direct product, metabelian, supersoluble, monomial, A-group
Aliases: C12×C3⋊Dic3, C33⋊8C42, C62.140D6, (C3×C12)⋊10C12, C6.25(S3×C12), (C6×C12).45C6, (C6×C12).53S3, (C32×C12)⋊9C4, C32⋊6(C4×C12), C3⋊2(Dic3×C12), (C3×C12)⋊9Dic3, C12⋊2(C3×Dic3), C62.64(C2×C6), C6.26(C6×Dic3), C32⋊9(C4×Dic3), (C3×C62).46C22, C6.26(C4×C3⋊S3), C2.2(C12×C3⋊S3), (C3×C6×C12).16C2, (C3×C6).77(C4×S3), (C2×C6).67(S3×C6), C22.3(C6×C3⋊S3), C2.2(C6×C3⋊Dic3), (C2×C12).39(C3×S3), (C3×C6).63(C2×C12), C6.22(C2×C3⋊Dic3), (C2×C12).32(C3⋊S3), (C6×C3⋊Dic3).20C2, (C2×C3⋊Dic3).14C6, (C32×C6).56(C2×C4), (C3×C6).68(C2×Dic3), (C2×C4).6(C3×C3⋊S3), (C2×C6).61(C2×C3⋊S3), SmallGroup(432,487)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C12×C3⋊Dic3 |
Generators and relations for C12×C3⋊Dic3
G = < a,b,c,d | a12=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 532 in 244 conjugacy classes, 110 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C62, C4×Dic3, C4×C12, C32×C6, C32×C6, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C6×C12, C3×C3⋊Dic3, C32×C12, C3×C62, Dic3×C12, C4×C3⋊Dic3, C6×C3⋊Dic3, C3×C6×C12, C12×C3⋊Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C42, C3×S3, C3⋊S3, C4×S3, C2×Dic3, C2×C12, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C4×Dic3, C4×C12, C3×C3⋊S3, S3×C12, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C3×C3⋊Dic3, C6×C3⋊S3, Dic3×C12, C4×C3⋊Dic3, C12×C3⋊S3, C6×C3⋊Dic3, C12×C3⋊Dic3
►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 40 56)(2 41 57)(3 42 58)(4 43 59)(5 44 60)(6 45 49)(7 46 50)(8 47 51)(9 48 52)(10 37 53)(11 38 54)(12 39 55)(13 87 35)(14 88 36)(15 89 25)(16 90 26)(17 91 27)(18 92 28)(19 93 29)(20 94 30)(21 95 31)(22 96 32)(23 85 33)(24 86 34)(61 76 123)(62 77 124)(63 78 125)(64 79 126)(65 80 127)(66 81 128)(67 82 129)(68 83 130)(69 84 131)(70 73 132)(71 74 121)(72 75 122)(97 142 114)(98 143 115)(99 144 116)(100 133 117)(101 134 118)(102 135 119)(103 136 120)(104 137 109)(105 138 110)(106 139 111)(107 140 112)(108 141 113)
(1 144 60 103 48 112)(2 133 49 104 37 113)(3 134 50 105 38 114)(4 135 51 106 39 115)(5 136 52 107 40 116)(6 137 53 108 41 117)(7 138 54 97 42 118)(8 139 55 98 43 119)(9 140 56 99 44 120)(10 141 57 100 45 109)(11 142 58 101 46 110)(12 143 59 102 47 111)(13 71 31 129 91 78)(14 72 32 130 92 79)(15 61 33 131 93 80)(16 62 34 132 94 81)(17 63 35 121 95 82)(18 64 36 122 96 83)(19 65 25 123 85 84)(20 66 26 124 86 73)(21 67 27 125 87 74)(22 68 28 126 88 75)(23 69 29 127 89 76)(24 70 30 128 90 77)
(1 131 103 15)(2 132 104 16)(3 121 105 17)(4 122 106 18)(5 123 107 19)(6 124 108 20)(7 125 97 21)(8 126 98 22)(9 127 99 23)(10 128 100 24)(11 129 101 13)(12 130 102 14)(25 40 84 136)(26 41 73 137)(27 42 74 138)(28 43 75 139)(29 44 76 140)(30 45 77 141)(31 46 78 142)(32 47 79 143)(33 48 80 144)(34 37 81 133)(35 38 82 134)(36 39 83 135)(49 62 113 94)(50 63 114 95)(51 64 115 96)(52 65 116 85)(53 66 117 86)(54 67 118 87)(55 68 119 88)(56 69 120 89)(57 70 109 90)(58 71 110 91)(59 72 111 92)(60 61 112 93)
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), (1,40,56)(2,41,57)(3,42,58)(4,43,59)(5,44,60)(6,45,49)(7,46,50)(8,47,51)(9,48,52)(10,37,53)(11,38,54)(12,39,55)(13,87,35)(14,88,36)(15,89,25)(16,90,26)(17,91,27)(18,92,28)(19,93,29)(20,94,30)(21,95,31)(22,96,32)(23,85,33)(24,86,34)(61,76,123)(62,77,124)(63,78,125)(64,79,126)(65,80,127)(66,81,128)(67,82,129)(68,83,130)(69,84,131)(70,73,132)(71,74,121)(72,75,122)(97,142,114)(98,143,115)(99,144,116)(100,133,117)(101,134,118)(102,135,119)(103,136,120)(104,137,109)(105,138,110)(106,139,111)(107,140,112)(108,141,113), (1,144,60,103,48,112)(2,133,49,104,37,113)(3,134,50,105,38,114)(4,135,51,106,39,115)(5,136,52,107,40,116)(6,137,53,108,41,117)(7,138,54,97,42,118)(8,139,55,98,43,119)(9,140,56,99,44,120)(10,141,57,100,45,109)(11,142,58,101,46,110)(12,143,59,102,47,111)(13,71,31,129,91,78)(14,72,32,130,92,79)(15,61,33,131,93,80)(16,62,34,132,94,81)(17,63,35,121,95,82)(18,64,36,122,96,83)(19,65,25,123,85,84)(20,66,26,124,86,73)(21,67,27,125,87,74)(22,68,28,126,88,75)(23,69,29,127,89,76)(24,70,30,128,90,77), (1,131,103,15)(2,132,104,16)(3,121,105,17)(4,122,106,18)(5,123,107,19)(6,124,108,20)(7,125,97,21)(8,126,98,22)(9,127,99,23)(10,128,100,24)(11,129,101,13)(12,130,102,14)(25,40,84,136)(26,41,73,137)(27,42,74,138)(28,43,75,139)(29,44,76,140)(30,45,77,141)(31,46,78,142)(32,47,79,143)(33,48,80,144)(34,37,81,133)(35,38,82,134)(36,39,83,135)(49,62,113,94)(50,63,114,95)(51,64,115,96)(52,65,116,85)(53,66,117,86)(54,67,118,87)(55,68,119,88)(56,69,120,89)(57,70,109,90)(58,71,110,91)(59,72,111,92)(60,61,112,93)>;
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), (1,40,56)(2,41,57)(3,42,58)(4,43,59)(5,44,60)(6,45,49)(7,46,50)(8,47,51)(9,48,52)(10,37,53)(11,38,54)(12,39,55)(13,87,35)(14,88,36)(15,89,25)(16,90,26)(17,91,27)(18,92,28)(19,93,29)(20,94,30)(21,95,31)(22,96,32)(23,85,33)(24,86,34)(61,76,123)(62,77,124)(63,78,125)(64,79,126)(65,80,127)(66,81,128)(67,82,129)(68,83,130)(69,84,131)(70,73,132)(71,74,121)(72,75,122)(97,142,114)(98,143,115)(99,144,116)(100,133,117)(101,134,118)(102,135,119)(103,136,120)(104,137,109)(105,138,110)(106,139,111)(107,140,112)(108,141,113), (1,144,60,103,48,112)(2,133,49,104,37,113)(3,134,50,105,38,114)(4,135,51,106,39,115)(5,136,52,107,40,116)(6,137,53,108,41,117)(7,138,54,97,42,118)(8,139,55,98,43,119)(9,140,56,99,44,120)(10,141,57,100,45,109)(11,142,58,101,46,110)(12,143,59,102,47,111)(13,71,31,129,91,78)(14,72,32,130,92,79)(15,61,33,131,93,80)(16,62,34,132,94,81)(17,63,35,121,95,82)(18,64,36,122,96,83)(19,65,25,123,85,84)(20,66,26,124,86,73)(21,67,27,125,87,74)(22,68,28,126,88,75)(23,69,29,127,89,76)(24,70,30,128,90,77), (1,131,103,15)(2,132,104,16)(3,121,105,17)(4,122,106,18)(5,123,107,19)(6,124,108,20)(7,125,97,21)(8,126,98,22)(9,127,99,23)(10,128,100,24)(11,129,101,13)(12,130,102,14)(25,40,84,136)(26,41,73,137)(27,42,74,138)(28,43,75,139)(29,44,76,140)(30,45,77,141)(31,46,78,142)(32,47,79,143)(33,48,80,144)(34,37,81,133)(35,38,82,134)(36,39,83,135)(49,62,113,94)(50,63,114,95)(51,64,115,96)(52,65,116,85)(53,66,117,86)(54,67,118,87)(55,68,119,88)(56,69,120,89)(57,70,109,90)(58,71,110,91)(59,72,111,92)(60,61,112,93) );
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)], [(1,40,56),(2,41,57),(3,42,58),(4,43,59),(5,44,60),(6,45,49),(7,46,50),(8,47,51),(9,48,52),(10,37,53),(11,38,54),(12,39,55),(13,87,35),(14,88,36),(15,89,25),(16,90,26),(17,91,27),(18,92,28),(19,93,29),(20,94,30),(21,95,31),(22,96,32),(23,85,33),(24,86,34),(61,76,123),(62,77,124),(63,78,125),(64,79,126),(65,80,127),(66,81,128),(67,82,129),(68,83,130),(69,84,131),(70,73,132),(71,74,121),(72,75,122),(97,142,114),(98,143,115),(99,144,116),(100,133,117),(101,134,118),(102,135,119),(103,136,120),(104,137,109),(105,138,110),(106,139,111),(107,140,112),(108,141,113)], [(1,144,60,103,48,112),(2,133,49,104,37,113),(3,134,50,105,38,114),(4,135,51,106,39,115),(5,136,52,107,40,116),(6,137,53,108,41,117),(7,138,54,97,42,118),(8,139,55,98,43,119),(9,140,56,99,44,120),(10,141,57,100,45,109),(11,142,58,101,46,110),(12,143,59,102,47,111),(13,71,31,129,91,78),(14,72,32,130,92,79),(15,61,33,131,93,80),(16,62,34,132,94,81),(17,63,35,121,95,82),(18,64,36,122,96,83),(19,65,25,123,85,84),(20,66,26,124,86,73),(21,67,27,125,87,74),(22,68,28,126,88,75),(23,69,29,127,89,76),(24,70,30,128,90,77)], [(1,131,103,15),(2,132,104,16),(3,121,105,17),(4,122,106,18),(5,123,107,19),(6,124,108,20),(7,125,97,21),(8,126,98,22),(9,127,99,23),(10,128,100,24),(11,129,101,13),(12,130,102,14),(25,40,84,136),(26,41,73,137),(27,42,74,138),(28,43,75,139),(29,44,76,140),(30,45,77,141),(31,46,78,142),(32,47,79,143),(33,48,80,144),(34,37,81,133),(35,38,82,134),(36,39,83,135),(49,62,113,94),(50,63,114,95),(51,64,115,96),(52,65,116,85),(53,66,117,86),(54,67,118,87),(55,68,119,88),(56,69,120,89),(57,70,109,90),(58,71,110,91),(59,72,111,92),(60,61,112,93)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6AP | 12A | ··· | 12H | 12I | ··· | 12BD | 12BE | ··· | 12BT |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | C3×S3 | C4×S3 | C3×Dic3 | S3×C6 | S3×C12 |
| kernel | C12×C3⋊Dic3 | C6×C3⋊Dic3 | C3×C6×C12 | C4×C3⋊Dic3 | C3×C3⋊Dic3 | C32×C12 | C2×C3⋊Dic3 | C6×C12 | C3⋊Dic3 | C3×C12 | C6×C12 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 16 | 8 | 4 | 8 | 4 | 8 | 16 | 16 | 8 | 32 |
Matrix representation of C12×C3⋊Dic3 ►in GL5(𝔽13)
| 7 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(5,GF(13))| [7,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,11,0,0,0,0,0,11],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,9],[1,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0,0,4,0,0,0,3,0] >;
C12×C3⋊Dic3 in GAP, Magma, Sage, TeX
C_{12}\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("C12xC3:Dic3"); // GroupNames label
G:=SmallGroup(432,487);
// by ID
G=gap.SmallGroup(432,487);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,176,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^3=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations