direct product, metabelian, supersoluble, monomial
Aliases: C2×C12.26D6, C62.281C23, (C6×Q8)⋊10S3, (C3×Q8)⋊19D6, C6⋊3(Q8⋊3S3), (C2×C12).171D6, C6.62(S3×C23), (C3×C6).61C24, C12⋊S3⋊27C22, C12.113(C22×S3), (C3×C12).132C23, (C6×C12).170C22, C3⋊Dic3.54C23, (Q8×C32)⋊22C22, Q8⋊6(C2×C3⋊S3), (Q8×C3×C6)⋊13C2, (C2×Q8)⋊8(C3⋊S3), C3⋊4(C2×Q8⋊3S3), C32⋊18(C2×C4○D4), (C3×C6)⋊12(C4○D4), (C4×C3⋊S3)⋊16C22, (C2×C12⋊S3)⋊21C2, C4.23(C22×C3⋊S3), C2.10(C23×C3⋊S3), (C2×C3⋊S3).53C23, (C2×C6).289(C22×S3), C22.32(C22×C3⋊S3), (C22×C3⋊S3).108C22, (C2×C3⋊Dic3).188C22, (C2×C4×C3⋊S3)⋊9C2, (C2×C4).62(C2×C3⋊S3), SmallGroup(288,1011)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C12.26D6
G = < a,b,c,d | a2=b12=1, c6=d2=b6, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=c5 >
Subgroups: 1764 in 492 conjugacy classes, 165 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×4], C4 [×6], C4 [×2], C22, C22 [×12], S3 [×24], C6 [×12], C2×C4 [×3], C2×C4 [×13], D4 [×12], Q8 [×4], C23 [×3], C32, Dic3 [×8], C12 [×24], D6 [×48], C2×C6 [×4], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3⋊S3 [×6], C3×C6, C3×C6 [×2], C4×S3 [×48], D12 [×48], C2×Dic3 [×4], C2×C12 [×12], C3×Q8 [×16], C22×S3 [×12], C2×C4○D4, C3⋊Dic3 [×2], C3×C12 [×6], C2×C3⋊S3 [×6], C2×C3⋊S3 [×6], C62, S3×C2×C4 [×12], C2×D12 [×12], Q8⋊3S3 [×32], C6×Q8 [×4], C4×C3⋊S3 [×12], C12⋊S3 [×12], C2×C3⋊Dic3, C6×C12 [×3], Q8×C32 [×4], C22×C3⋊S3 [×3], C2×Q8⋊3S3 [×4], C2×C4×C3⋊S3 [×3], C2×C12⋊S3 [×3], C12.26D6 [×8], Q8×C3×C6, C2×C12.26D6
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], Q8⋊3S3 [×8], S3×C23 [×4], C22×C3⋊S3 [×7], C2×Q8⋊3S3 [×4], C12.26D6 [×2], C23×C3⋊S3, C2×C12.26D6
►On 144 points
Generators in S144
(1 106)(2 107)(3 108)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 117)(26 118)(27 119)(28 120)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 84)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(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)
(1 134 49 109 24 44 7 140 55 115 18 38)(2 141 50 116 13 39 8 135 56 110 19 45)(3 136 51 111 14 46 9 142 57 117 20 40)(4 143 52 118 15 41 10 137 58 112 21 47)(5 138 53 113 16 48 11 144 59 119 22 42)(6 133 54 120 17 43 12 139 60 114 23 37)(25 92 75 108 64 123 31 86 81 102 70 129)(26 87 76 103 65 130 32 93 82 97 71 124)(27 94 77 98 66 125 33 88 83 104 72 131)(28 89 78 105 67 132 34 95 84 99 61 126)(29 96 79 100 68 127 35 90 73 106 62 121)(30 91 80 107 69 122 36 85 74 101 63 128)
(1 76 7 82)(2 81 8 75)(3 74 9 80)(4 79 10 73)(5 84 11 78)(6 77 12 83)(13 64 19 70)(14 69 20 63)(15 62 21 68)(16 67 22 61)(17 72 23 66)(18 65 24 71)(25 50 31 56)(26 55 32 49)(27 60 33 54)(28 53 34 59)(29 58 35 52)(30 51 36 57)(37 104 43 98)(38 97 44 103)(39 102 45 108)(40 107 46 101)(41 100 47 106)(42 105 48 99)(85 136 91 142)(86 141 92 135)(87 134 93 140)(88 139 94 133)(89 144 95 138)(90 137 96 143)(109 130 115 124)(110 123 116 129)(111 128 117 122)(112 121 118 127)(113 126 119 132)(114 131 120 125)
G:=sub<Sym(144)| (1,106)(2,107)(3,108)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,117)(26,118)(27,119)(28,120)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,84)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,134,49,109,24,44,7,140,55,115,18,38)(2,141,50,116,13,39,8,135,56,110,19,45)(3,136,51,111,14,46,9,142,57,117,20,40)(4,143,52,118,15,41,10,137,58,112,21,47)(5,138,53,113,16,48,11,144,59,119,22,42)(6,133,54,120,17,43,12,139,60,114,23,37)(25,92,75,108,64,123,31,86,81,102,70,129)(26,87,76,103,65,130,32,93,82,97,71,124)(27,94,77,98,66,125,33,88,83,104,72,131)(28,89,78,105,67,132,34,95,84,99,61,126)(29,96,79,100,68,127,35,90,73,106,62,121)(30,91,80,107,69,122,36,85,74,101,63,128), (1,76,7,82)(2,81,8,75)(3,74,9,80)(4,79,10,73)(5,84,11,78)(6,77,12,83)(13,64,19,70)(14,69,20,63)(15,62,21,68)(16,67,22,61)(17,72,23,66)(18,65,24,71)(25,50,31,56)(26,55,32,49)(27,60,33,54)(28,53,34,59)(29,58,35,52)(30,51,36,57)(37,104,43,98)(38,97,44,103)(39,102,45,108)(40,107,46,101)(41,100,47,106)(42,105,48,99)(85,136,91,142)(86,141,92,135)(87,134,93,140)(88,139,94,133)(89,144,95,138)(90,137,96,143)(109,130,115,124)(110,123,116,129)(111,128,117,122)(112,121,118,127)(113,126,119,132)(114,131,120,125)>;
G:=Group( (1,106)(2,107)(3,108)(4,97)(5,98)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,117)(26,118)(27,119)(28,120)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,84)(38,73)(39,74)(40,75)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82)(48,83)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,134,49,109,24,44,7,140,55,115,18,38)(2,141,50,116,13,39,8,135,56,110,19,45)(3,136,51,111,14,46,9,142,57,117,20,40)(4,143,52,118,15,41,10,137,58,112,21,47)(5,138,53,113,16,48,11,144,59,119,22,42)(6,133,54,120,17,43,12,139,60,114,23,37)(25,92,75,108,64,123,31,86,81,102,70,129)(26,87,76,103,65,130,32,93,82,97,71,124)(27,94,77,98,66,125,33,88,83,104,72,131)(28,89,78,105,67,132,34,95,84,99,61,126)(29,96,79,100,68,127,35,90,73,106,62,121)(30,91,80,107,69,122,36,85,74,101,63,128), (1,76,7,82)(2,81,8,75)(3,74,9,80)(4,79,10,73)(5,84,11,78)(6,77,12,83)(13,64,19,70)(14,69,20,63)(15,62,21,68)(16,67,22,61)(17,72,23,66)(18,65,24,71)(25,50,31,56)(26,55,32,49)(27,60,33,54)(28,53,34,59)(29,58,35,52)(30,51,36,57)(37,104,43,98)(38,97,44,103)(39,102,45,108)(40,107,46,101)(41,100,47,106)(42,105,48,99)(85,136,91,142)(86,141,92,135)(87,134,93,140)(88,139,94,133)(89,144,95,138)(90,137,96,143)(109,130,115,124)(110,123,116,129)(111,128,117,122)(112,121,118,127)(113,126,119,132)(114,131,120,125) );
G=PermutationGroup([(1,106),(2,107),(3,108),(4,97),(5,98),(6,99),(7,100),(8,101),(9,102),(10,103),(11,104),(12,105),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,117),(26,118),(27,119),(28,120),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,84),(38,73),(39,74),(40,75),(41,76),(42,77),(43,78),(44,79),(45,80),(46,81),(47,82),(48,83),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)], [(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)], [(1,134,49,109,24,44,7,140,55,115,18,38),(2,141,50,116,13,39,8,135,56,110,19,45),(3,136,51,111,14,46,9,142,57,117,20,40),(4,143,52,118,15,41,10,137,58,112,21,47),(5,138,53,113,16,48,11,144,59,119,22,42),(6,133,54,120,17,43,12,139,60,114,23,37),(25,92,75,108,64,123,31,86,81,102,70,129),(26,87,76,103,65,130,32,93,82,97,71,124),(27,94,77,98,66,125,33,88,83,104,72,131),(28,89,78,105,67,132,34,95,84,99,61,126),(29,96,79,100,68,127,35,90,73,106,62,121),(30,91,80,107,69,122,36,85,74,101,63,128)], [(1,76,7,82),(2,81,8,75),(3,74,9,80),(4,79,10,73),(5,84,11,78),(6,77,12,83),(13,64,19,70),(14,69,20,63),(15,62,21,68),(16,67,22,61),(17,72,23,66),(18,65,24,71),(25,50,31,56),(26,55,32,49),(27,60,33,54),(28,53,34,59),(29,58,35,52),(30,51,36,57),(37,104,43,98),(38,97,44,103),(39,102,45,108),(40,107,46,101),(41,100,47,106),(42,105,48,99),(85,136,91,142),(86,141,92,135),(87,134,93,140),(88,139,94,133),(89,144,95,138),(90,137,96,143),(109,130,115,124),(110,123,116,129),(111,128,117,122),(112,121,118,127),(113,126,119,132),(114,131,120,125)])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6L | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | ··· | 18 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | Q8⋊3S3 |
| kernel | C2×C12.26D6 | C2×C4×C3⋊S3 | C2×C12⋊S3 | C12.26D6 | Q8×C3×C6 | C6×Q8 | C2×C12 | C3×Q8 | C3×C6 | C6 |
| # reps | 1 | 3 | 3 | 8 | 1 | 4 | 12 | 16 | 4 | 8 |
Matrix representation of C2×C12.26D6 ►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 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 8 | 12 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
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],[0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,8,0,0,0,0,3,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,2,8],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;
C2×C12.26D6 in GAP, Magma, Sage, TeX
C_2\times C_{12}._{26}D_6 % in TeX
G:=Group("C2xC12.26D6"); // GroupNames label
G:=SmallGroup(288,1011);
// by ID
G=gap.SmallGroup(288,1011);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=1,c^6=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^5,d*c*d^-1=c^5>;
// generators/relations