direct product, metacyclic, supersoluble, monomial
Aliases: C3×C8⋊D9, C24⋊7D9, C72⋊12C6, D18.3C12, C12.76D18, Dic9.3C12, C9⋊C8⋊11C6, C8⋊3(C3×D9), (C3×C72)⋊7C2, C6.6(S3×C12), (C6×D9).3C4, (C4×D9).5C6, C2.3(C12×D9), C4.13(C6×D9), C6.22(C4×D9), (C3×C9)⋊5M4(2), C12.85(S3×C6), C24.18(C3×S3), C36.36(C2×C6), (C3×C24).27S3, (C12×D9).5C2, C9⋊4(C3×M4(2)), C18.16(C2×C12), (C3×C12).213D6, (C3×Dic9).3C4, (C3×C36).66C22, C32.4(C8⋊S3), (C3×C9⋊C8)⋊11C2, C3.1(C3×C8⋊S3), (C3×C6).64(C4×S3), (C3×C18).19(C2×C4), SmallGroup(432,106)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊D9
G = < a,b,c,d | a3=b8=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 222 in 68 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, M4(2), D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, C8⋊S3, C3×M4(2), C3×D9, C3×C18, C9⋊C8, C72, C72, C4×D9, C3×C3⋊C8, C3×C24, S3×C12, C3×Dic9, C3×C36, C6×D9, C8⋊D9, C3×C8⋊S3, C3×C9⋊C8, C3×C72, C12×D9, C3×C8⋊D9
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, M4(2), D9, C3×S3, C4×S3, C2×C12, D18, S3×C6, C8⋊S3, C3×M4(2), C3×D9, C4×D9, S3×C12, C6×D9, C8⋊D9, C3×C8⋊S3, C12×D9, C3×C8⋊D9
►On 144 points
Generators in S144
(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 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(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)(109 112 115)(110 113 116)(111 114 117)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)
(1 64 28 46 10 55 19 37)(2 65 29 47 11 56 20 38)(3 66 30 48 12 57 21 39)(4 67 31 49 13 58 22 40)(5 68 32 50 14 59 23 41)(6 69 33 51 15 60 24 42)(7 70 34 52 16 61 25 43)(8 71 35 53 17 62 26 44)(9 72 36 54 18 63 27 45)(73 127 100 109 82 136 91 118)(74 128 101 110 83 137 92 119)(75 129 102 111 84 138 93 120)(76 130 103 112 85 139 94 121)(77 131 104 113 86 140 95 122)(78 132 105 114 87 141 96 123)(79 133 106 115 88 142 97 124)(80 134 107 116 89 143 98 125)(81 135 108 117 90 144 99 126)
(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 79)(2 78)(3 77)(4 76)(5 75)(6 74)(7 73)(8 81)(9 80)(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)
G:=sub<Sym(144)| (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)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(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)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135)(136,139,142)(137,140,143)(138,141,144), (1,64,28,46,10,55,19,37)(2,65,29,47,11,56,20,38)(3,66,30,48,12,57,21,39)(4,67,31,49,13,58,22,40)(5,68,32,50,14,59,23,41)(6,69,33,51,15,60,24,42)(7,70,34,52,16,61,25,43)(8,71,35,53,17,62,26,44)(9,72,36,54,18,63,27,45)(73,127,100,109,82,136,91,118)(74,128,101,110,83,137,92,119)(75,129,102,111,84,138,93,120)(76,130,103,112,85,139,94,121)(77,131,104,113,86,140,95,122)(78,132,105,114,87,141,96,123)(79,133,106,115,88,142,97,124)(80,134,107,116,89,143,98,125)(81,135,108,117,90,144,99,126), (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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,81)(9,80)(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)>;
G:=Group( (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)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(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)(109,112,115)(110,113,116)(111,114,117)(118,121,124)(119,122,125)(120,123,126)(127,130,133)(128,131,134)(129,132,135)(136,139,142)(137,140,143)(138,141,144), (1,64,28,46,10,55,19,37)(2,65,29,47,11,56,20,38)(3,66,30,48,12,57,21,39)(4,67,31,49,13,58,22,40)(5,68,32,50,14,59,23,41)(6,69,33,51,15,60,24,42)(7,70,34,52,16,61,25,43)(8,71,35,53,17,62,26,44)(9,72,36,54,18,63,27,45)(73,127,100,109,82,136,91,118)(74,128,101,110,83,137,92,119)(75,129,102,111,84,138,93,120)(76,130,103,112,85,139,94,121)(77,131,104,113,86,140,95,122)(78,132,105,114,87,141,96,123)(79,133,106,115,88,142,97,124)(80,134,107,116,89,143,98,125)(81,135,108,117,90,144,99,126), (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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,81)(9,80)(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) );
G=PermutationGroup([[(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),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51),(55,61,58),(56,62,59),(57,63,60),(64,70,67),(65,71,68),(66,72,69),(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),(109,112,115),(110,113,116),(111,114,117),(118,121,124),(119,122,125),(120,123,126),(127,130,133),(128,131,134),(129,132,135),(136,139,142),(137,140,143),(138,141,144)], [(1,64,28,46,10,55,19,37),(2,65,29,47,11,56,20,38),(3,66,30,48,12,57,21,39),(4,67,31,49,13,58,22,40),(5,68,32,50,14,59,23,41),(6,69,33,51,15,60,24,42),(7,70,34,52,16,61,25,43),(8,71,35,53,17,62,26,44),(9,72,36,54,18,63,27,45),(73,127,100,109,82,136,91,118),(74,128,101,110,83,137,92,119),(75,129,102,111,84,138,93,120),(76,130,103,112,85,139,94,121),(77,131,104,113,86,140,95,122),(78,132,105,114,87,141,96,123),(79,133,106,115,88,142,97,124),(80,134,107,116,89,143,98,125),(81,135,108,117,90,144,99,126)], [(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)], [(1,79),(2,78),(3,77),(4,76),(5,75),(6,74),(7,73),(8,81),(9,80),(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)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 9A | ··· | 9I | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 18A | ··· | 18I | 24A | ··· | 24P | 24Q | 24R | 24S | 24T | 36A | ··· | 36R | 72A | ··· | 72AJ |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 18 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 18 | 1 | 1 | 2 | 2 | 2 | 18 | 18 | 2 | 2 | 18 | 18 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | S3 | D6 | M4(2) | D9 | C3×S3 | C4×S3 | D18 | S3×C6 | C3×M4(2) | C8⋊S3 | C3×D9 | C4×D9 | S3×C12 | C6×D9 | C8⋊D9 | C3×C8⋊S3 | C12×D9 | C3×C8⋊D9 |
| kernel | C3×C8⋊D9 | C3×C9⋊C8 | C3×C72 | C12×D9 | C8⋊D9 | C3×Dic9 | C6×D9 | C9⋊C8 | C72 | C4×D9 | Dic9 | D18 | C3×C24 | C3×C12 | C3×C9 | C24 | C24 | C3×C6 | C12 | C12 | C9 | C32 | C8 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 3 | 2 | 2 | 3 | 2 | 4 | 4 | 6 | 6 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C3×C8⋊D9 ►in GL2(𝔽73) generated by
| 8 | 0 |
| 0 | 8 |
| 22 | 0 |
| 0 | 51 |
| 32 | 0 |
| 0 | 16 |
| 0 | 16 |
| 32 | 0 |
G:=sub<GL(2,GF(73))| [8,0,0,8],[22,0,0,51],[32,0,0,16],[0,32,16,0] >;
C3×C8⋊D9 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes D_9
% in TeX
G:=Group("C3xC8:D9"); // GroupNames label
G:=SmallGroup(432,106);
// by ID
G=gap.SmallGroup(432,106);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,80,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations