direct product, metabelian, supersoluble, monomial
Aliases: C9×Q8⋊2S3, C36.46D6, D12.2C18, C3⋊C8⋊3C18, (Q8×C9)⋊6S3, Q8⋊3(S3×C9), C6.9(D4×C9), C4.3(S3×C18), (C3×Q8)⋊3C18, C3⋊3(C9×SD16), (C3×C9)⋊13SD16, C12.51(S3×C6), C12.3(C2×C18), (C3×D12).2C6, (C9×D12).4C2, (C3×C18).36D4, C18.33(C3⋊D4), (C3×C36).45C22, (Q8×C32).18C6, C32.4(C3×SD16), (C9×C3⋊C8)⋊10C2, (Q8×C3×C9)⋊7C2, (C3×C3⋊C8).6C6, C2.6(C9×C3⋊D4), (C3×C6).57(C3×D4), C6.47(C3×C3⋊D4), (C3×Q8⋊2S3).C3, (C3×C12).29(C2×C6), (C3×Q8).34(C3×S3), C3.4(C3×Q8⋊2S3), SmallGroup(432,158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×Q8⋊2S3
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, ece=b-1c, ede=d-1 >
Subgroups: 168 in 72 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, C12, C12, D6, C2×C6, SD16, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×Q8, C3×Q8, C3×C9, C36, C36, C2×C18, C3×C12, C3×C12, S3×C6, Q8⋊2S3, C3×SD16, S3×C9, C3×C18, C72, D4×C9, Q8×C9, Q8×C9, C3×C3⋊C8, C3×D12, Q8×C32, C3×C36, C3×C36, S3×C18, C9×SD16, C3×Q8⋊2S3, C9×C3⋊C8, C9×D12, Q8×C3×C9, C9×Q8⋊2S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C9, D6, C2×C6, SD16, C18, C3×S3, C3⋊D4, C3×D4, C2×C18, S3×C6, Q8⋊2S3, C3×SD16, S3×C9, D4×C9, C3×C3⋊D4, S3×C18, C9×SD16, C3×Q8⋊2S3, C9×C3⋊D4, C9×Q8⋊2S3
►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 51 42 32)(2 52 43 33)(3 53 44 34)(4 54 45 35)(5 46 37 36)(6 47 38 28)(7 48 39 29)(8 49 40 30)(9 50 41 31)(10 21 138 134)(11 22 139 135)(12 23 140 127)(13 24 141 128)(14 25 142 129)(15 26 143 130)(16 27 144 131)(17 19 136 132)(18 20 137 133)(55 68 88 79)(56 69 89 80)(57 70 90 81)(58 71 82 73)(59 72 83 74)(60 64 84 75)(61 65 85 76)(62 66 86 77)(63 67 87 78)(91 104 124 115)(92 105 125 116)(93 106 126 117)(94 107 118 109)(95 108 119 110)(96 100 120 111)(97 101 121 112)(98 102 122 113)(99 103 123 114)
(1 67 42 78)(2 68 43 79)(3 69 44 80)(4 70 45 81)(5 71 37 73)(6 72 38 74)(7 64 39 75)(8 65 40 76)(9 66 41 77)(10 122 138 98)(11 123 139 99)(12 124 140 91)(13 125 141 92)(14 126 142 93)(15 118 143 94)(16 119 144 95)(17 120 136 96)(18 121 137 97)(19 100 132 111)(20 101 133 112)(21 102 134 113)(22 103 135 114)(23 104 127 115)(24 105 128 116)(25 106 129 117)(26 107 130 109)(27 108 131 110)(28 83 47 59)(29 84 48 60)(30 85 49 61)(31 86 50 62)(32 87 51 63)(33 88 52 55)(34 89 53 56)(35 90 54 57)(36 82 46 58)
(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)
(1 114)(2 115)(3 116)(4 117)(5 109)(6 110)(7 111)(8 112)(9 113)(10 77)(11 78)(12 79)(13 80)(14 81)(15 73)(16 74)(17 75)(18 76)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 82)(27 83)(28 95)(29 96)(30 97)(31 98)(32 99)(33 91)(34 92)(35 93)(36 94)(37 107)(38 108)(39 100)(40 101)(41 102)(42 103)(43 104)(44 105)(45 106)(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), (1,51,42,32)(2,52,43,33)(3,53,44,34)(4,54,45,35)(5,46,37,36)(6,47,38,28)(7,48,39,29)(8,49,40,30)(9,50,41,31)(10,21,138,134)(11,22,139,135)(12,23,140,127)(13,24,141,128)(14,25,142,129)(15,26,143,130)(16,27,144,131)(17,19,136,132)(18,20,137,133)(55,68,88,79)(56,69,89,80)(57,70,90,81)(58,71,82,73)(59,72,83,74)(60,64,84,75)(61,65,85,76)(62,66,86,77)(63,67,87,78)(91,104,124,115)(92,105,125,116)(93,106,126,117)(94,107,118,109)(95,108,119,110)(96,100,120,111)(97,101,121,112)(98,102,122,113)(99,103,123,114), (1,67,42,78)(2,68,43,79)(3,69,44,80)(4,70,45,81)(5,71,37,73)(6,72,38,74)(7,64,39,75)(8,65,40,76)(9,66,41,77)(10,122,138,98)(11,123,139,99)(12,124,140,91)(13,125,141,92)(14,126,142,93)(15,118,143,94)(16,119,144,95)(17,120,136,96)(18,121,137,97)(19,100,132,111)(20,101,133,112)(21,102,134,113)(22,103,135,114)(23,104,127,115)(24,105,128,116)(25,106,129,117)(26,107,130,109)(27,108,131,110)(28,83,47,59)(29,84,48,60)(30,85,49,61)(31,86,50,62)(32,87,51,63)(33,88,52,55)(34,89,53,56)(35,90,54,57)(36,82,46,58), (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), (1,114)(2,115)(3,116)(4,117)(5,109)(6,110)(7,111)(8,112)(9,113)(10,77)(11,78)(12,79)(13,80)(14,81)(15,73)(16,74)(17,75)(18,76)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,82)(27,83)(28,95)(29,96)(30,97)(31,98)(32,99)(33,91)(34,92)(35,93)(36,94)(37,107)(38,108)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(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), (1,51,42,32)(2,52,43,33)(3,53,44,34)(4,54,45,35)(5,46,37,36)(6,47,38,28)(7,48,39,29)(8,49,40,30)(9,50,41,31)(10,21,138,134)(11,22,139,135)(12,23,140,127)(13,24,141,128)(14,25,142,129)(15,26,143,130)(16,27,144,131)(17,19,136,132)(18,20,137,133)(55,68,88,79)(56,69,89,80)(57,70,90,81)(58,71,82,73)(59,72,83,74)(60,64,84,75)(61,65,85,76)(62,66,86,77)(63,67,87,78)(91,104,124,115)(92,105,125,116)(93,106,126,117)(94,107,118,109)(95,108,119,110)(96,100,120,111)(97,101,121,112)(98,102,122,113)(99,103,123,114), (1,67,42,78)(2,68,43,79)(3,69,44,80)(4,70,45,81)(5,71,37,73)(6,72,38,74)(7,64,39,75)(8,65,40,76)(9,66,41,77)(10,122,138,98)(11,123,139,99)(12,124,140,91)(13,125,141,92)(14,126,142,93)(15,118,143,94)(16,119,144,95)(17,120,136,96)(18,121,137,97)(19,100,132,111)(20,101,133,112)(21,102,134,113)(22,103,135,114)(23,104,127,115)(24,105,128,116)(25,106,129,117)(26,107,130,109)(27,108,131,110)(28,83,47,59)(29,84,48,60)(30,85,49,61)(31,86,50,62)(32,87,51,63)(33,88,52,55)(34,89,53,56)(35,90,54,57)(36,82,46,58), (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), (1,114)(2,115)(3,116)(4,117)(5,109)(6,110)(7,111)(8,112)(9,113)(10,77)(11,78)(12,79)(13,80)(14,81)(15,73)(16,74)(17,75)(18,76)(19,84)(20,85)(21,86)(22,87)(23,88)(24,89)(25,90)(26,82)(27,83)(28,95)(29,96)(30,97)(31,98)(32,99)(33,91)(34,92)(35,93)(36,94)(37,107)(38,108)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(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)], [(1,51,42,32),(2,52,43,33),(3,53,44,34),(4,54,45,35),(5,46,37,36),(6,47,38,28),(7,48,39,29),(8,49,40,30),(9,50,41,31),(10,21,138,134),(11,22,139,135),(12,23,140,127),(13,24,141,128),(14,25,142,129),(15,26,143,130),(16,27,144,131),(17,19,136,132),(18,20,137,133),(55,68,88,79),(56,69,89,80),(57,70,90,81),(58,71,82,73),(59,72,83,74),(60,64,84,75),(61,65,85,76),(62,66,86,77),(63,67,87,78),(91,104,124,115),(92,105,125,116),(93,106,126,117),(94,107,118,109),(95,108,119,110),(96,100,120,111),(97,101,121,112),(98,102,122,113),(99,103,123,114)], [(1,67,42,78),(2,68,43,79),(3,69,44,80),(4,70,45,81),(5,71,37,73),(6,72,38,74),(7,64,39,75),(8,65,40,76),(9,66,41,77),(10,122,138,98),(11,123,139,99),(12,124,140,91),(13,125,141,92),(14,126,142,93),(15,118,143,94),(16,119,144,95),(17,120,136,96),(18,121,137,97),(19,100,132,111),(20,101,133,112),(21,102,134,113),(22,103,135,114),(23,104,127,115),(24,105,128,116),(25,106,129,117),(26,107,130,109),(27,108,131,110),(28,83,47,59),(29,84,48,60),(30,85,49,61),(31,86,50,62),(32,87,51,63),(33,88,52,55),(34,89,53,56),(35,90,54,57),(36,82,46,58)], [(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)], [(1,114),(2,115),(3,116),(4,117),(5,109),(6,110),(7,111),(8,112),(9,113),(10,77),(11,78),(12,79),(13,80),(14,81),(15,73),(16,74),(17,75),(18,76),(19,84),(20,85),(21,86),(22,87),(23,88),(24,89),(25,90),(26,82),(27,83),(28,95),(29,96),(30,97),(31,98),(32,99),(33,91),(34,92),(35,93),(36,94),(37,107),(38,108),(39,100),(40,101),(41,102),(42,103),(43,104),(44,105),(45,106),(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)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | ··· | 12M | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18R | 24A | 24B | 24C | 24D | 36A | ··· | 36F | 36G | ··· | 36AD | 72A | ··· | 72L |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | ··· | 12 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
108 irreducible representations
| dim | 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 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | S3 | D4 | D6 | SD16 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×SD16 | S3×C9 | D4×C9 | C3×C3⋊D4 | S3×C18 | C9×SD16 | C9×C3⋊D4 | Q8⋊2S3 | C3×Q8⋊2S3 | C9×Q8⋊2S3 |
| kernel | C9×Q8⋊2S3 | C9×C3⋊C8 | C9×D12 | Q8×C3×C9 | C3×Q8⋊2S3 | C3×C3⋊C8 | C3×D12 | Q8×C32 | Q8⋊2S3 | C3⋊C8 | D12 | C3×Q8 | Q8×C9 | C3×C18 | C36 | C3×C9 | C3×Q8 | C18 | C3×C6 | C12 | C32 | Q8 | C6 | C6 | C4 | C3 | C2 | C9 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 6 | 6 | 4 | 6 | 12 | 12 | 1 | 2 | 6 |
Matrix representation of C9×Q8⋊2S3 ►in GL4(𝔽73) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 25 | 1 | 0 | 0 |
| 0 | 0 | 60 | 66 |
| 0 | 0 | 66 | 13 |
| 64 | 0 | 0 | 0 |
| 30 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 21 | 63 | 0 | 0 |
| 44 | 52 | 0 | 0 |
| 0 | 0 | 26 | 36 |
| 0 | 0 | 36 | 47 |
G:=sub<GL(4,GF(73))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[72,25,0,0,0,1,0,0,0,0,60,66,0,0,66,13],[64,30,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[21,44,0,0,63,52,0,0,0,0,26,36,0,0,36,47] >;
C9×Q8⋊2S3 in GAP, Magma, Sage, TeX
C_9\times Q_8\rtimes_2S_3
% in TeX
G:=Group("C9xQ8:2S3"); // GroupNames label
G:=SmallGroup(432,158);
// by ID
G=gap.SmallGroup(432,158);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,197,512,142,2355,1186,192,14118]);
// 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,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations