3×C3⋊C8 | Dic9 | C36 | D18 | C3×C12 | C3⋊C8 | D9 | C3×C6 | C12 | C32 | C6 | C3 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 2 | 3 | 4 | 6 | 12 | 1 | 1 | 3 | 2 | 3 | 6 |
Matrix representation of D9×C3⋊C8 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 70 |
| 0 | 0 | 3 | 28 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 70 | 45 |
| 0 | 0 | 42 | 3 |
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 51 | 0 |
| 0 | 0 | 0 | 51 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,31,3,0,0,70,28],[72,0,0,0,0,72,0,0,0,0,70,42,0,0,45,3],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[1,72,0,0,0,72,0,0,0,0,51,0,0,0,0,51] >;
D9×C3⋊C8 in GAP, Magma, Sage, TeX
D_9\times C_3\rtimes C_8
% in TeX
G:=Group("D9xC3:C8"); // GroupNames label
G:=SmallGroup(432,58);
// by ID
G=gap.SmallGroup(432,58);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,36,58,571,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^2=c^3=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations