direct product, non-abelian, soluble
Aliases: S3×Q8⋊C9, (S3×Q8)⋊C9, (C3×Q8)⋊C18, Q8⋊2(S3×C9), (S3×C6).1A4, C6.17(S3×A4), D6.(C3.A4), (Q8×C32).3C6, (C3×S3).SL2(𝔽3), C3.4(S3×SL2(𝔽3)), C32.2(C2×SL2(𝔽3)), C3⋊(C2×Q8⋊C9), (C3×S3×Q8).C3, (C3×Q8⋊C9)⋊2C2, C6.2(C2×C3.A4), C2.3(S3×C3.A4), (C3×C6).12(C2×A4), (C3×Q8).19(C3×S3), SmallGroup(432,268)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×Q8 — S3×Q8⋊C9 |
Generators and relations for S3×Q8⋊C9
G = < a,b,c,d,e | a3=b2=c4=e9=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >
Subgroups: 228 in 65 conjugacy classes, 23 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C9, C32, Dic3, C12, D6, C2×C6, C2×Q8, C18, C3×S3, C3×C6, Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, C3×C9, C2×C18, C3×Dic3, C3×C12, S3×C6, S3×Q8, C6×Q8, S3×C9, C3×C18, Q8⋊C9, Q8⋊C9, C3×Dic6, S3×C12, Q8×C32, S3×C18, C2×Q8⋊C9, C3×S3×Q8, C3×Q8⋊C9, S3×Q8⋊C9
Quotients: C1, C2, C3, S3, C6, C9, A4, C18, C3×S3, SL2(𝔽3), C2×A4, C3.A4, C2×SL2(𝔽3), S3×C9, Q8⋊C9, C2×C3.A4, S3×A4, C2×Q8⋊C9, S3×SL2(𝔽3), S3×C3.A4, S3×Q8⋊C9
►On 144 points
Generators in S144
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(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 52 49)(47 53 50)(48 54 51)(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 97 94)(92 98 95)(93 99 96)(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 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 91)(8 92)(9 93)(10 111)(11 112)(12 113)(13 114)(14 115)(15 116)(16 117)(17 109)(18 110)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 100)(26 101)(27 102)(28 77)(29 78)(30 79)(31 80)(32 81)(33 73)(34 74)(35 75)(36 76)(37 122)(38 123)(39 124)(40 125)(41 126)(42 118)(43 119)(44 120)(45 121)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 135)(62 127)(63 128)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(1 39 74 58)(2 84 75 71)(3 11 76 103)(4 42 77 61)(5 87 78 65)(6 14 79 106)(7 45 80 55)(8 90 81 68)(9 17 73 100)(10 59 102 40)(12 86 104 64)(13 62 105 43)(15 89 107 67)(16 56 108 37)(18 83 101 70)(19 96 112 36)(20 136 113 50)(21 119 114 127)(22 99 115 30)(23 139 116 53)(24 122 117 130)(25 93 109 33)(26 142 110 47)(27 125 111 133)(28 135 97 118)(29 137 98 51)(31 129 91 121)(32 140 92 54)(34 132 94 124)(35 143 95 48)(38 69 57 82)(41 72 60 85)(44 66 63 88)(46 123 141 131)(49 126 144 134)(52 120 138 128)
(1 83 74 70)(2 10 75 102)(3 41 76 60)(4 86 77 64)(5 13 78 105)(6 44 79 63)(7 89 80 67)(8 16 81 108)(9 38 73 57)(11 85 103 72)(12 61 104 42)(14 88 106 66)(15 55 107 45)(17 82 100 69)(18 58 101 39)(19 144 112 49)(20 118 113 135)(21 98 114 29)(22 138 115 52)(23 121 116 129)(24 92 117 32)(25 141 109 46)(26 124 110 132)(27 95 111 35)(28 136 97 50)(30 128 99 120)(31 139 91 53)(33 131 93 123)(34 142 94 47)(36 134 96 126)(37 68 56 90)(40 71 59 84)(43 65 62 87)(48 125 143 133)(51 119 137 127)(54 122 140 130)
(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)
G:=sub<Sym(144)| (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(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)(46,52,49)(47,53,50)(48,54,51)(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)(91,97,94)(92,98,95)(93,99,96)(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), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,91)(8,92)(9,93)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,109)(18,110)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,100)(26,101)(27,102)(28,77)(29,78)(30,79)(31,80)(32,81)(33,73)(34,74)(35,75)(36,76)(37,122)(38,123)(39,124)(40,125)(41,126)(42,118)(43,119)(44,120)(45,121)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,127)(63,128)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (1,39,74,58)(2,84,75,71)(3,11,76,103)(4,42,77,61)(5,87,78,65)(6,14,79,106)(7,45,80,55)(8,90,81,68)(9,17,73,100)(10,59,102,40)(12,86,104,64)(13,62,105,43)(15,89,107,67)(16,56,108,37)(18,83,101,70)(19,96,112,36)(20,136,113,50)(21,119,114,127)(22,99,115,30)(23,139,116,53)(24,122,117,130)(25,93,109,33)(26,142,110,47)(27,125,111,133)(28,135,97,118)(29,137,98,51)(31,129,91,121)(32,140,92,54)(34,132,94,124)(35,143,95,48)(38,69,57,82)(41,72,60,85)(44,66,63,88)(46,123,141,131)(49,126,144,134)(52,120,138,128), (1,83,74,70)(2,10,75,102)(3,41,76,60)(4,86,77,64)(5,13,78,105)(6,44,79,63)(7,89,80,67)(8,16,81,108)(9,38,73,57)(11,85,103,72)(12,61,104,42)(14,88,106,66)(15,55,107,45)(17,82,100,69)(18,58,101,39)(19,144,112,49)(20,118,113,135)(21,98,114,29)(22,138,115,52)(23,121,116,129)(24,92,117,32)(25,141,109,46)(26,124,110,132)(27,95,111,35)(28,136,97,50)(30,128,99,120)(31,139,91,53)(33,131,93,123)(34,142,94,47)(36,134,96,126)(37,68,56,90)(40,71,59,84)(43,65,62,87)(48,125,143,133)(51,119,137,127)(54,122,140,130), (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)>;
G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(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)(46,52,49)(47,53,50)(48,54,51)(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)(91,97,94)(92,98,95)(93,99,96)(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), (1,94)(2,95)(3,96)(4,97)(5,98)(6,99)(7,91)(8,92)(9,93)(10,111)(11,112)(12,113)(13,114)(14,115)(15,116)(16,117)(17,109)(18,110)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,100)(26,101)(27,102)(28,77)(29,78)(30,79)(31,80)(32,81)(33,73)(34,74)(35,75)(36,76)(37,122)(38,123)(39,124)(40,125)(41,126)(42,118)(43,119)(44,120)(45,121)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,127)(63,128)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (1,39,74,58)(2,84,75,71)(3,11,76,103)(4,42,77,61)(5,87,78,65)(6,14,79,106)(7,45,80,55)(8,90,81,68)(9,17,73,100)(10,59,102,40)(12,86,104,64)(13,62,105,43)(15,89,107,67)(16,56,108,37)(18,83,101,70)(19,96,112,36)(20,136,113,50)(21,119,114,127)(22,99,115,30)(23,139,116,53)(24,122,117,130)(25,93,109,33)(26,142,110,47)(27,125,111,133)(28,135,97,118)(29,137,98,51)(31,129,91,121)(32,140,92,54)(34,132,94,124)(35,143,95,48)(38,69,57,82)(41,72,60,85)(44,66,63,88)(46,123,141,131)(49,126,144,134)(52,120,138,128), (1,83,74,70)(2,10,75,102)(3,41,76,60)(4,86,77,64)(5,13,78,105)(6,44,79,63)(7,89,80,67)(8,16,81,108)(9,38,73,57)(11,85,103,72)(12,61,104,42)(14,88,106,66)(15,55,107,45)(17,82,100,69)(18,58,101,39)(19,144,112,49)(20,118,113,135)(21,98,114,29)(22,138,115,52)(23,121,116,129)(24,92,117,32)(25,141,109,46)(26,124,110,132)(27,95,111,35)(28,136,97,50)(30,128,99,120)(31,139,91,53)(33,131,93,123)(34,142,94,47)(36,134,96,126)(37,68,56,90)(40,71,59,84)(43,65,62,87)(48,125,143,133)(51,119,137,127)(54,122,140,130), (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) );
