direct product, metabelian, supersoluble, monomial
Aliases: C32×D6⋊C4, C62.165D6, D6⋊(C3×C12), (C6×C12)⋊9S3, (C6×C12)⋊9C6, (S3×C6)⋊2C12, C6.5(C6×C12), C6.43(S3×C12), (C6×Dic3)⋊3C6, (C3×C6).88D12, C6.41(C3×D12), C6.6(D4×C32), (S3×C62).3C2, (C2×C6).17C62, C62.62(C2×C6), C2.2(C32×D12), (C32×C6).56D4, C33⋊14(C22⋊C4), (C3×C62).44C22, (S3×C3×C6)⋊5C4, (C3×C6×C12)⋊1C2, (S3×C2×C6).7C6, C2.5(S3×C3×C12), (C2×C12)⋊1(C3×C6), (C2×C12)⋊1(C3×S3), (Dic3×C3×C6)⋊5C2, C22.6(S3×C3×C6), (C2×C6).92(S3×C6), (C3×C6).97(C4×S3), (C2×C4)⋊1(S3×C32), (C3×C6).48(C3×D4), (C22×S3).(C3×C6), C6.52(C3×C3⋊D4), (C3×C6).46(C2×C12), (C2×Dic3)⋊1(C3×C6), C32⋊8(C3×C22⋊C4), C3⋊1(C32×C22⋊C4), C2.2(C32×C3⋊D4), (C32×C6).53(C2×C4), (C3×C6).121(C3⋊D4), SmallGroup(432,474)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×D6⋊C4
G = < a,b,c,d,e | a3=b3=c6=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c3d >
Subgroups: 584 in 268 conjugacy classes, 102 normal (30 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C32, C32, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C33, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, D6⋊C4, C3×C22⋊C4, S3×C32, C32×C6, C6×Dic3, C6×C12, C6×C12, C6×C12, S3×C2×C6, C2×C62, C32×Dic3, C32×C12, S3×C3×C6, S3×C3×C6, C3×C62, C3×D6⋊C4, C32×C22⋊C4, Dic3×C3×C6, C3×C6×C12, S3×C62, C32×D6⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C32, C12, D6, C2×C6, C22⋊C4, C3×S3, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×C12, S3×C6, C62, D6⋊C4, C3×C22⋊C4, S3×C32, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, D4×C32, S3×C3×C6, C3×D6⋊C4, C32×C22⋊C4, S3×C3×C12, C32×D12, C32×C3⋊D4, C32×D6⋊C4
►On 144 points
Generators in S144
(1 29 17)(2 30 18)(3 25 13)(4 26 14)(5 27 15)(6 28 16)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 61 49)(38 62 50)(39 63 51)(40 64 52)(41 65 53)(42 66 54)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)(73 97 85)(74 98 86)(75 99 87)(76 100 88)(77 101 89)(78 102 90)(79 103 91)(80 104 92)(81 105 93)(82 106 94)(83 107 95)(84 108 96)(109 133 121)(110 134 122)(111 135 123)(112 136 124)(113 137 125)(114 138 126)(115 139 127)(116 140 128)(117 141 129)(118 142 130)(119 143 131)(120 144 132)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(73 77 75)(74 78 76)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)(97 101 99)(98 102 100)(103 107 105)(104 108 106)(109 113 111)(110 114 112)(115 119 117)(116 120 118)(121 125 123)(122 126 124)(127 131 129)(128 132 130)(133 137 135)(134 138 136)(139 143 141)(140 144 142)
(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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 82)(8 81)(9 80)(10 79)(11 84)(12 83)(13 88)(14 87)(15 86)(16 85)(17 90)(18 89)(19 94)(20 93)(21 92)(22 91)(23 96)(24 95)(25 100)(26 99)(27 98)(28 97)(29 102)(30 101)(31 106)(32 105)(33 104)(34 103)(35 108)(36 107)(37 112)(38 111)(39 110)(40 109)(41 114)(42 113)(43 118)(44 117)(45 116)(46 115)(47 120)(48 119)(49 124)(50 123)(51 122)(52 121)(53 126)(54 125)(55 130)(56 129)(57 128)(58 127)(59 132)(60 131)(61 136)(62 135)(63 134)(64 133)(65 138)(66 137)(67 142)(68 141)(69 140)(70 139)(71 144)(72 143)
(1 47 11 41)(2 48 12 42)(3 43 7 37)(4 44 8 38)(5 45 9 39)(6 46 10 40)(13 55 19 49)(14 56 20 50)(15 57 21 51)(16 58 22 52)(17 59 23 53)(18 60 24 54)(25 67 31 61)(26 68 32 62)(27 69 33 63)(28 70 34 64)(29 71 35 65)(30 72 36 66)(73 118 79 112)(74 119 80 113)(75 120 81 114)(76 115 82 109)(77 116 83 110)(78 117 84 111)(85 130 91 124)(86 131 92 125)(87 132 93 126)(88 127 94 121)(89 128 95 122)(90 129 96 123)(97 142 103 136)(98 143 104 137)(99 144 105 138)(100 139 106 133)(101 140 107 134)(102 141 108 135)
G:=sub<Sym(144)| (1,29,17)(2,30,18)(3,25,13)(4,26,14)(5,27,15)(6,28,16)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60)(73,97,85)(74,98,86)(75,99,87)(76,100,88)(77,101,89)(78,102,90)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96)(109,133,121)(110,134,122)(111,135,123)(112,136,124)(113,137,125)(114,138,126)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72)(73,77,75)(74,78,76)(79,83,81)(80,84,82)(85,89,87)(86,90,88)(91,95,93)(92,96,94)(97,101,99)(98,102,100)(103,107,105)(104,108,106)(109,113,111)(110,114,112)(115,119,117)(116,120,118)(121,125,123)(122,126,124)(127,131,129)(128,132,130)(133,137,135)(134,138,136)(139,143,141)(140,144,142), (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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,82)(8,81)(9,80)(10,79)(11,84)(12,83)(13,88)(14,87)(15,86)(16,85)(17,90)(18,89)(19,94)(20,93)(21,92)(22,91)(23,96)(24,95)(25,100)(26,99)(27,98)(28,97)(29,102)(30,101)(31,106)(32,105)(33,104)(34,103)(35,108)(36,107)(37,112)(38,111)(39,110)(40,109)(41,114)(42,113)(43,118)(44,117)(45,116)(46,115)(47,120)(48,119)(49,124)(50,123)(51,122)(52,121)(53,126)(54,125)(55,130)(56,129)(57,128)(58,127)(59,132)(60,131)(61,136)(62,135)(63,134)(64,133)(65,138)(66,137)(67,142)(68,141)(69,140)(70,139)(71,144)(72,143), (1,47,11,41)(2,48,12,42)(3,43,7,37)(4,44,8,38)(5,45,9,39)(6,46,10,40)(13,55,19,49)(14,56,20,50)(15,57,21,51)(16,58,22,52)(17,59,23,53)(18,60,24,54)(25,67,31,61)(26,68,32,62)(27,69,33,63)(28,70,34,64)(29,71,35,65)(30,72,36,66)(73,118,79,112)(74,119,80,113)(75,120,81,114)(76,115,82,109)(77,116,83,110)(78,117,84,111)(85,130,91,124)(86,131,92,125)(87,132,93,126)(88,127,94,121)(89,128,95,122)(90,129,96,123)(97,142,103,136)(98,143,104,137)(99,144,105,138)(100,139,106,133)(101,140,107,134)(102,141,108,135)>;
G:=Group( (1,29,17)(2,30,18)(3,25,13)(4,26,14)(5,27,15)(6,28,16)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60)(73,97,85)(74,98,86)(75,99,87)(76,100,88)(77,101,89)(78,102,90)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96)(109,133,121)(110,134,122)(111,135,123)(112,136,124)(113,137,125)(114,138,126)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48)(49,51,53)(50,52,54)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72)(73,77,75)(74,78,76)(79,83,81)(80,84,82)(85,89,87)(86,90,88)(91,95,93)(92,96,94)(97,101,99)(98,102,100)(103,107,105)(104,108,106)(109,113,111)(110,114,112)(115,119,117)(116,120,118)(121,125,123)(122,126,124)(127,131,129)(128,132,130)(133,137,135)(134,138,136)(139,143,141)(140,144,142), (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), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,82)(8,81)(9,80)(10,79)(11,84)(12,83)(13,88)(14,87)(15,86)(16,85)(17,90)(18,89)(19,94)(20,93)(21,92)(22,91)(23,96)(24,95)(25,100)(26,99)(27,98)(28,97)(29,102)(30,101)(31,106)(32,105)(33,104)(34,103)(35,108)(36,107)(37,112)(38,111)(39,110)(40,109)(41,114)(42,113)(43,118)(44,117)(45,116)(46,115)(47,120)(48,119)(49,124)(50,123)(51,122)(52,121)(53,126)(54,125)(55,130)(56,129)(57,128)(58,127)(59,132)(60,131)(61,136)(62,135)(63,134)(64,133)(65,138)(66,137)(67,142)(68,141)(69,140)(70,139)(71,144)(72,143), (1,47,11,41)(2,48,12,42)(3,43,7,37)(4,44,8,38)(5,45,9,39)(6,46,10,40)(13,55,19,49)(14,56,20,50)(15,57,21,51)(16,58,22,52)(17,59,23,53)(18,60,24,54)(25,67,31,61)(26,68,32,62)(27,69,33,63)(28,70,34,64)(29,71,35,65)(30,72,36,66)(73,118,79,112)(74,119,80,113)(75,120,81,114)(76,115,82,109)(77,116,83,110)(78,117,84,111)(85,130,91,124)(86,131,92,125)(87,132,93,126)(88,127,94,121)(89,128,95,122)(90,129,96,123)(97,142,103,136)(98,143,104,137)(99,144,105,138)(100,139,106,133)(101,140,107,134)(102,141,108,135) );
G=PermutationGroup([[(1,29,17),(2,30,18),(3,25,13),(4,26,14),(5,27,15),(6,28,16),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24),(37,61,49),(38,62,50),(39,63,51),(40,64,52),(41,65,53),(42,66,54),(43,67,55),(44,68,56),(45,69,57),(46,70,58),(47,71,59),(48,72,60),(73,97,85),(74,98,86),(75,99,87),(76,100,88),(77,101,89),(78,102,90),(79,103,91),(80,104,92),(81,105,93),(82,106,94),(83,107,95),(84,108,96),(109,133,121),(110,134,122),(111,135,123),(112,136,124),(113,137,125),(114,138,126),(115,139,127),(116,140,128),(117,141,129),(118,142,130),(119,143,131),(120,144,132)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48),(49,51,53),(50,52,54),(55,57,59),(56,58,60),(61,63,65),(62,64,66),(67,69,71),(68,70,72),(73,77,75),(74,78,76),(79,83,81),(80,84,82),(85,89,87),(86,90,88),(91,95,93),(92,96,94),(97,101,99),(98,102,100),(103,107,105),(104,108,106),(109,113,111),(110,114,112),(115,119,117),(116,120,118),(121,125,123),(122,126,124),(127,131,129),(128,132,130),(133,137,135),(134,138,136),(139,143,141),(140,144,142)], [(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)], [(1,78),(2,77),(3,76),(4,75),(5,74),(6,73),(7,82),(8,81),(9,80),(10,79),(11,84),(12,83),(13,88),(14,87),(15,86),(16,85),(17,90),(18,89),(19,94),(20,93),(21,92),(22,91),(23,96),(24,95),(25,100),(26,99),(27,98),(28,97),(29,102),(30,101),(31,106),(32,105),(33,104),(34,103),(35,108),(36,107),(37,112),(38,111),(39,110),(40,109),(41,114),(42,113),(43,118),(44,117),(45,116),(46,115),(47,120),(48,119),(49,124),(50,123),(51,122),(52,121),(53,126),(54,125),(55,130),(56,129),(57,128),(58,127),(59,132),(60,131),(61,136),(62,135),(63,134),(64,133),(65,138),(66,137),(67,142),(68,141),(69,140),(70,139),(71,144),(72,143)], [(1,47,11,41),(2,48,12,42),(3,43,7,37),(4,44,8,38),(5,45,9,39),(6,46,10,40),(13,55,19,49),(14,56,20,50),(15,57,21,51),(16,58,22,52),(17,59,23,53),(18,60,24,54),(25,67,31,61),(26,68,32,62),(27,69,33,63),(28,70,34,64),(29,71,35,65),(30,72,36,66),(73,118,79,112),(74,119,80,113),(75,120,81,114),(76,115,82,109),(77,116,83,110),(78,117,84,111),(85,130,91,124),(86,131,92,125),(87,132,93,126),(88,127,94,121),(89,128,95,122),(90,129,96,123),(97,142,103,136),(98,143,104,137),(99,144,105,138),(100,139,106,133),(101,140,107,134),(102,141,108,135)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 4D | 6A | ··· | 6X | 6Y | ··· | 6AY | 6AZ | ··· | 6BO | 12A | ··· | 12AZ | 12BA | ··· | 12BP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | D6 | C3×S3 | C4×S3 | D12 | C3⋊D4 | C3×D4 | S3×C6 | S3×C12 | C3×D12 | C3×C3⋊D4 |
| kernel | C32×D6⋊C4 | Dic3×C3×C6 | C3×C6×C12 | S3×C62 | C3×D6⋊C4 | S3×C3×C6 | C6×Dic3 | C6×C12 | S3×C2×C6 | S3×C6 | C6×C12 | C32×C6 | C62 | C2×C12 | C3×C6 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 |
| # reps | 1 | 1 | 1 | 1 | 8 | 4 | 8 | 8 | 8 | 32 | 1 | 2 | 1 | 8 | 2 | 2 | 2 | 16 | 8 | 16 | 16 | 16 |
Matrix representation of C32×D6⋊C4 ►in GL3(𝔽13) generated by
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 9 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 10 |
| 12 | 0 | 0 |
| 0 | 0 | 10 |
| 0 | 4 | 0 |
| 8 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 12 |
G:=sub<GL(3,GF(13))| [1,0,0,0,3,0,0,0,3],[9,0,0,0,3,0,0,0,3],[1,0,0,0,4,0,0,0,10],[12,0,0,0,0,4,0,10,0],[8,0,0,0,1,0,0,0,12] >;
C32×D6⋊C4 in GAP, Magma, Sage, TeX
C_3^2\times D_6\rtimes C_4
% in TeX
G:=Group("C3^2xD6:C4"); // GroupNames label
G:=SmallGroup(432,474);
// by ID
G=gap.SmallGroup(432,474);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,1037,260,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^6=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^3*d>;
// generators/relations