direct product, metabelian, supersoluble, monomial
Aliases: S3×Q8×C9, C36.49D6, Dic6⋊4C18, C3⋊2(Q8×C18), C4.6(S3×C18), (C3×Q8)⋊4C18, (C4×S3).1C18, (S3×C36).5C2, (S3×C12).4C6, C12.55(S3×C6), C12.6(C2×C18), D6.5(C2×C18), (C9×Dic6)⋊10C2, C32.3(C6×Q8), C6.7(C22×C18), (C3×Dic6).4C6, (C3×C18).34C23, (C3×C36).48C22, C18.55(C22×S3), Dic3.4(C2×C18), (Q8×C32).20C6, (S3×C18).16C22, (C9×Dic3).15C22, (Q8×C3×C9)⋊9C2, (C3×C9)⋊8(C2×Q8), C3.4(C3×S3×Q8), C6.68(S3×C2×C6), C2.8(S3×C2×C18), (C3×S3).(C3×Q8), (C3×S3×Q8).2C3, (S3×C6).20(C2×C6), (C3×C12).33(C2×C6), (C3×Q8).37(C3×S3), (C3×C6).44(C22×C6), (C3×Dic3).14(C2×C6), SmallGroup(432,366)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×Q8×C9
G = < a,b,c,d,e | a9=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 216 in 126 conjugacy classes, 75 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, C18, C18, C3×S3, C3×C6, Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, C3×C9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, S3×Q8, C6×Q8, S3×C9, C3×C18, C2×C36, Q8×C9, Q8×C9, C3×Dic6, S3×C12, Q8×C32, C9×Dic3, C3×C36, S3×C18, Q8×C18, C3×S3×Q8, C9×Dic6, S3×C36, Q8×C3×C9, S3×Q8×C9
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, C9, D6, C2×C6, C2×Q8, C18, C3×S3, C3×Q8, C22×S3, C22×C6, C2×C18, S3×C6, S3×Q8, C6×Q8, S3×C9, Q8×C9, C22×C18, S3×C2×C6, S3×C18, Q8×C18, C3×S3×Q8, S3×C2×C18, S3×Q8×C9
►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 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(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 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)(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 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 91)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 64)(17 65)(18 66)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 82)(27 83)(28 116)(29 117)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 100)(45 101)(46 134)(47 135)(48 127)(49 128)(50 129)(51 130)(52 131)(53 132)(54 133)(55 120)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 118)(63 119)(73 138)(74 139)(75 140)(76 141)(77 142)(78 143)(79 144)(80 136)(81 137)
(1 38 20 31)(2 39 21 32)(3 40 22 33)(4 41 23 34)(5 42 24 35)(6 43 25 36)(7 44 26 28)(8 45 27 29)(9 37 19 30)(10 121 139 128)(11 122 140 129)(12 123 141 130)(13 124 142 131)(14 125 143 132)(15 126 144 133)(16 118 136 134)(17 119 137 135)(18 120 138 127)(46 64 62 80)(47 65 63 81)(48 66 55 73)(49 67 56 74)(50 68 57 75)(51 69 58 76)(52 70 59 77)(53 71 60 78)(54 72 61 79)(82 116 98 100)(83 117 99 101)(84 109 91 102)(85 110 92 103)(86 111 93 104)(87 112 94 105)(88 113 95 106)(89 114 96 107)(90 115 97 108)
(1 56 20 49)(2 57 21 50)(3 58 22 51)(4 59 23 52)(5 60 24 53)(6 61 25 54)(7 62 26 46)(8 63 27 47)(9 55 19 48)(10 110 139 103)(11 111 140 104)(12 112 141 105)(13 113 142 106)(14 114 143 107)(15 115 144 108)(16 116 136 100)(17 117 137 101)(18 109 138 102)(28 80 44 64)(29 81 45 65)(30 73 37 66)(31 74 38 67)(32 75 39 68)(33 76 40 69)(34 77 41 70)(35 78 42 71)(36 79 43 72)(82 134 98 118)(83 135 99 119)(84 127 91 120)(85 128 92 121)(86 129 93 122)(87 130 94 123)(88 131 95 124)(89 132 96 125)(90 133 97 126)
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), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(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)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(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), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,91)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,64)(17,65)(18,66)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,82)(27,83)(28,116)(29,117)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,100)(45,101)(46,134)(47,135)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,118)(63,119)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(79,144)(80,136)(81,137), (1,38,20,31)(2,39,21,32)(3,40,22,33)(4,41,23,34)(5,42,24,35)(6,43,25,36)(7,44,26,28)(8,45,27,29)(9,37,19,30)(10,121,139,128)(11,122,140,129)(12,123,141,130)(13,124,142,131)(14,125,143,132)(15,126,144,133)(16,118,136,134)(17,119,137,135)(18,120,138,127)(46,64,62,80)(47,65,63,81)(48,66,55,73)(49,67,56,74)(50,68,57,75)(51,69,58,76)(52,70,59,77)(53,71,60,78)(54,72,61,79)(82,116,98,100)(83,117,99,101)(84,109,91,102)(85,110,92,103)(86,111,93,104)(87,112,94,105)(88,113,95,106)(89,114,96,107)(90,115,97,108), (1,56,20,49)(2,57,21,50)(3,58,22,51)(4,59,23,52)(5,60,24,53)(6,61,25,54)(7,62,26,46)(8,63,27,47)(9,55,19,48)(10,110,139,103)(11,111,140,104)(12,112,141,105)(13,113,142,106)(14,114,143,107)(15,115,144,108)(16,116,136,100)(17,117,137,101)(18,109,138,102)(28,80,44,64)(29,81,45,65)(30,73,37,66)(31,74,38,67)(32,75,39,68)(33,76,40,69)(34,77,41,70)(35,78,42,71)(36,79,43,72)(82,134,98,118)(83,135,99,119)(84,127,91,120)(85,128,92,121)(86,129,93,122)(87,130,94,123)(88,131,95,124)(89,132,96,125)(90,133,97,126)>;
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), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36)(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)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(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), (1,92)(2,93)(3,94)(4,95)(5,96)(6,97)(7,98)(8,99)(9,91)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,64)(17,65)(18,66)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,82)(27,83)(28,116)(29,117)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,100)(45,101)(46,134)(47,135)(48,127)(49,128)(50,129)(51,130)(52,131)(53,132)(54,133)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,118)(63,119)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(79,144)(80,136)(81,137), (1,38,20,31)(2,39,21,32)(3,40,22,33)(4,41,23,34)(5,42,24,35)(6,43,25,36)(7,44,26,28)(8,45,27,29)(9,37,19,30)(10,121,139,128)(11,122,140,129)(12,123,141,130)(13,124,142,131)(14,125,143,132)(15,126,144,133)(16,118,136,134)(17,119,137,135)(18,120,138,127)(46,64,62,80)(47,65,63,81)(48,66,55,73)(49,67,56,74)(50,68,57,75)(51,69,58,76)(52,70,59,77)(53,71,60,78)(54,72,61,79)(82,116,98,100)(83,117,99,101)(84,109,91,102)(85,110,92,103)(86,111,93,104)(87,112,94,105)(88,113,95,106)(89,114,96,107)(90,115,97,108), (1,56,20,49)(2,57,21,50)(3,58,22,51)(4,59,23,52)(5,60,24,53)(6,61,25,54)(7,62,26,46)(8,63,27,47)(9,55,19,48)(10,110,139,103)(11,111,140,104)(12,112,141,105)(13,113,142,106)(14,114,143,107)(15,115,144,108)(16,116,136,100)(17,117,137,101)(18,109,138,102)(28,80,44,64)(29,81,45,65)(30,73,37,66)(31,74,38,67)(32,75,39,68)(33,76,40,69)(34,77,41,70)(35,78,42,71)(36,79,43,72)(82,134,98,118)(83,135,99,119)(84,127,91,120)(85,128,92,121)(86,129,93,122)(87,130,94,123)(88,131,95,124)(89,132,96,125)(90,133,97,126) );
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)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36),(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),(82,88,85),(83,89,86),(84,90,87),(91,97,94),(92,98,95),(93,99,96),(100,106,103),(101,107,104),(102,108,105),(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)], [(1,92),(2,93),(3,94),(4,95),(5,96),(6,97),(7,98),(8,99),(9,91),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,64),(17,65),(18,66),(19,84),(20,85),(21,86),(22,87),(23,88),(24,89),(25,90),(26,82),(27,83),(28,116),(29,117),(30,109),(31,110),(32,111),(33,112),(34,113),(35,114),(36,115),(37,102),(38,103),(39,104),(40,105),(41,106),(42,107),(43,108),(44,100),(45,101),(46,134),(47,135),(48,127),(49,128),(50,129),(51,130),(52,131),(53,132),(54,133),(55,120),(56,121),(57,122),(58,123),(59,124),(60,125),(61,126),(62,118),(63,119),(73,138),(74,139),(75,140),(76,141),(77,142),(78,143),(79,144),(80,136),(81,137)], [(1,38,20,31),(2,39,21,32),(3,40,22,33),(4,41,23,34),(5,42,24,35),(6,43,25,36),(7,44,26,28),(8,45,27,29),(9,37,19,30),(10,121,139,128),(11,122,140,129),(12,123,141,130),(13,124,142,131),(14,125,143,132),(15,126,144,133),(16,118,136,134),(17,119,137,135),(18,120,138,127),(46,64,62,80),(47,65,63,81),(48,66,55,73),(49,67,56,74),(50,68,57,75),(51,69,58,76),(52,70,59,77),(53,71,60,78),(54,72,61,79),(82,116,98,100),(83,117,99,101),(84,109,91,102),(85,110,92,103),(86,111,93,104),(87,112,94,105),(88,113,95,106),(89,114,96,107),(90,115,97,108)], [(1,56,20,49),(2,57,21,50),(3,58,22,51),(4,59,23,52),(5,60,24,53),(6,61,25,54),(7,62,26,46),(8,63,27,47),(9,55,19,48),(10,110,139,103),(11,111,140,104),(12,112,141,105),(13,113,142,106),(14,114,143,107),(15,115,144,108),(16,116,136,100),(17,117,137,101),(18,109,138,102),(28,80,44,64),(29,81,45,65),(30,73,37,66),(31,74,38,67),(32,75,39,68),(33,76,40,69),(34,77,41,70),(35,78,42,71),(36,79,43,72),(82,134,98,118),(83,135,99,119),(84,127,91,120),(85,128,92,121),(86,129,93,122),(87,130,94,123),(88,131,95,124),(89,132,96,125),(90,133,97,126)]])
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12F | 12G | ··· | 12O | 12P | ··· | 12U | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18X | 36A | ··· | 36R | 36S | ··· | 36AJ | 36AK | ··· | 36BB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | S3 | Q8 | D6 | C3×S3 | C3×Q8 | S3×C6 | S3×C9 | Q8×C9 | S3×C18 | S3×Q8 | C3×S3×Q8 | S3×Q8×C9 |
| kernel | S3×Q8×C9 | C9×Dic6 | S3×C36 | Q8×C3×C9 | C3×S3×Q8 | C3×Dic6 | S3×C12 | Q8×C32 | S3×Q8 | Dic6 | C4×S3 | C3×Q8 | Q8×C9 | S3×C9 | C36 | C3×Q8 | C3×S3 | C12 | Q8 | S3 | C4 | C9 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 6 | 18 | 18 | 6 | 1 | 2 | 3 | 2 | 4 | 6 | 6 | 12 | 18 | 1 | 2 | 6 |
Matrix representation of S3×Q8×C9 ►in GL4(𝔽37) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 26 | 0 |
| 0 | 0 | 27 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 36 |
| 0 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 27 | 0 | 0 |
| 27 | 26 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
G:=sub<GL(4,GF(37))| [10,0,0,0,0,10,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,26,27,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,9,36],[0,36,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[11,27,0,0,27,26,0,0,0,0,36,0,0,0,0,36] >;
S3×Q8×C9 in GAP, Magma, Sage, TeX
S_3\times Q_8\times C_9
% in TeX
G:=Group("S3xQ8xC9"); // GroupNames label
G:=SmallGroup(432,366);
// by ID
G=gap.SmallGroup(432,366);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,303,142,192,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^3=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations