direct product, metabelian, supersoluble, monomial
Aliases: C9×Q8⋊3S3, D12⋊4C18, C36.50D6, Q8⋊4(S3×C9), (Q8×C9)⋊7S3, (C4×S3)⋊3C18, (S3×C36)⋊9C2, C4.7(S3×C18), (C3×Q8)⋊5C18, (C9×D12)⋊10C2, (S3×C12).5C6, C12.56(S3×C6), C12.7(C2×C18), D6.3(C2×C18), (C3×D12).4C6, C6.8(C22×C18), (S3×C18).7C22, (C3×C18).35C23, (C3×C36).49C22, C18.56(C22×S3), Dic3.5(C2×C18), (Q8×C32).21C6, (C9×Dic3).17C22, C3⋊3(C9×C4○D4), C2.9(S3×C2×C18), C6.69(S3×C2×C6), (Q8×C3×C9)⋊10C2, (C3×C9)⋊17(C4○D4), (S3×C6).9(C2×C6), (C3×C12).34(C2×C6), (C3×Q8).38(C3×S3), C3.4(C3×Q8⋊3S3), C32.4(C3×C4○D4), (C3×C6).45(C22×C6), (C3×Q8⋊3S3).2C3, (C3×Dic3).20(C2×C6), SmallGroup(432,367)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×Q8⋊3S3
G = < a,b,c,d,e | a9=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >
Subgroups: 264 in 132 conjugacy classes, 69 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, C18, C18, C3×S3, C3×C6, C4×S3, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, C3×C9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, Q8⋊3S3, C3×C4○D4, S3×C9, C3×C18, C2×C36, D4×C9, Q8×C9, Q8×C9, S3×C12, C3×D12, Q8×C32, C9×Dic3, C3×C36, S3×C18, C9×C4○D4, C3×Q8⋊3S3, S3×C36, C9×D12, Q8×C3×C9, C9×Q8⋊3S3
Quotients: C1, C2, C3, C22, S3, C6, C23, C9, D6, C2×C6, C4○D4, C18, C3×S3, C22×S3, C22×C6, C2×C18, S3×C6, Q8⋊3S3, C3×C4○D4, S3×C9, C22×C18, S3×C2×C6, S3×C18, C9×C4○D4, C3×Q8⋊3S3, S3×C2×C18, C9×Q8⋊3S3
►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 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)(10 128 139 121)(11 129 140 122)(12 130 141 123)(13 131 142 124)(14 132 143 125)(15 133 144 126)(16 134 136 118)(17 135 137 119)(18 127 138 120)(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)(82 100 98 116)(83 101 99 117)(84 102 91 109)(85 103 92 110)(86 104 93 111)(87 105 94 112)(88 106 95 113)(89 107 96 114)(90 108 97 115)
(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)
(1 7 4)(2 8 5)(3 9 6)(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 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 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 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 82)(8 83)(9 84)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 81)(18 73)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(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), (1,38,20,31)(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)(10,128,139,121)(11,129,140,122)(12,130,141,123)(13,131,142,124)(14,132,143,125)(15,133,144,126)(16,134,136,118)(17,135,137,119)(18,127,138,120)(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)(82,100,98,116)(83,101,99,117)(84,102,91,109)(85,103,92,110)(86,104,93,111)(87,105,94,112)(88,106,95,113)(89,107,96,114)(90,108,97,115), (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), (1,7,4)(2,8,5)(3,9,6)(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)(37,43,40)(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)(82,85,88)(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), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,82)(8,83)(9,84)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,73)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(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), (1,38,20,31)(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)(10,128,139,121)(11,129,140,122)(12,130,141,123)(13,131,142,124)(14,132,143,125)(15,133,144,126)(16,134,136,118)(17,135,137,119)(18,127,138,120)(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)(82,100,98,116)(83,101,99,117)(84,102,91,109)(85,103,92,110)(86,104,93,111)(87,105,94,112)(88,106,95,113)(89,107,96,114)(90,108,97,115), (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), (1,7,4)(2,8,5)(3,9,6)(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)(37,43,40)(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)(82,85,88)(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), (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,82)(8,83)(9,84)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,81)(18,73)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(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)], [(1,38,20,31),(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),(10,128,139,121),(11,129,140,122),(12,130,141,123),(13,131,142,124),(14,132,143,125),(15,133,144,126),(16,134,136,118),(17,135,137,119),(18,127,138,120),(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),(82,100,98,116),(83,101,99,117),(84,102,91,109),(85,103,92,110),(86,104,93,111),(87,105,94,112),(88,106,95,113),(89,107,96,114),(90,108,97,115)], [(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)], [(1,7,4),(2,8,5),(3,9,6),(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),(37,43,40),(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),(82,85,88),(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)], [(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,82),(8,83),(9,84),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,81),(18,73),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(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)]])
135 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | ··· | 6K | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12F | 12G | 12H | 12I | 12J | 12K | ··· | 12S | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18AD | 36A | ··· | 36R | 36S | ··· | 36AD | 36AE | ··· | 36AV |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 |
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 | D6 | C4○D4 | C3×S3 | S3×C6 | C3×C4○D4 | S3×C9 | S3×C18 | C9×C4○D4 | Q8⋊3S3 | C3×Q8⋊3S3 | C9×Q8⋊3S3 |
| kernel | C9×Q8⋊3S3 | S3×C36 | C9×D12 | Q8×C3×C9 | C3×Q8⋊3S3 | S3×C12 | C3×D12 | Q8×C32 | Q8⋊3S3 | C4×S3 | D12 | C3×Q8 | Q8×C9 | C36 | C3×C9 | C3×Q8 | C12 | C32 | Q8 | C4 | C3 | C9 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 6 | 18 | 18 | 6 | 1 | 3 | 2 | 2 | 6 | 4 | 6 | 18 | 12 | 1 | 2 | 6 |
Matrix representation of C9×Q8⋊3S3 ►in GL4(𝔽37) generated by
| 26 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 35 | 0 | 0 |
| 1 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 26 | 16 | 0 | 0 |
| 34 | 11 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 26 |
| 8 | 22 | 0 | 0 |
| 19 | 29 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(37))| [26,0,0,0,0,26,0,0,0,0,16,0,0,0,0,16],[1,1,0,0,35,36,0,0,0,0,1,0,0,0,0,1],[26,34,0,0,16,11,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,26],[8,19,0,0,22,29,0,0,0,0,0,1,0,0,1,0] >;
C9×Q8⋊3S3 in GAP, Magma, Sage, TeX
C_9\times Q_8\rtimes_3S_3
% in TeX
G:=Group("C9xQ8:3S3"); // GroupNames label
G:=SmallGroup(432,367);
// by ID
G=gap.SmallGroup(432,367);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,142,192,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations