direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C6×C3⋊Dic3, C63.6C2, C62⋊17C12, C62⋊16Dic3, C62.156D6, C6⋊2(C6×Dic3), (C3×C62)⋊10C4, (C2×C62).31S3, (C2×C62).30C6, C62.79(C2×C6), C33⋊20(C22×C4), C32⋊11(C22×C12), (C32×C6).93C23, (C3×C62).65C22, C32⋊10(C22×Dic3), C6.60(S3×C2×C6), C3⋊2(Dic3×C2×C6), (C3×C6)⋊10(C2×C12), (C2×C6).79(S3×C6), (C3×C6)⋊9(C2×Dic3), (C2×C6)⋊7(C3×Dic3), C23.4(C3×C3⋊S3), (C32×C6)⋊14(C2×C4), C6.60(C22×C3⋊S3), C22.11(C6×C3⋊S3), (C22×C6).37(C3×S3), (C3×C6).67(C22×C6), (C22×C6).20(C3⋊S3), (C3×C6).182(C22×S3), C2.2(C2×C6×C3⋊S3), (C2×C6).70(C2×C3⋊S3), SmallGroup(432,718)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C32×C6 — C3×C3⋊Dic3 — C6×C3⋊Dic3 — C2×C6×C3⋊Dic3 |
| C32 — C2×C6×C3⋊Dic3 |
Generators and relations for C2×C6×C3⋊Dic3
G = < a,b,c,d,e | a2=b6=c3=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 932 in 452 conjugacy classes, 214 normal (14 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C33, C3×Dic3, C3⋊Dic3, C62, C62, C22×Dic3, C22×C12, C32×C6, C32×C6, C6×Dic3, C2×C3⋊Dic3, C2×C62, C2×C62, C2×C62, C3×C3⋊Dic3, C3×C62, Dic3×C2×C6, C22×C3⋊Dic3, C6×C3⋊Dic3, C63, C2×C6×C3⋊Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, C3×S3, C3⋊S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C22×Dic3, C22×C12, C3×C3⋊S3, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C22×C3⋊S3, C3×C3⋊Dic3, C6×C3⋊S3, Dic3×C2×C6, C22×C3⋊Dic3, C6×C3⋊Dic3, C2×C6×C3⋊S3, C2×C6×C3⋊Dic3
►On 144 points
Generators in S144
(1 66)(2 61)(3 62)(4 63)(5 64)(6 65)(7 125)(8 126)(9 121)(10 122)(11 123)(12 124)(13 41)(14 42)(15 37)(16 38)(17 39)(18 40)(19 70)(20 71)(21 72)(22 67)(23 68)(24 69)(25 132)(26 127)(27 128)(28 129)(29 130)(30 131)(31 116)(32 117)(33 118)(34 119)(35 120)(36 115)(43 82)(44 83)(45 84)(46 79)(47 80)(48 81)(49 88)(50 89)(51 90)(52 85)(53 86)(54 87)(55 94)(56 95)(57 96)(58 91)(59 92)(60 93)(73 112)(74 113)(75 114)(76 109)(77 110)(78 111)(97 136)(98 137)(99 138)(100 133)(101 134)(102 135)(103 142)(104 143)(105 144)(106 139)(107 140)(108 141)
(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 57 44)(2 58 45)(3 59 46)(4 60 47)(5 55 48)(6 56 43)(7 144 110)(8 139 111)(9 140 112)(10 141 113)(11 142 114)(12 143 109)(13 24 50)(14 19 51)(15 20 52)(16 21 53)(17 22 54)(18 23 49)(25 135 34)(26 136 35)(27 137 36)(28 138 31)(29 133 32)(30 134 33)(37 71 85)(38 72 86)(39 67 87)(40 68 88)(41 69 89)(42 70 90)(61 91 84)(62 92 79)(63 93 80)(64 94 81)(65 95 82)(66 96 83)(73 121 107)(74 122 108)(75 123 103)(76 124 104)(77 125 105)(78 126 106)(97 120 127)(98 115 128)(99 116 129)(100 117 130)(101 118 131)(102 119 132)
(1 13 46 52 55 22)(2 14 47 53 56 23)(3 15 48 54 57 24)(4 16 43 49 58 19)(5 17 44 50 59 20)(6 18 45 51 60 21)(7 34 114 133 140 27)(8 35 109 134 141 28)(9 36 110 135 142 29)(10 31 111 136 143 30)(11 32 112 137 144 25)(12 33 113 138 139 26)(37 81 87 96 69 62)(38 82 88 91 70 63)(39 83 89 92 71 64)(40 84 90 93 72 65)(41 79 85 94 67 66)(42 80 86 95 68 61)(73 98 105 132 123 117)(74 99 106 127 124 118)(75 100 107 128 125 119)(76 101 108 129 126 120)(77 102 103 130 121 115)(78 97 104 131 122 116)
(1 124 52 99)(2 125 53 100)(3 126 54 101)(4 121 49 102)(5 122 50 97)(6 123 51 98)(7 86 133 61)(8 87 134 62)(9 88 135 63)(10 89 136 64)(11 90 137 65)(12 85 138 66)(13 127 55 74)(14 128 56 75)(15 129 57 76)(16 130 58 77)(17 131 59 78)(18 132 60 73)(19 115 43 103)(20 116 44 104)(21 117 45 105)(22 118 46 106)(23 119 47 107)(24 120 48 108)(25 93 112 40)(26 94 113 41)(27 95 114 42)(28 96 109 37)(29 91 110 38)(30 92 111 39)(31 83 143 71)(32 84 144 72)(33 79 139 67)(34 80 140 68)(35 81 141 69)(36 82 142 70)
G:=sub<Sym(144)| (1,66)(2,61)(3,62)(4,63)(5,64)(6,65)(7,125)(8,126)(9,121)(10,122)(11,123)(12,124)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40)(19,70)(20,71)(21,72)(22,67)(23,68)(24,69)(25,132)(26,127)(27,128)(28,129)(29,130)(30,131)(31,116)(32,117)(33,118)(34,119)(35,120)(36,115)(43,82)(44,83)(45,84)(46,79)(47,80)(48,81)(49,88)(50,89)(51,90)(52,85)(53,86)(54,87)(55,94)(56,95)(57,96)(58,91)(59,92)(60,93)(73,112)(74,113)(75,114)(76,109)(77,110)(78,111)(97,136)(98,137)(99,138)(100,133)(101,134)(102,135)(103,142)(104,143)(105,144)(106,139)(107,140)(108,141), (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), (1,57,44)(2,58,45)(3,59,46)(4,60,47)(5,55,48)(6,56,43)(7,144,110)(8,139,111)(9,140,112)(10,141,113)(11,142,114)(12,143,109)(13,24,50)(14,19,51)(15,20,52)(16,21,53)(17,22,54)(18,23,49)(25,135,34)(26,136,35)(27,137,36)(28,138,31)(29,133,32)(30,134,33)(37,71,85)(38,72,86)(39,67,87)(40,68,88)(41,69,89)(42,70,90)(61,91,84)(62,92,79)(63,93,80)(64,94,81)(65,95,82)(66,96,83)(73,121,107)(74,122,108)(75,123,103)(76,124,104)(77,125,105)(78,126,106)(97,120,127)(98,115,128)(99,116,129)(100,117,130)(101,118,131)(102,119,132), (1,13,46,52,55,22)(2,14,47,53,56,23)(3,15,48,54,57,24)(4,16,43,49,58,19)(5,17,44,50,59,20)(6,18,45,51,60,21)(7,34,114,133,140,27)(8,35,109,134,141,28)(9,36,110,135,142,29)(10,31,111,136,143,30)(11,32,112,137,144,25)(12,33,113,138,139,26)(37,81,87,96,69,62)(38,82,88,91,70,63)(39,83,89,92,71,64)(40,84,90,93,72,65)(41,79,85,94,67,66)(42,80,86,95,68,61)(73,98,105,132,123,117)(74,99,106,127,124,118)(75,100,107,128,125,119)(76,101,108,129,126,120)(77,102,103,130,121,115)(78,97,104,131,122,116), (1,124,52,99)(2,125,53,100)(3,126,54,101)(4,121,49,102)(5,122,50,97)(6,123,51,98)(7,86,133,61)(8,87,134,62)(9,88,135,63)(10,89,136,64)(11,90,137,65)(12,85,138,66)(13,127,55,74)(14,128,56,75)(15,129,57,76)(16,130,58,77)(17,131,59,78)(18,132,60,73)(19,115,43,103)(20,116,44,104)(21,117,45,105)(22,118,46,106)(23,119,47,107)(24,120,48,108)(25,93,112,40)(26,94,113,41)(27,95,114,42)(28,96,109,37)(29,91,110,38)(30,92,111,39)(31,83,143,71)(32,84,144,72)(33,79,139,67)(34,80,140,68)(35,81,141,69)(36,82,142,70)>;
G:=Group( (1,66)(2,61)(3,62)(4,63)(5,64)(6,65)(7,125)(8,126)(9,121)(10,122)(11,123)(12,124)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40)(19,70)(20,71)(21,72)(22,67)(23,68)(24,69)(25,132)(26,127)(27,128)(28,129)(29,130)(30,131)(31,116)(32,117)(33,118)(34,119)(35,120)(36,115)(43,82)(44,83)(45,84)(46,79)(47,80)(48,81)(49,88)(50,89)(51,90)(52,85)(53,86)(54,87)(55,94)(56,95)(57,96)(58,91)(59,92)(60,93)(73,112)(74,113)(75,114)(76,109)(77,110)(78,111)(97,136)(98,137)(99,138)(100,133)(101,134)(102,135)(103,142)(104,143)(105,144)(106,139)(107,140)(108,141), (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), (1,57,44)(2,58,45)(3,59,46)(4,60,47)(5,55,48)(6,56,43)(7,144,110)(8,139,111)(9,140,112)(10,141,113)(11,142,114)(12,143,109)(13,24,50)(14,19,51)(15,20,52)(16,21,53)(17,22,54)(18,23,49)(25,135,34)(26,136,35)(27,137,36)(28,138,31)(29,133,32)(30,134,33)(37,71,85)(38,72,86)(39,67,87)(40,68,88)(41,69,89)(42,70,90)(61,91,84)(62,92,79)(63,93,80)(64,94,81)(65,95,82)(66,96,83)(73,121,107)(74,122,108)(75,123,103)(76,124,104)(77,125,105)(78,126,106)(97,120,127)(98,115,128)(99,116,129)(100,117,130)(101,118,131)(102,119,132), (1,13,46,52,55,22)(2,14,47,53,56,23)(3,15,48,54,57,24)(4,16,43,49,58,19)(5,17,44,50,59,20)(6,18,45,51,60,21)(7,34,114,133,140,27)(8,35,109,134,141,28)(9,36,110,135,142,29)(10,31,111,136,143,30)(11,32,112,137,144,25)(12,33,113,138,139,26)(37,81,87,96,69,62)(38,82,88,91,70,63)(39,83,89,92,71,64)(40,84,90,93,72,65)(41,79,85,94,67,66)(42,80,86,95,68,61)(73,98,105,132,123,117)(74,99,106,127,124,118)(75,100,107,128,125,119)(76,101,108,129,126,120)(77,102,103,130,121,115)(78,97,104,131,122,116), (1,124,52,99)(2,125,53,100)(3,126,54,101)(4,121,49,102)(5,122,50,97)(6,123,51,98)(7,86,133,61)(8,87,134,62)(9,88,135,63)(10,89,136,64)(11,90,137,65)(12,85,138,66)(13,127,55,74)(14,128,56,75)(15,129,57,76)(16,130,58,77)(17,131,59,78)(18,132,60,73)(19,115,43,103)(20,116,44,104)(21,117,45,105)(22,118,46,106)(23,119,47,107)(24,120,48,108)(25,93,112,40)(26,94,113,41)(27,95,114,42)(28,96,109,37)(29,91,110,38)(30,92,111,39)(31,83,143,71)(32,84,144,72)(33,79,139,67)(34,80,140,68)(35,81,141,69)(36,82,142,70) );
G=PermutationGroup([[(1,66),(2,61),(3,62),(4,63),(5,64),(6,65),(7,125),(8,126),(9,121),(10,122),(11,123),(12,124),(13,41),(14,42),(15,37),(16,38),(17,39),(18,40),(19,70),(20,71),(21,72),(22,67),(23,68),(24,69),(25,132),(26,127),(27,128),(28,129),(29,130),(30,131),(31,116),(32,117),(33,118),(34,119),(35,120),(36,115),(43,82),(44,83),(45,84),(46,79),(47,80),(48,81),(49,88),(50,89),(51,90),(52,85),(53,86),(54,87),(55,94),(56,95),(57,96),(58,91),(59,92),(60,93),(73,112),(74,113),(75,114),(76,109),(77,110),(78,111),(97,136),(98,137),(99,138),(100,133),(101,134),(102,135),(103,142),(104,143),(105,144),(106,139),(107,140),(108,141)], [(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)], [(1,57,44),(2,58,45),(3,59,46),(4,60,47),(5,55,48),(6,56,43),(7,144,110),(8,139,111),(9,140,112),(10,141,113),(11,142,114),(12,143,109),(13,24,50),(14,19,51),(15,20,52),(16,21,53),(17,22,54),(18,23,49),(25,135,34),(26,136,35),(27,137,36),(28,138,31),(29,133,32),(30,134,33),(37,71,85),(38,72,86),(39,67,87),(40,68,88),(41,69,89),(42,70,90),(61,91,84),(62,92,79),(63,93,80),(64,94,81),(65,95,82),(66,96,83),(73,121,107),(74,122,108),(75,123,103),(76,124,104),(77,125,105),(78,126,106),(97,120,127),(98,115,128),(99,116,129),(100,117,130),(101,118,131),(102,119,132)], [(1,13,46,52,55,22),(2,14,47,53,56,23),(3,15,48,54,57,24),(4,16,43,49,58,19),(5,17,44,50,59,20),(6,18,45,51,60,21),(7,34,114,133,140,27),(8,35,109,134,141,28),(9,36,110,135,142,29),(10,31,111,136,143,30),(11,32,112,137,144,25),(12,33,113,138,139,26),(37,81,87,96,69,62),(38,82,88,91,70,63),(39,83,89,92,71,64),(40,84,90,93,72,65),(41,79,85,94,67,66),(42,80,86,95,68,61),(73,98,105,132,123,117),(74,99,106,127,124,118),(75,100,107,128,125,119),(76,101,108,129,126,120),(77,102,103,130,121,115),(78,97,104,131,122,116)], [(1,124,52,99),(2,125,53,100),(3,126,54,101),(4,121,49,102),(5,122,50,97),(6,123,51,98),(7,86,133,61),(8,87,134,62),(9,88,135,63),(10,89,136,64),(11,90,137,65),(12,85,138,66),(13,127,55,74),(14,128,56,75),(15,129,57,76),(16,130,58,77),(17,131,59,78),(18,132,60,73),(19,115,43,103),(20,116,44,104),(21,117,45,105),(22,118,46,106),(23,119,47,107),(24,120,48,108),(25,93,112,40),(26,94,113,41),(27,95,114,42),(28,96,109,37),(29,91,110,38),(30,92,111,39),(31,83,143,71),(32,84,144,72),(33,79,139,67),(34,80,140,68),(35,81,141,69),(36,82,142,70)]])
144 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | ··· | 3N | 4A | ··· | 4H | 6A | ··· | 6N | 6O | ··· | 6CT | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | Dic3 | D6 | C3×S3 | C3×Dic3 | S3×C6 |
| kernel | C2×C6×C3⋊Dic3 | C6×C3⋊Dic3 | C63 | C22×C3⋊Dic3 | C3×C62 | C2×C3⋊Dic3 | C2×C62 | C62 | C2×C62 | C62 | C62 | C22×C6 | C2×C6 | C2×C6 |
| # reps | 1 | 6 | 1 | 2 | 8 | 12 | 2 | 16 | 4 | 16 | 12 | 8 | 32 | 24 |
Matrix representation of C2×C6×C3⋊Dic3 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 7 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 8 | 0 | 0 |
| 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[4,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,9,4,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,9,7,0,0,0,0,3],[1,0,0,0,0,0,12,3,0,0,0,8,1,0,0,0,0,0,12,0,0,0,0,12,1] >;
C2×C6×C3⋊Dic3 in GAP, Magma, Sage, TeX
C_2\times C_6\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("C2xC6xC3:Dic3"); // GroupNames label
G:=SmallGroup(432,718);
// by ID
G=gap.SmallGroup(432,718);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^3=d^6=1,e^2=d^3,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,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations