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 [×2], C2 [×6], C3 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], S3 [×8], C6 [×12], C6 [×16], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, C32, Dic3 [×24], C12 [×8], D6 [×16], C2×C6 [×20], C2×C6 [×16], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×4], Dic6 [×16], C4×S3 [×16], C2×Dic3 [×44], C3⋊D4 [×32], C2×C12 [×4], C3×D4 [×16], C22×S3 [×4], C22×C6 [×8], C2×C4○D4, C3⋊Dic3 [×6], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×4], C62 [×4], C2×Dic6 [×4], S3×C2×C4 [×4], D4⋊2S3 [×32], C22×Dic3 [×8], C2×C3⋊D4 [×8], C6×D4 [×4], C32⋊4Q8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×10], C32⋊7D4 [×8], C6×C12, D4×C32 [×4], C22×C3⋊S3, C2×C62 [×2], C2×D4⋊2S3 [×4], C2×C32⋊4Q8, C2×C4×C3⋊S3, C12.D6 [×8], C22×C3⋊Dic3 [×2], C2×C32⋊7D4 [×2], D4×C3×C6, C2×C12.D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×4], C23 [×15], D6 [×28], C4○D4 [×2], C24, C3⋊S3, C22×S3 [×28], C2×C4○D4, C2×C3⋊S3 [×7], D4⋊2S3 [×8], S3×C23 [×4], C22×C3⋊S3 [×7], C2×D4⋊2S3 [×4], C12.D6 [×2], C23×C3⋊S3, C2×C12.D6
►On 144 points
Generators in S144
(1 104)(2 105)(3 106)(4 107)(5 108)(6 97)(7 98)(8 99)(9 100)(10 101)(11 102)(12 103)(13 144)(14 133)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 141)(23 142)(24 143)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 61)(34 62)(35 63)(36 64)(37 129)(38 130)(39 131)(40 132)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 81)(50 82)(51 83)(52 84)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(85 117)(86 118)(87 119)(88 120)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)
(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 81 137 115 43 25)(2 76 138 110 44 32)(3 83 139 117 45 27)(4 78 140 112 46 34)(5 73 141 119 47 29)(6 80 142 114 48 36)(7 75 143 109 37 31)(8 82 144 116 38 26)(9 77 133 111 39 33)(10 84 134 118 40 28)(11 79 135 113 41 35)(12 74 136 120 42 30)(13 96 130 66 99 50)(14 91 131 61 100 57)(15 86 132 68 101 52)(16 93 121 63 102 59)(17 88 122 70 103 54)(18 95 123 65 104 49)(19 90 124 72 105 56)(20 85 125 67 106 51)(21 92 126 62 107 58)(22 87 127 69 108 53)(23 94 128 64 97 60)(24 89 129 71 98 55)
(1 25 7 31)(2 36 8 30)(3 35 9 29)(4 34 10 28)(5 33 11 27)(6 32 12 26)(13 88 19 94)(14 87 20 93)(15 86 21 92)(16 85 22 91)(17 96 23 90)(18 95 24 89)(37 75 43 81)(38 74 44 80)(39 73 45 79)(40 84 46 78)(41 83 47 77)(42 82 48 76)(49 129 55 123)(50 128 56 122)(51 127 57 121)(52 126 58 132)(53 125 59 131)(54 124 60 130)(61 102 67 108)(62 101 68 107)(63 100 69 106)(64 99 70 105)(65 98 71 104)(66 97 72 103)(109 137 115 143)(110 136 116 142)(111 135 117 141)(112 134 118 140)(113 133 119 139)(114 144 120 138)
G:=sub<Sym(144)| (1,104)(2,105)(3,106)(4,107)(5,108)(6,97)(7,98)(8,99)(9,100)(10,101)(11,102)(12,103)(13,144)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,141)(23,142)(24,143)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,61)(34,62)(35,63)(36,64)(37,129)(38,130)(39,131)(40,132)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(85,117)(86,118)(87,119)(88,120)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116), (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), (1,81,137,115,43,25)(2,76,138,110,44,32)(3,83,139,117,45,27)(4,78,140,112,46,34)(5,73,141,119,47,29)(6,80,142,114,48,36)(7,75,143,109,37,31)(8,82,144,116,38,26)(9,77,133,111,39,33)(10,84,134,118,40,28)(11,79,135,113,41,35)(12,74,136,120,42,30)(13,96,130,66,99,50)(14,91,131,61,100,57)(15,86,132,68,101,52)(16,93,121,63,102,59)(17,88,122,70,103,54)(18,95,123,65,104,49)(19,90,124,72,105,56)(20,85,125,67,106,51)(21,92,126,62,107,58)(22,87,127,69,108,53)(23,94,128,64,97,60)(24,89,129,71,98,55), (1,25,7,31)(2,36,8,30)(3,35,9,29)(4,34,10,28)(5,33,11,27)(6,32,12,26)(13,88,19,94)(14,87,20,93)(15,86,21,92)(16,85,22,91)(17,96,23,90)(18,95,24,89)(37,75,43,81)(38,74,44,80)(39,73,45,79)(40,84,46,78)(41,83,47,77)(42,82,48,76)(49,129,55,123)(50,128,56,122)(51,127,57,121)(52,126,58,132)(53,125,59,131)(54,124,60,130)(61,102,67,108)(62,101,68,107)(63,100,69,106)(64,99,70,105)(65,98,71,104)(66,97,72,103)(109,137,115,143)(110,136,116,142)(111,135,117,141)(112,134,118,140)(113,133,119,139)(114,144,120,138)>;
G:=Group( (1,104)(2,105)(3,106)(4,107)(5,108)(6,97)(7,98)(8,99)(9,100)(10,101)(11,102)(12,103)(13,144)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,141)(23,142)(24,143)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,61)(34,62)(35,63)(36,64)(37,129)(38,130)(39,131)(40,132)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(85,117)(86,118)(87,119)(88,120)(89,109)(90,110)(91,111)(92,112)(93,113)(94,114)(95,115)(96,116), (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), (1,81,137,115,43,25)(2,76,138,110,44,32)(3,83,139,117,45,27)(4,78,140,112,46,34)(5,73,141,119,47,29)(6,80,142,114,48,36)(7,75,143,109,37,31)(8,82,144,116,38,26)(9,77,133,111,39,33)(10,84,134,118,40,28)(11,79,135,113,41,35)(12,74,136,120,42,30)(13,96,130,66,99,50)(14,91,131,61,100,57)(15,86,132,68,101,52)(16,93,121,63,102,59)(17,88,122,70,103,54)(18,95,123,65,104,49)(19,90,124,72,105,56)(20,85,125,67,106,51)(21,92,126,62,107,58)(22,87,127,69,108,53)(23,94,128,64,97,60)(24,89,129,71,98,55), (1,25,7,31)(2,36,8,30)(3,35,9,29)(4,34,10,28)(5,33,11,27)(6,32,12,26)(13,88,19,94)(14,87,20,93)(15,86,21,92)(16,85,22,91)(17,96,23,90)(18,95,24,89)(37,75,43,81)(38,74,44,80)(39,73,45,79)(40,84,46,78)(41,83,47,77)(42,82,48,76)(49,129,55,123)(50,128,56,122)(51,127,57,121)(52,126,58,132)(53,125,59,131)(54,124,60,130)(61,102,67,108)(62,101,68,107)(63,100,69,106)(64,99,70,105)(65,98,71,104)(66,97,72,103)(109,137,115,143)(110,136,116,142)(111,135,117,141)(112,134,118,140)(113,133,119,139)(114,144,120,138) );
G=PermutationGroup([(1,104),(2,105),(3,106),(4,107),(5,108),(6,97),(7,98),(8,99),(9,100),(10,101),(11,102),(12,103),(13,144),(14,133),(15,134),(16,135),(17,136),(18,137),(19,138),(20,139),(21,140),(22,141),(23,142),(24,143),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,61),(34,62),(35,63),(36,64),(37,129),(38,130),(39,131),(40,132),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,81),(50,82),(51,83),(52,84),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(85,117),(86,118),(87,119),(88,120),(89,109),(90,110),(91,111),(92,112),(93,113),(94,114),(95,115),(96,116)], [(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)], [(1,81,137,115,43,25),(2,76,138,110,44,32),(3,83,139,117,45,27),(4,78,140,112,46,34),(5,73,141,119,47,29),(6,80,142,114,48,36),(7,75,143,109,37,31),(8,82,144,116,38,26),(9,77,133,111,39,33),(10,84,134,118,40,28),(11,79,135,113,41,35),(12,74,136,120,42,30),(13,96,130,66,99,50),(14,91,131,61,100,57),(15,86,132,68,101,52),(16,93,121,63,102,59),(17,88,122,70,103,54),(18,95,123,65,104,49),(19,90,124,72,105,56),(20,85,125,67,106,51),(21,92,126,62,107,58),(22,87,127,69,108,53),(23,94,128,64,97,60),(24,89,129,71,98,55)], [(1,25,7,31),(2,36,8,30),(3,35,9,29),(4,34,10,28),(5,33,11,27),(6,32,12,26),(13,88,19,94),(14,87,20,93),(15,86,21,92),(16,85,22,91),(17,96,23,90),(18,95,24,89),(37,75,43,81),(38,74,44,80),(39,73,45,79),(40,84,46,78),(41,83,47,77),(42,82,48,76),(49,129,55,123),(50,128,56,122),(51,127,57,121),(52,126,58,132),(53,125,59,131),(54,124,60,130),(61,102,67,108),(62,101,68,107),(63,100,69,106),(64,99,70,105),(65,98,71,104),(66,97,72,103),(109,137,115,143),(110,136,116,142),(111,135,117,141),(112,134,118,140),(113,133,119,139),(114,144,120,138)])
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