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