direct product, metabelian, supersoluble, monomial
Aliases: C3×Q8⋊3D9, D36⋊10C6, C12.50D18, (C4×D9)⋊8C6, Q8⋊4(C3×D9), (C3×Q8)⋊7D9, C4.7(C6×D9), (C12×D9)⋊6C2, (Q8×C9)⋊13C6, (C3×D36)⋊10C2, C12.20(S3×C6), C36.27(C2×C6), D18.8(C2×C6), (C3×C12).104D6, C6.56(C22×D9), (C3×C36).32C22, C18.22(C22×C6), (C3×C18).45C23, Dic9.10(C2×C6), (C6×D9).14C22, (Q8×C32).23S3, C32.4(Q8⋊3S3), (C3×Dic9).17C22, (Q8×C3×C9)⋊4C2, C2.9(C2×C6×D9), C9⋊7(C3×C4○D4), C6.38(S3×C2×C6), (C3×C9)⋊18(C4○D4), (C3×Q8).28(C3×S3), C3.1(C3×Q8⋊3S3), (C3×C6).159(C22×S3), SmallGroup(432,365)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8⋊3D9
G = < a,b,c,d,e | a3=b4=d9=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: 502 in 132 conjugacy classes, 58 normal (22 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, D9, C18, C18, C3×S3, C3×C6, C4×S3, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, Q8⋊3S3, C3×C4○D4, C3×D9, C3×C18, C4×D9, D36, Q8×C9, Q8×C9, S3×C12, C3×D12, Q8×C32, C3×Dic9, C3×C36, C6×D9, Q8⋊3D9, C3×Q8⋊3S3, C12×D9, C3×D36, Q8×C3×C9, C3×Q8⋊3D9
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, D9, C3×S3, C22×S3, C22×C6, D18, S3×C6, Q8⋊3S3, C3×C4○D4, C3×D9, C22×D9, S3×C2×C6, C6×D9, Q8⋊3D9, C3×Q8⋊3S3, C2×C6×D9, C3×Q8⋊3D9
►On 144 points
Generators in S144
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(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 76 79)(74 77 80)(75 78 81)(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 28 10 19)(2 29 11 20)(3 30 12 21)(4 31 13 22)(5 32 14 23)(6 33 15 24)(7 34 16 25)(8 35 17 26)(9 36 18 27)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)(73 91 82 100)(74 92 83 101)(75 93 84 102)(76 94 85 103)(77 95 86 104)(78 96 87 105)(79 97 88 106)(80 98 89 107)(81 99 90 108)(109 136 118 127)(110 137 119 128)(111 138 120 129)(112 139 121 130)(113 140 122 131)(114 141 123 132)(115 142 124 133)(116 143 125 134)(117 144 126 135)
(1 46 10 37)(2 47 11 38)(3 48 12 39)(4 49 13 40)(5 50 14 41)(6 51 15 42)(7 52 16 43)(8 53 17 44)(9 54 18 45)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)(73 118 82 109)(74 119 83 110)(75 120 84 111)(76 121 85 112)(77 122 86 113)(78 123 87 114)(79 124 88 115)(80 125 89 116)(81 126 90 117)(91 136 100 127)(92 137 101 128)(93 138 102 129)(94 139 103 130)(95 140 104 131)(96 141 105 132)(97 142 106 133)(98 143 107 134)(99 144 108 135)
(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 79)(2 78)(3 77)(4 76)(5 75)(6 74)(7 73)(8 81)(9 80)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 90)(18 89)(19 97)(20 96)(21 95)(22 94)(23 93)(24 92)(25 91)(26 99)(27 98)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 108)(36 107)(37 115)(38 114)(39 113)(40 112)(41 111)(42 110)(43 109)(44 117)(45 116)(46 124)(47 123)(48 122)(49 121)(50 120)(51 119)(52 118)(53 126)(54 125)(55 133)(56 132)(57 131)(58 130)(59 129)(60 128)(61 127)(62 135)(63 134)(64 142)(65 141)(66 140)(67 139)(68 138)(69 137)(70 136)(71 144)(72 143)
G:=sub<Sym(144)| (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(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)(73,76,79)(74,77,80)(75,78,81)(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), (1,28,10,19)(2,29,11,20)(3,30,12,21)(4,31,13,22)(5,32,14,23)(6,33,15,24)(7,34,16,25)(8,35,17,26)(9,36,18,27)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72)(73,91,82,100)(74,92,83,101)(75,93,84,102)(76,94,85,103)(77,95,86,104)(78,96,87,105)(79,97,88,106)(80,98,89,107)(81,99,90,108)(109,136,118,127)(110,137,119,128)(111,138,120,129)(112,139,121,130)(113,140,122,131)(114,141,123,132)(115,142,124,133)(116,143,125,134)(117,144,126,135), (1,46,10,37)(2,47,11,38)(3,48,12,39)(4,49,13,40)(5,50,14,41)(6,51,15,42)(7,52,16,43)(8,53,17,44)(9,54,18,45)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117)(91,136,100,127)(92,137,101,128)(93,138,102,129)(94,139,103,130)(95,140,104,131)(96,141,105,132)(97,142,106,133)(98,143,107,134)(99,144,108,135), (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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,81)(9,80)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,90)(18,89)(19,97)(20,96)(21,95)(22,94)(23,93)(24,92)(25,91)(26,99)(27,98)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,108)(36,107)(37,115)(38,114)(39,113)(40,112)(41,111)(42,110)(43,109)(44,117)(45,116)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,126)(54,125)(55,133)(56,132)(57,131)(58,130)(59,129)(60,128)(61,127)(62,135)(63,134)(64,142)(65,141)(66,140)(67,139)(68,138)(69,137)(70,136)(71,144)(72,143)>;
G:=Group( (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(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)(73,76,79)(74,77,80)(75,78,81)(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), (1,28,10,19)(2,29,11,20)(3,30,12,21)(4,31,13,22)(5,32,14,23)(6,33,15,24)(7,34,16,25)(8,35,17,26)(9,36,18,27)(37,55,46,64)(38,56,47,65)(39,57,48,66)(40,58,49,67)(41,59,50,68)(42,60,51,69)(43,61,52,70)(44,62,53,71)(45,63,54,72)(73,91,82,100)(74,92,83,101)(75,93,84,102)(76,94,85,103)(77,95,86,104)(78,96,87,105)(79,97,88,106)(80,98,89,107)(81,99,90,108)(109,136,118,127)(110,137,119,128)(111,138,120,129)(112,139,121,130)(113,140,122,131)(114,141,123,132)(115,142,124,133)(116,143,125,134)(117,144,126,135), (1,46,10,37)(2,47,11,38)(3,48,12,39)(4,49,13,40)(5,50,14,41)(6,51,15,42)(7,52,16,43)(8,53,17,44)(9,54,18,45)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63)(73,118,82,109)(74,119,83,110)(75,120,84,111)(76,121,85,112)(77,122,86,113)(78,123,87,114)(79,124,88,115)(80,125,89,116)(81,126,90,117)(91,136,100,127)(92,137,101,128)(93,138,102,129)(94,139,103,130)(95,140,104,131)(96,141,105,132)(97,142,106,133)(98,143,107,134)(99,144,108,135), (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), (1,79)(2,78)(3,77)(4,76)(5,75)(6,74)(7,73)(8,81)(9,80)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,90)(18,89)(19,97)(20,96)(21,95)(22,94)(23,93)(24,92)(25,91)(26,99)(27,98)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,108)(36,107)(37,115)(38,114)(39,113)(40,112)(41,111)(42,110)(43,109)(44,117)(45,116)(46,124)(47,123)(48,122)(49,121)(50,120)(51,119)(52,118)(53,126)(54,125)(55,133)(56,132)(57,131)(58,130)(59,129)(60,128)(61,127)(62,135)(63,134)(64,142)(65,141)(66,140)(67,139)(68,138)(69,137)(70,136)(71,144)(72,143) );
G=PermutationGroup([[(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(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),(73,76,79),(74,77,80),(75,78,81),(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)], [(1,28,10,19),(2,29,11,20),(3,30,12,21),(4,31,13,22),(5,32,14,23),(6,33,15,24),(7,34,16,25),(8,35,17,26),(9,36,18,27),(37,55,46,64),(38,56,47,65),(39,57,48,66),(40,58,49,67),(41,59,50,68),(42,60,51,69),(43,61,52,70),(44,62,53,71),(45,63,54,72),(73,91,82,100),(74,92,83,101),(75,93,84,102),(76,94,85,103),(77,95,86,104),(78,96,87,105),(79,97,88,106),(80,98,89,107),(81,99,90,108),(109,136,118,127),(110,137,119,128),(111,138,120,129),(112,139,121,130),(113,140,122,131),(114,141,123,132),(115,142,124,133),(116,143,125,134),(117,144,126,135)], [(1,46,10,37),(2,47,11,38),(3,48,12,39),(4,49,13,40),(5,50,14,41),(6,51,15,42),(7,52,16,43),(8,53,17,44),(9,54,18,45),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63),(73,118,82,109),(74,119,83,110),(75,120,84,111),(76,121,85,112),(77,122,86,113),(78,123,87,114),(79,124,88,115),(80,125,89,116),(81,126,90,117),(91,136,100,127),(92,137,101,128),(93,138,102,129),(94,139,103,130),(95,140,104,131),(96,141,105,132),(97,142,106,133),(98,143,107,134),(99,144,108,135)], [(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)], [(1,79),(2,78),(3,77),(4,76),(5,75),(6,74),(7,73),(8,81),(9,80),(10,88),(11,87),(12,86),(13,85),(14,84),(15,83),(16,82),(17,90),(18,89),(19,97),(20,96),(21,95),(22,94),(23,93),(24,92),(25,91),(26,99),(27,98),(28,106),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,108),(36,107),(37,115),(38,114),(39,113),(40,112),(41,111),(42,110),(43,109),(44,117),(45,116),(46,124),(47,123),(48,122),(49,121),(50,120),(51,119),(52,118),(53,126),(54,125),(55,133),(56,132),(57,131),(58,130),(59,129),(60,128),(61,127),(62,135),(63,134),(64,142),(65,141),(66,140),(67,139),(68,138),(69,137),(70,136),(71,144),(72,143)]])
90 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 | ··· | 9I | 12A | ··· | 12F | 12G | ··· | 12O | 12P | 12Q | 12R | 12S | 18A | ··· | 18I | 36A | ··· | 36AA |
| 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 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 18 | 18 | 18 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | 1 | 1 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D6 | C4○D4 | D9 | C3×S3 | D18 | S3×C6 | C3×C4○D4 | C3×D9 | C6×D9 | Q8⋊3S3 | Q8⋊3D9 | C3×Q8⋊3S3 | C3×Q8⋊3D9 |
| kernel | C3×Q8⋊3D9 | C12×D9 | C3×D36 | Q8×C3×C9 | Q8⋊3D9 | C4×D9 | D36 | Q8×C9 | Q8×C32 | C3×C12 | C3×C9 | C3×Q8 | C3×Q8 | C12 | C12 | C9 | Q8 | C4 | C32 | C3 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 1 | 3 | 2 | 3 | 2 | 9 | 6 | 4 | 6 | 18 | 1 | 3 | 2 | 6 |
Matrix representation of C3×Q8⋊3D9 ►in GL4(𝔽37) generated by
| 26 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 18 |
| 0 | 0 | 0 | 31 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 33 |
| 0 | 0 | 3 | 23 |
| 33 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 7 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 27 | 24 |
| 0 | 0 | 19 | 10 |
G:=sub<GL(4,GF(37))| [26,0,0,0,0,26,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,0,0,0,18,31],[1,0,0,0,0,1,0,0,0,0,14,3,0,0,33,23],[33,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,7,0,0,0,0,0,27,19,0,0,24,10] >;
C3×Q8⋊3D9 in GAP, Magma, Sage, TeX
C_3\times Q_8\rtimes_3D_9
% in TeX
G:=Group("C3xQ8:3D9"); // GroupNames label
G:=SmallGroup(432,365);
// by ID
G=gap.SmallGroup(432,365);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,142,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^9=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