direct product, metacyclic, supersoluble, monomial
Aliases: C9×C8⋊S3, C72⋊7S3, D6.C36, C24⋊5C18, C36.77D6, Dic3.C36, C3⋊C8⋊4C18, C8⋊3(S3×C9), (C3×C72)⋊14C2, C2.3(S3×C36), C6.2(C2×C36), (C3×C9)⋊4M4(2), (C4×S3).2C18, (S3×C36).4C2, (S3×C18).1C4, (S3×C6).1C12, (S3×C12).9C6, C18.22(C4×S3), C4.13(S3×C18), C24.25(C3×S3), C6.34(S3×C12), (C3×C24).23C6, C3⋊1(C9×M4(2)), C12.117(S3×C6), C12.13(C2×C18), (C9×Dic3).3C4, (C3×C36).51C22, (C3×Dic3).3C12, C32.2(C3×M4(2)), (C9×C3⋊C8)⋊11C2, (C3×C3⋊C8).7C6, (C3×C8⋊S3).C3, C3.4(C3×C8⋊S3), (C3×C6).38(C2×C12), (C3×C18).15(C2×C4), (C3×C12).88(C2×C6), SmallGroup(432,110)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C8⋊S3
G = < a,b,c,d | a9=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 124 in 68 conjugacy classes, 39 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), C18, C18, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, C3×C9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, C8⋊S3, C3×M4(2), S3×C9, C3×C18, C72, C72, C2×C36, C3×C3⋊C8, C3×C24, S3×C12, C9×Dic3, C3×C36, S3×C18, C9×M4(2), C3×C8⋊S3, C9×C3⋊C8, C3×C72, S3×C36, C9×C8⋊S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C9, C12, D6, C2×C6, M4(2), C18, C3×S3, C4×S3, C2×C12, C36, C2×C18, S3×C6, C8⋊S3, C3×M4(2), S3×C9, C2×C36, S3×C12, S3×C18, C9×M4(2), C3×C8⋊S3, S3×C36, C9×C8⋊S3
►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 89 53 67 31 63 44 80)(2 90 54 68 32 55 45 81)(3 82 46 69 33 56 37 73)(4 83 47 70 34 57 38 74)(5 84 48 71 35 58 39 75)(6 85 49 72 36 59 40 76)(7 86 50 64 28 60 41 77)(8 87 51 65 29 61 42 78)(9 88 52 66 30 62 43 79)(10 103 22 99 139 116 135 125)(11 104 23 91 140 117 127 126)(12 105 24 92 141 109 128 118)(13 106 25 93 142 110 129 119)(14 107 26 94 143 111 130 120)(15 108 27 95 144 112 131 121)(16 100 19 96 136 113 132 122)(17 101 20 97 137 114 133 123)(18 102 21 98 138 115 134 124)
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(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 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(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)
(1 116)(2 117)(3 109)(4 110)(5 111)(6 112)(7 113)(8 114)(9 115)(10 80)(11 81)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 86)(20 87)(21 88)(22 89)(23 90)(24 82)(25 83)(26 84)(27 85)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 91)(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)
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), (1,89,53,67,31,63,44,80)(2,90,54,68,32,55,45,81)(3,82,46,69,33,56,37,73)(4,83,47,70,34,57,38,74)(5,84,48,71,35,58,39,75)(6,85,49,72,36,59,40,76)(7,86,50,64,28,60,41,77)(8,87,51,65,29,61,42,78)(9,88,52,66,30,62,43,79)(10,103,22,99,139,116,135,125)(11,104,23,91,140,117,127,126)(12,105,24,92,141,109,128,118)(13,106,25,93,142,110,129,119)(14,107,26,94,143,111,130,120)(15,108,27,95,144,112,131,121)(16,100,19,96,136,113,132,122)(17,101,20,97,137,114,133,123)(18,102,21,98,138,115,134,124), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(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)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(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), (1,116)(2,117)(3,109)(4,110)(5,111)(6,112)(7,113)(8,114)(9,115)(10,80)(11,81)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,86)(20,87)(21,88)(22,89)(23,90)(24,82)(25,83)(26,84)(27,85)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,91)(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)>;
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), (1,89,53,67,31,63,44,80)(2,90,54,68,32,55,45,81)(3,82,46,69,33,56,37,73)(4,83,47,70,34,57,38,74)(5,84,48,71,35,58,39,75)(6,85,49,72,36,59,40,76)(7,86,50,64,28,60,41,77)(8,87,51,65,29,61,42,78)(9,88,52,66,30,62,43,79)(10,103,22,99,139,116,135,125)(11,104,23,91,140,117,127,126)(12,105,24,92,141,109,128,118)(13,106,25,93,142,110,129,119)(14,107,26,94,143,111,130,120)(15,108,27,95,144,112,131,121)(16,100,19,96,136,113,132,122)(17,101,20,97,137,114,133,123)(18,102,21,98,138,115,134,124), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(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)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(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), (1,116)(2,117)(3,109)(4,110)(5,111)(6,112)(7,113)(8,114)(9,115)(10,80)(11,81)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,86)(20,87)(21,88)(22,89)(23,90)(24,82)(25,83)(26,84)(27,85)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,91)(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) );
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)], [(1,89,53,67,31,63,44,80),(2,90,54,68,32,55,45,81),(3,82,46,69,33,56,37,73),(4,83,47,70,34,57,38,74),(5,84,48,71,35,58,39,75),(6,85,49,72,36,59,40,76),(7,86,50,64,28,60,41,77),(8,87,51,65,29,61,42,78),(9,88,52,66,30,62,43,79),(10,103,22,99,139,116,135,125),(11,104,23,91,140,117,127,126),(12,105,24,92,141,109,128,118),(13,106,25,93,142,110,129,119),(14,107,26,94,143,111,130,120),(15,108,27,95,144,112,131,121),(16,100,19,96,136,113,132,122),(17,101,20,97,137,114,133,123),(18,102,21,98,138,115,134,124)], [(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(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),(73,79,76),(74,80,77),(75,81,78),(82,88,85),(83,89,86),(84,90,87),(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)], [(1,116),(2,117),(3,109),(4,110),(5,111),(6,112),(7,113),(8,114),(9,115),(10,80),(11,81),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,86),(20,87),(21,88),(22,89),(23,90),(24,82),(25,83),(26,84),(27,85),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(44,99),(45,91),(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)]])
162 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 | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18R | 24A | ··· | 24P | 24Q | 24R | 24S | 24T | 36A | ··· | 36L | 36M | ··· | 36X | 36Y | ··· | 36AD | 72A | ··· | 72AJ | 72AK | ··· | 72AV |
| 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 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 | 72 | ··· | 72 |
| size | 1 | 1 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 6 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 2 | 2 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 |
| type | + | + | + | + | + | + | |||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C9 | C12 | C12 | C18 | C18 | C18 | C36 | C36 | S3 | D6 | M4(2) | C3×S3 | C4×S3 | S3×C6 | C8⋊S3 | C3×M4(2) | S3×C9 | S3×C12 | S3×C18 | C9×M4(2) | C3×C8⋊S3 | S3×C36 | C9×C8⋊S3 |
| kernel | C9×C8⋊S3 | C9×C3⋊C8 | C3×C72 | S3×C36 | C3×C8⋊S3 | C9×Dic3 | S3×C18 | C3×C3⋊C8 | C3×C24 | S3×C12 | C8⋊S3 | C3×Dic3 | S3×C6 | C3⋊C8 | C24 | C4×S3 | Dic3 | D6 | C72 | C36 | C3×C9 | C24 | C18 | C12 | C9 | C32 | C8 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 4 | 4 | 6 | 6 | 6 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 4 | 6 | 12 | 8 | 12 | 24 |
Matrix representation of C9×C8⋊S3 ►in GL2(𝔽73) generated by
| 2 | 0 |
| 0 | 2 |
| 10 | 0 |
| 0 | 63 |
| 64 | 0 |
| 0 | 8 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(73))| [2,0,0,2],[10,0,0,63],[64,0,0,8],[0,1,1,0] >;
C9×C8⋊S3 in GAP, Magma, Sage, TeX
C_9\times C_8\rtimes S_3
% in TeX
G:=Group("C9xC8:S3"); // GroupNames label
G:=SmallGroup(432,110);
// by ID
G=gap.SmallGroup(432,110);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,1037,92,142,192,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^8=c^3=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