direct product, metabelian, supersoluble, monomial
Aliases: C2×C12.D6, C62.278C23, (C6×D4)⋊6S3, (C3×D4)⋊17D6, C6⋊5(D4⋊2S3), (C2×C12).169D6, C6.59(S3×C23), (C3×C6).58C24, (C22×C6).101D6, C12.110(C22×S3), (C3×C12).129C23, (C6×C12).168C22, (D4×C32)⋊24C22, C32⋊7D4⋊12C22, C3⋊Dic3.47C23, (C2×C62).84C22, C32⋊4Q8⋊23C22, D4⋊5(C2×C3⋊S3), (D4×C3×C6)⋊13C2, (C2×D4)⋊8(C3⋊S3), C3⋊6(C2×D4⋊2S3), C32⋊16(C2×C4○D4), C2.7(C23×C3⋊S3), (C3×C6)⋊10(C4○D4), (C4×C3⋊S3)⋊15C22, C23.24(C2×C3⋊S3), C4.20(C22×C3⋊S3), (C2×C3⋊S3).51C23, (C2×C32⋊7D4)⋊19C2, (C2×C32⋊4Q8)⋊21C2, C22.1(C22×C3⋊S3), (C2×C6).287(C22×S3), (C22×C3⋊Dic3)⋊15C2, (C2×C3⋊Dic3)⋊27C22, (C22×C3⋊S3).107C22, (C2×C4×C3⋊S3)⋊8C2, (C2×C4).60(C2×C3⋊S3), SmallGroup(288,1008)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.D6
G = < a,b,c,d | a2=b12=c6=1, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b-1, dcd-1=b6c-1 >
Subgroups: 1508 in 492 conjugacy classes, 165 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C2×C4○D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C2×Dic6, S3×C2×C4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C2×C3⋊Dic3, C32⋊7D4, C6×C12, D4×C32, C22×C3⋊S3, C2×C62, C2×D4⋊2S3, C2×C32⋊4Q8, C2×C4×C3⋊S3, C12.D6, C22×C3⋊Dic3, C2×C32⋊7D4, D4×C3×C6, C2×C12.D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C3⋊S3, C22×S3, C2×C4○D4, C2×C3⋊S3, D4⋊2S3, S3×C23, C22×C3⋊S3, C2×D4⋊2S3, C12.D6, C23×C3⋊S3, C2×C12.D6
►On 144 points
Generators in S144
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 49)(10 50)(11 51)(12 52)(13 80)(14 81)(15 82)(16 83)(17 84)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 141)(32 142)(33 143)(34 144)(35 133)(36 134)(37 131)(38 132)(39 121)(40 122)(41 123)(42 124)(43 125)(44 126)(45 127)(46 128)(47 129)(48 130)(61 106)(62 107)(63 108)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(85 119)(86 120)(87 109)(88 110)(89 111)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)
(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 23 38 110 25 103)(2 18 39 117 26 98)(3 13 40 112 27 105)(4 20 41 119 28 100)(5 15 42 114 29 107)(6 22 43 109 30 102)(7 17 44 116 31 97)(8 24 45 111 32 104)(9 19 46 118 33 99)(10 14 47 113 34 106)(11 21 48 120 35 101)(12 16 37 115 36 108)(49 74 128 96 143 66)(50 81 129 91 144 61)(51 76 130 86 133 68)(52 83 131 93 134 63)(53 78 132 88 135 70)(54 73 121 95 136 65)(55 80 122 90 137 72)(56 75 123 85 138 67)(57 82 124 92 139 62)(58 77 125 87 140 69)(59 84 126 94 141 64)(60 79 127 89 142 71)
(1 103 7 97)(2 102 8 108)(3 101 9 107)(4 100 10 106)(5 99 11 105)(6 98 12 104)(13 29 19 35)(14 28 20 34)(15 27 21 33)(16 26 22 32)(17 25 23 31)(18 36 24 30)(37 111 43 117)(38 110 44 116)(39 109 45 115)(40 120 46 114)(41 119 47 113)(42 118 48 112)(49 62 55 68)(50 61 56 67)(51 72 57 66)(52 71 58 65)(53 70 59 64)(54 69 60 63)(73 134 79 140)(74 133 80 139)(75 144 81 138)(76 143 82 137)(77 142 83 136)(78 141 84 135)(85 129 91 123)(86 128 92 122)(87 127 93 121)(88 126 94 132)(89 125 95 131)(90 124 96 130)
G:=sub<Sym(144)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,49)(10,50)(11,51)(12,52)(13,80)(14,81)(15,82)(16,83)(17,84)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,133)(36,134)(37,131)(38,132)(39,121)(40,122)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(61,106)(62,107)(63,108)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(85,119)(86,120)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118), (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), (1,23,38,110,25,103)(2,18,39,117,26,98)(3,13,40,112,27,105)(4,20,41,119,28,100)(5,15,42,114,29,107)(6,22,43,109,30,102)(7,17,44,116,31,97)(8,24,45,111,32,104)(9,19,46,118,33,99)(10,14,47,113,34,106)(11,21,48,120,35,101)(12,16,37,115,36,108)(49,74,128,96,143,66)(50,81,129,91,144,61)(51,76,130,86,133,68)(52,83,131,93,134,63)(53,78,132,88,135,70)(54,73,121,95,136,65)(55,80,122,90,137,72)(56,75,123,85,138,67)(57,82,124,92,139,62)(58,77,125,87,140,69)(59,84,126,94,141,64)(60,79,127,89,142,71), (1,103,7,97)(2,102,8,108)(3,101,9,107)(4,100,10,106)(5,99,11,105)(6,98,12,104)(13,29,19,35)(14,28,20,34)(15,27,21,33)(16,26,22,32)(17,25,23,31)(18,36,24,30)(37,111,43,117)(38,110,44,116)(39,109,45,115)(40,120,46,114)(41,119,47,113)(42,118,48,112)(49,62,55,68)(50,61,56,67)(51,72,57,66)(52,71,58,65)(53,70,59,64)(54,69,60,63)(73,134,79,140)(74,133,80,139)(75,144,81,138)(76,143,82,137)(77,142,83,136)(78,141,84,135)(85,129,91,123)(86,128,92,122)(87,127,93,121)(88,126,94,132)(89,125,95,131)(90,124,96,130)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,49)(10,50)(11,51)(12,52)(13,80)(14,81)(15,82)(16,83)(17,84)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,135)(26,136)(27,137)(28,138)(29,139)(30,140)(31,141)(32,142)(33,143)(34,144)(35,133)(36,134)(37,131)(38,132)(39,121)(40,122)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(61,106)(62,107)(63,108)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(85,119)(86,120)(87,109)(88,110)(89,111)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118), (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), (1,23,38,110,25,103)(2,18,39,117,26,98)(3,13,40,112,27,105)(4,20,41,119,28,100)(5,15,42,114,29,107)(6,22,43,109,30,102)(7,17,44,116,31,97)(8,24,45,111,32,104)(9,19,46,118,33,99)(10,14,47,113,34,106)(11,21,48,120,35,101)(12,16,37,115,36,108)(49,74,128,96,143,66)(50,81,129,91,144,61)(51,76,130,86,133,68)(52,83,131,93,134,63)(53,78,132,88,135,70)(54,73,121,95,136,65)(55,80,122,90,137,72)(56,75,123,85,138,67)(57,82,124,92,139,62)(58,77,125,87,140,69)(59,84,126,94,141,64)(60,79,127,89,142,71), (1,103,7,97)(2,102,8,108)(3,101,9,107)(4,100,10,106)(5,99,11,105)(6,98,12,104)(13,29,19,35)(14,28,20,34)(15,27,21,33)(16,26,22,32)(17,25,23,31)(18,36,24,30)(37,111,43,117)(38,110,44,116)(39,109,45,115)(40,120,46,114)(41,119,47,113)(42,118,48,112)(49,62,55,68)(50,61,56,67)(51,72,57,66)(52,71,58,65)(53,70,59,64)(54,69,60,63)(73,134,79,140)(74,133,80,139)(75,144,81,138)(76,143,82,137)(77,142,83,136)(78,141,84,135)(85,129,91,123)(86,128,92,122)(87,127,93,121)(88,126,94,132)(89,125,95,131)(90,124,96,130) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,49),(10,50),(11,51),(12,52),(13,80),(14,81),(15,82),(16,83),(17,84),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,135),(26,136),(27,137),(28,138),(29,139),(30,140),(31,141),(32,142),(33,143),(34,144),(35,133),(36,134),(37,131),(38,132),(39,121),(40,122),(41,123),(42,124),(43,125),(44,126),(45,127),(46,128),(47,129),(48,130),(61,106),(62,107),(63,108),(64,97),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(85,119),(86,120),(87,109),(88,110),(89,111),(90,112),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118)], [(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)], [(1,23,38,110,25,103),(2,18,39,117,26,98),(3,13,40,112,27,105),(4,20,41,119,28,100),(5,15,42,114,29,107),(6,22,43,109,30,102),(7,17,44,116,31,97),(8,24,45,111,32,104),(9,19,46,118,33,99),(10,14,47,113,34,106),(11,21,48,120,35,101),(12,16,37,115,36,108),(49,74,128,96,143,66),(50,81,129,91,144,61),(51,76,130,86,133,68),(52,83,131,93,134,63),(53,78,132,88,135,70),(54,73,121,95,136,65),(55,80,122,90,137,72),(56,75,123,85,138,67),(57,82,124,92,139,62),(58,77,125,87,140,69),(59,84,126,94,141,64),(60,79,127,89,142,71)], [(1,103,7,97),(2,102,8,108),(3,101,9,107),(4,100,10,106),(5,99,11,105),(6,98,12,104),(13,29,19,35),(14,28,20,34),(15,27,21,33),(16,26,22,32),(17,25,23,31),(18,36,24,30),(37,111,43,117),(38,110,44,116),(39,109,45,115),(40,120,46,114),(41,119,47,113),(42,118,48,112),(49,62,55,68),(50,61,56,67),(51,72,57,66),(52,71,58,65),(53,70,59,64),(54,69,60,63),(73,134,79,140),(74,133,80,139),(75,144,81,138),(76,143,82,137),(77,142,83,136),(78,141,84,135),(85,129,91,123),(86,128,92,122),(87,127,93,121),(88,126,94,132),(89,125,95,131),(90,124,96,130)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6L | 6M | ··· | 6AB | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | D4⋊2S3 |
| kernel | C2×C12.D6 | C2×C32⋊4Q8 | C2×C4×C3⋊S3 | C12.D6 | C22×C3⋊Dic3 | C2×C32⋊7D4 | D4×C3×C6 | C6×D4 | C2×C12 | C3×D4 | C22×C6 | C3×C6 | C6 |
| # reps | 1 | 1 | 1 | 8 | 2 | 2 | 1 | 4 | 4 | 16 | 8 | 4 | 8 |
Matrix representation of C2×C12.D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C2×C12.D6 in GAP, Magma, Sage, TeX
C_2\times C_{12}.D_6 % in TeX
G:=Group("C2xC12.D6"); // GroupNames label
G:=SmallGroup(288,1008);
// by ID
G=gap.SmallGroup(288,1008);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,185,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^6=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations