Aliases: Q8⋊C9⋊3S3, Q8⋊3S3⋊C9, (C3×Q8).C18, C6.16(S3×A4), C3⋊(Q8.C18), Q8.2(S3×C9), (C3×Dic3).1A4, Dic3.(C3.A4), (Q8×C32).2C6, C32.2(C4.A4), C3.4(Dic3.A4), (C3×Q8⋊C9)⋊1C2, C6.1(C2×C3.A4), C2.2(S3×C3.A4), (C3×C6).11(C2×A4), (C3×Q8⋊3S3).C3, (C3×Q8).18(C3×S3), SmallGroup(432,267)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×Q8 — Q8⋊C9⋊3S3 |
Generators and relations for Q8⋊C9⋊3S3
G = < a,b,c,d,e | a4=c9=d3=e2=1, b2=a2, bab-1=a-1, cac-1=b, ad=da, ae=ea, cbc-1=ab, bd=db, ebe=a2b, cd=dc, ece=a-1bc, ede=d-1 >
Subgroups: 228 in 57 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C32, Dic3, C12, D6, C2×C6, C4○D4, C18, C3×S3, C3×C6, C4×S3, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, C3×C9, C36, C3×Dic3, C3×C12, S3×C6, Q8⋊3S3, C3×C4○D4, C3×C18, Q8⋊C9, Q8⋊C9, S3×C12, C3×D12, Q8×C32, C9×Dic3, Q8.C18, C3×Q8⋊3S3, C3×Q8⋊C9, Q8⋊C9⋊3S3
Quotients: C1, C2, C3, S3, C6, C9, A4, C18, C3×S3, C2×A4, C3.A4, C4.A4, S3×C9, C2×C3.A4, S3×A4, Q8.C18, Dic3.A4, S3×C3.A4, Q8⋊C9⋊3S3
►On 144 points
Generators in S144
(1 59 74 101)(2 40 75 10)(3 85 76 72)(4 62 77 104)(5 43 78 13)(6 88 79 66)(7 56 80 107)(8 37 81 16)(9 82 73 69)(11 103 41 61)(12 86 42 64)(14 106 44 55)(15 89 45 67)(17 100 38 58)(18 83 39 70)(19 126 92 134)(20 117 93 28)(21 137 94 51)(22 120 95 128)(23 111 96 31)(24 140 97 54)(25 123 98 131)(26 114 99 34)(27 143 91 48)(29 127 109 119)(30 138 110 52)(32 130 112 122)(33 141 113 46)(35 133 115 125)(36 144 116 49)(47 124 142 132)(50 118 136 135)(53 121 139 129)(57 90 108 68)(60 84 102 71)(63 87 105 65)
(1 39 74 18)(2 84 75 71)(3 61 76 103)(4 42 77 12)(5 87 78 65)(6 55 79 106)(7 45 80 15)(8 90 81 68)(9 58 73 100)(10 102 40 60)(11 85 41 72)(13 105 43 63)(14 88 44 66)(16 108 37 57)(17 82 38 69)(19 116 92 36)(20 136 93 50)(21 119 94 127)(22 110 95 30)(23 139 96 53)(24 122 97 130)(25 113 98 33)(26 142 99 47)(27 125 91 133)(28 135 117 118)(29 137 109 51)(31 129 111 121)(32 140 112 54)(34 132 114 124)(35 143 115 48)(46 123 141 131)(49 126 144 134)(52 120 138 128)(56 89 107 67)(59 83 101 70)(62 86 104 64)
(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 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 26)(2 133)(3 144)(4 20)(5 127)(6 138)(7 23)(8 130)(9 141)(10 35)(11 92)(12 136)(13 29)(14 95)(15 139)(16 32)(17 98)(18 142)(19 41)(21 65)(22 44)(24 68)(25 38)(27 71)(28 104)(30 66)(31 107)(33 69)(34 101)(36 72)(37 112)(39 47)(40 115)(42 50)(43 109)(45 53)(46 73)(48 102)(49 76)(51 105)(52 79)(54 108)(55 120)(56 111)(57 140)(58 123)(59 114)(60 143)(61 126)(62 117)(63 137)(64 118)(67 121)(70 124)(74 99)(75 125)(77 93)(78 119)(80 96)(81 122)(82 113)(83 132)(84 91)(85 116)(86 135)(87 94)(88 110)(89 129)(90 97)(100 131)(103 134)(106 128)
G:=sub<Sym(144)| (1,59,74,101)(2,40,75,10)(3,85,76,72)(4,62,77,104)(5,43,78,13)(6,88,79,66)(7,56,80,107)(8,37,81,16)(9,82,73,69)(11,103,41,61)(12,86,42,64)(14,106,44,55)(15,89,45,67)(17,100,38,58)(18,83,39,70)(19,126,92,134)(20,117,93,28)(21,137,94,51)(22,120,95,128)(23,111,96,31)(24,140,97,54)(25,123,98,131)(26,114,99,34)(27,143,91,48)(29,127,109,119)(30,138,110,52)(32,130,112,122)(33,141,113,46)(35,133,115,125)(36,144,116,49)(47,124,142,132)(50,118,136,135)(53,121,139,129)(57,90,108,68)(60,84,102,71)(63,87,105,65), (1,39,74,18)(2,84,75,71)(3,61,76,103)(4,42,77,12)(5,87,78,65)(6,55,79,106)(7,45,80,15)(8,90,81,68)(9,58,73,100)(10,102,40,60)(11,85,41,72)(13,105,43,63)(14,88,44,66)(16,108,37,57)(17,82,38,69)(19,116,92,36)(20,136,93,50)(21,119,94,127)(22,110,95,30)(23,139,96,53)(24,122,97,130)(25,113,98,33)(26,142,99,47)(27,125,91,133)(28,135,117,118)(29,137,109,51)(31,129,111,121)(32,140,112,54)(34,132,114,124)(35,143,115,48)(46,123,141,131)(49,126,144,134)(52,120,138,128)(56,89,107,67)(59,83,101,70)(62,86,104,64), (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), (1,4,7)(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), (1,26)(2,133)(3,144)(4,20)(5,127)(6,138)(7,23)(8,130)(9,141)(10,35)(11,92)(12,136)(13,29)(14,95)(15,139)(16,32)(17,98)(18,142)(19,41)(21,65)(22,44)(24,68)(25,38)(27,71)(28,104)(30,66)(31,107)(33,69)(34,101)(36,72)(37,112)(39,47)(40,115)(42,50)(43,109)(45,53)(46,73)(48,102)(49,76)(51,105)(52,79)(54,108)(55,120)(56,111)(57,140)(58,123)(59,114)(60,143)(61,126)(62,117)(63,137)(64,118)(67,121)(70,124)(74,99)(75,125)(77,93)(78,119)(80,96)(81,122)(82,113)(83,132)(84,91)(85,116)(86,135)(87,94)(88,110)(89,129)(90,97)(100,131)(103,134)(106,128)>;
G:=Group( (1,59,74,101)(2,40,75,10)(3,85,76,72)(4,62,77,104)(5,43,78,13)(6,88,79,66)(7,56,80,107)(8,37,81,16)(9,82,73,69)(11,103,41,61)(12,86,42,64)(14,106,44,55)(15,89,45,67)(17,100,38,58)(18,83,39,70)(19,126,92,134)(20,117,93,28)(21,137,94,51)(22,120,95,128)(23,111,96,31)(24,140,97,54)(25,123,98,131)(26,114,99,34)(27,143,91,48)(29,127,109,119)(30,138,110,52)(32,130,112,122)(33,141,113,46)(35,133,115,125)(36,144,116,49)(47,124,142,132)(50,118,136,135)(53,121,139,129)(57,90,108,68)(60,84,102,71)(63,87,105,65), (1,39,74,18)(2,84,75,71)(3,61,76,103)(4,42,77,12)(5,87,78,65)(6,55,79,106)(7,45,80,15)(8,90,81,68)(9,58,73,100)(10,102,40,60)(11,85,41,72)(13,105,43,63)(14,88,44,66)(16,108,37,57)(17,82,38,69)(19,116,92,36)(20,136,93,50)(21,119,94,127)(22,110,95,30)(23,139,96,53)(24,122,97,130)(25,113,98,33)(26,142,99,47)(27,125,91,133)(28,135,117,118)(29,137,109,51)(31,129,111,121)(32,140,112,54)(34,132,114,124)(35,143,115,48)(46,123,141,131)(49,126,144,134)(52,120,138,128)(56,89,107,67)(59,83,101,70)(62,86,104,64), (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), (1,4,7)(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), (1,26)(2,133)(3,144)(4,20)(5,127)(6,138)(7,23)(8,130)(9,141)(10,35)(11,92)(12,136)(13,29)(14,95)(15,139)(16,32)(17,98)(18,142)(19,41)(21,65)(22,44)(24,68)(25,38)(27,71)(28,104)(30,66)(31,107)(33,69)(34,101)(36,72)(37,112)(39,47)(40,115)(42,50)(43,109)(45,53)(46,73)(48,102)(49,76)(51,105)(52,79)(54,108)(55,120)(56,111)(57,140)(58,123)(59,114)(60,143)(61,126)(62,117)(63,137)(64,118)(67,121)(70,124)(74,99)(75,125)(77,93)(78,119)(80,96)(81,122)(82,113)(83,132)(84,91)(85,116)(86,135)(87,94)(88,110)(89,129)(90,97)(100,131)(103,134)(106,128) );
G=PermutationGroup([[(1,59,74,101),(2,40,75,10),(3,85,76,72),(4,62,77,104),(5,43,78,13),(6,88,79,66),(7,56,80,107),(8,37,81,16),(9,82,73,69),(11,103,41,61),(12,86,42,64),(14,106,44,55),(15,89,45,67),(17,100,38,58),(18,83,39,70),(19,126,92,134),(20,117,93,28),(21,137,94,51),(22,120,95,128),(23,111,96,31),(24,140,97,54),(25,123,98,131),(26,114,99,34),(27,143,91,48),(29,127,109,119),(30,138,110,52),(32,130,112,122),(33,141,113,46),(35,133,115,125),(36,144,116,49),(47,124,142,132),(50,118,136,135),(53,121,139,129),(57,90,108,68),(60,84,102,71),(63,87,105,65)], [(1,39,74,18),(2,84,75,71),(3,61,76,103),(4,42,77,12),(5,87,78,65),(6,55,79,106),(7,45,80,15),(8,90,81,68),(9,58,73,100),(10,102,40,60),(11,85,41,72),(13,105,43,63),(14,88,44,66),(16,108,37,57),(17,82,38,69),(19,116,92,36),(20,136,93,50),(21,119,94,127),(22,110,95,30),(23,139,96,53),(24,122,97,130),(25,113,98,33),(26,142,99,47),(27,125,91,133),(28,135,117,118),(29,137,109,51),(31,129,111,121),(32,140,112,54),(34,132,114,124),(35,143,115,48),(46,123,141,131),(49,126,144,134),(52,120,138,128),(56,89,107,67),(59,83,101,70),(62,86,104,64)], [(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)], [(1,4,7),(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)], [(1,26),(2,133),(3,144),(4,20),(5,127),(6,138),(7,23),(8,130),(9,141),(10,35),(11,92),(12,136),(13,29),(14,95),(15,139),(16,32),(17,98),(18,142),(19,41),(21,65),(22,44),(24,68),(25,38),(27,71),(28,104),(30,66),(31,107),(33,69),(34,101),(36,72),(37,112),(39,47),(40,115),(42,50),(43,109),(45,53),(46,73),(48,102),(49,76),(51,105),(52,79),(54,108),(55,120),(56,111),(57,140),(58,123),(59,114),(60,143),(61,126),(62,117),(63,137),(64,118),(67,121),(70,124),(74,99),(75,125),(77,93),(78,119),(80,96),(81,122),(82,113),(83,132),(84,91),(85,116),(86,135),(87,94),(88,110),(89,129),(90,97),(100,131),(103,134),(106,128)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 18A | ··· | 18F | 18G | ··· | 18L | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 18 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 1 | 1 | 2 | 2 | 2 | 18 | 18 | 4 | ··· | 4 | 8 | ··· | 8 | 3 | 3 | 3 | 3 | 6 | 6 | 12 | 12 | 12 | 4 | ··· | 4 | 8 | ··· | 8 | 12 | ··· | 12 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C4.A4 | S3×C9 | Q8.C18 | A4 | C2×A4 | C3.A4 | C2×C3.A4 | Dic3.A4 | Dic3.A4 | Q8⋊C9⋊3S3 | S3×A4 | S3×C3.A4 |
| kernel | Q8⋊C9⋊3S3 | C3×Q8⋊C9 | C3×Q8⋊3S3 | Q8×C32 | Q8⋊3S3 | C3×Q8 | Q8⋊C9 | C3×Q8 | C32 | Q8 | C3 | C3×Dic3 | C3×C6 | Dic3 | C6 | C3 | C3 | C1 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 6 | 6 | 12 | 1 | 1 | 2 | 2 | 1 | 2 | 6 | 1 | 2 |
Matrix representation of Q8⋊C9⋊3S3 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 17 |
| 0 | 0 | 26 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 35 |
| 0 | 0 | 1 | 36 |
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 26 | 22 |
| 0 | 0 | 0 | 10 |
| 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 | 6 | 9 |
| 0 | 0 | 29 | 31 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,36,26,0,0,17,1],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,35,36],[16,0,0,0,0,16,0,0,0,0,26,0,0,0,22,10],[26,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,6,29,0,0,9,31] >;
Q8⋊C9⋊3S3 in GAP, Magma, Sage, TeX
Q_8\rtimes C_9\rtimes_3S_3
% in TeX
G:=Group("Q8:C9:3S3"); // GroupNames label
G:=SmallGroup(432,267);
// by ID
G=gap.SmallGroup(432,267);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-3,-2,1512,50,766,360,326,515,242,6053]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^9=d^3=e^2=1,b^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,a*d=d*a,a*e=e*a,c*b*c^-1=a*b,b*d=d*b,e*b*e=a^2*b,c*d=d*c,e*c*e=a^-1*b*c,e*d*e=d^-1>;
// generators/relations