G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(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),(46,52,49),(47,53,50),(48,54,51),(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),(91,97,94),(92,98,95),(93,99,96),(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)], [(1,94),(2,95),(3,96),(4,97),(5,98),(6,99),(7,91),(8,92),(9,93),(10,111),(11,112),(12,113),(13,114),(14,115),(15,116),(16,117),(17,109),(18,110),(19,103),(20,104),(21,105),(22,106),(23,107),(24,108),(25,100),(26,101),(27,102),(28,77),(29,78),(30,79),(31,80),(32,81),(33,73),(34,74),(35,75),(36,76),(37,122),(38,123),(39,124),(40,125),(41,126),(42,118),(43,119),(44,120),(45,121),(46,82),(47,83),(48,84),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,129),(56,130),(57,131),(58,132),(59,133),(60,134),(61,135),(62,127),(63,128),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)], [(1,39,74,58),(2,84,75,71),(3,11,76,103),(4,42,77,61),(5,87,78,65),(6,14,79,106),(7,45,80,55),(8,90,81,68),(9,17,73,100),(10,59,102,40),(12,86,104,64),(13,62,105,43),(15,89,107,67),(16,56,108,37),(18,83,101,70),(19,96,112,36),(20,136,113,50),(21,119,114,127),(22,99,115,30),(23,139,116,53),(24,122,117,130),(25,93,109,33),(26,142,110,47),(27,125,111,133),(28,135,97,118),(29,137,98,51),(31,129,91,121),(32,140,92,54),(34,132,94,124),(35,143,95,48),(38,69,57,82),(41,72,60,85),(44,66,63,88),(46,123,141,131),(49,126,144,134),(52,120,138,128)], [(1,83,74,70),(2,10,75,102),(3,41,76,60),(4,86,77,64),(5,13,78,105),(6,44,79,63),(7,89,80,67),(8,16,81,108),(9,38,73,57),(11,85,103,72),(12,61,104,42),(14,88,106,66),(15,55,107,45),(17,82,100,69),(18,58,101,39),(19,144,112,49),(20,118,113,135),(21,98,114,29),(22,138,115,52),(23,121,116,129),(24,92,117,32),(25,141,109,46),(26,124,110,132),(27,95,111,35),(28,136,97,50),(30,128,99,120),(31,139,91,53),(33,131,93,123),(34,142,94,47),(36,134,96,126),(37,68,56,90),(40,71,59,84),(43,65,62,87),(48,125,143,133),(51,119,137,127),(54,122,140,130)], [(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)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18X |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 6 | 18 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 8 | ··· | 8 | 6 | 6 | 12 | 12 | 12 | 18 | 18 | 4 | ··· | 4 | 8 | ··· | 8 | 12 | ··· | 12 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 |
| type | + | + | + | - | + | + | - | + | |||||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | S3 | C3×S3 | SL2(𝔽3) | SL2(𝔽3) | S3×C9 | Q8⋊C9 | A4 | C2×A4 | C3.A4 | C2×C3.A4 | S3×SL2(𝔽3) | S3×SL2(𝔽3) | S3×Q8⋊C9 | S3×A4 | S3×C3.A4 |
| kernel | S3×Q8⋊C9 | C3×Q8⋊C9 | C3×S3×Q8 | Q8×C32 | S3×Q8 | C3×Q8 | Q8⋊C9 | C3×Q8 | C3×S3 | C3×S3 | Q8 | S3 | S3×C6 | C3×C6 | D6 | C6 | C3 | C3 | C1 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 2 | 4 | 6 | 12 | 1 | 1 | 2 | 2 | 1 | 2 | 6 | 1 | 2 |
Matrix representation of S3×Q8⋊C9 ►in GL4(𝔽37) generated by
| 26 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 14 |
| 0 | 0 | 14 | 32 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 27 |
| 0 | 0 | 28 | 32 |
G:=sub<GL(4,GF(37))| [26,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,0,36,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,5,14,0,0,14,32],[16,0,0,0,0,16,0,0,0,0,4,28,0,0,27,32] >;
S3×Q8⋊C9 in GAP, Magma, Sage, TeX
S_3\times Q_8\rtimes C_9
% in TeX
G:=Group("S3xQ8:C9"); // GroupNames label
G:=SmallGroup(432,268);
// by ID
G=gap.SmallGroup(432,268);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-3,-2,50,766,360,326,515,242,6053]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^9=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations