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, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C3×Q8, C22×S3, C2×C4○D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C2×D12, Q8⋊3S3, C6×Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C6×C12, Q8×C32, C22×C3⋊S3, C2×Q8⋊3S3, C2×C4×C3⋊S3, C2×C12⋊S3, C12.26D6, Q8×C3×C6, C2×C12.26D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C3⋊S3, C22×S3, C2×C4○D4, C2×C3⋊S3, Q8⋊3S3, S3×C23, C22×C3⋊S3, C2×Q8⋊3S3, C12.26D6, C23×C3⋊S3, C2×C12.26D6
►On 144 points
Generators in S144
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 73)(37 50)(38 51)(39 52)(40 53)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 49)(61 106)(62 107)(63 108)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 141)(93 142)(94 143)(95 144)(96 133)(109 130)(110 131)(111 132)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 129)
(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 58 94 121 14 67 7 52 88 127 20 61)(2 53 95 128 15 62 8 59 89 122 21 68)(3 60 96 123 16 69 9 54 90 129 22 63)(4 55 85 130 17 64 10 49 91 124 23 70)(5 50 86 125 18 71 11 56 92 131 24 65)(6 57 87 132 19 66 12 51 93 126 13 72)(25 39 137 118 81 106 31 45 143 112 75 100)(26 46 138 113 82 101 32 40 144 119 76 107)(27 41 139 120 83 108 33 47 133 114 77 102)(28 48 140 115 84 103 34 42 134 109 78 97)(29 43 141 110 73 98 35 37 135 116 79 104)(30 38 142 117 74 105 36 44 136 111 80 99)
(1 97 7 103)(2 102 8 108)(3 107 9 101)(4 100 10 106)(5 105 11 99)(6 98 12 104)(13 37 19 43)(14 42 20 48)(15 47 21 41)(16 40 22 46)(17 45 23 39)(18 38 24 44)(25 70 31 64)(26 63 32 69)(27 68 33 62)(28 61 34 67)(29 66 35 72)(30 71 36 65)(49 75 55 81)(50 80 56 74)(51 73 57 79)(52 78 58 84)(53 83 59 77)(54 76 60 82)(85 112 91 118)(86 117 92 111)(87 110 93 116)(88 115 94 109)(89 120 95 114)(90 113 96 119)(121 140 127 134)(122 133 128 139)(123 138 129 144)(124 143 130 137)(125 136 131 142)(126 141 132 135)
G:=sub<Sym(144)| (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,49)(61,106)(62,107)(63,108)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,133)(109,130)(110,131)(111,132)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129), (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), (1,58,94,121,14,67,7,52,88,127,20,61)(2,53,95,128,15,62,8,59,89,122,21,68)(3,60,96,123,16,69,9,54,90,129,22,63)(4,55,85,130,17,64,10,49,91,124,23,70)(5,50,86,125,18,71,11,56,92,131,24,65)(6,57,87,132,19,66,12,51,93,126,13,72)(25,39,137,118,81,106,31,45,143,112,75,100)(26,46,138,113,82,101,32,40,144,119,76,107)(27,41,139,120,83,108,33,47,133,114,77,102)(28,48,140,115,84,103,34,42,134,109,78,97)(29,43,141,110,73,98,35,37,135,116,79,104)(30,38,142,117,74,105,36,44,136,111,80,99), (1,97,7,103)(2,102,8,108)(3,107,9,101)(4,100,10,106)(5,105,11,99)(6,98,12,104)(13,37,19,43)(14,42,20,48)(15,47,21,41)(16,40,22,46)(17,45,23,39)(18,38,24,44)(25,70,31,64)(26,63,32,69)(27,68,33,62)(28,61,34,67)(29,66,35,72)(30,71,36,65)(49,75,55,81)(50,80,56,74)(51,73,57,79)(52,78,58,84)(53,83,59,77)(54,76,60,82)(85,112,91,118)(86,117,92,111)(87,110,93,116)(88,115,94,109)(89,120,95,114)(90,113,96,119)(121,140,127,134)(122,133,128,139)(123,138,129,144)(124,143,130,137)(125,136,131,142)(126,141,132,135)>;
G:=Group( (1,31)(2,32)(3,33)(4,34)(5,35)(6,36)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(37,50)(38,51)(39,52)(40,53)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,49)(61,106)(62,107)(63,108)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,141)(93,142)(94,143)(95,144)(96,133)(109,130)(110,131)(111,132)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129), (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), (1,58,94,121,14,67,7,52,88,127,20,61)(2,53,95,128,15,62,8,59,89,122,21,68)(3,60,96,123,16,69,9,54,90,129,22,63)(4,55,85,130,17,64,10,49,91,124,23,70)(5,50,86,125,18,71,11,56,92,131,24,65)(6,57,87,132,19,66,12,51,93,126,13,72)(25,39,137,118,81,106,31,45,143,112,75,100)(26,46,138,113,82,101,32,40,144,119,76,107)(27,41,139,120,83,108,33,47,133,114,77,102)(28,48,140,115,84,103,34,42,134,109,78,97)(29,43,141,110,73,98,35,37,135,116,79,104)(30,38,142,117,74,105,36,44,136,111,80,99), (1,97,7,103)(2,102,8,108)(3,107,9,101)(4,100,10,106)(5,105,11,99)(6,98,12,104)(13,37,19,43)(14,42,20,48)(15,47,21,41)(16,40,22,46)(17,45,23,39)(18,38,24,44)(25,70,31,64)(26,63,32,69)(27,68,33,62)(28,61,34,67)(29,66,35,72)(30,71,36,65)(49,75,55,81)(50,80,56,74)(51,73,57,79)(52,78,58,84)(53,83,59,77)(54,76,60,82)(85,112,91,118)(86,117,92,111)(87,110,93,116)(88,115,94,109)(89,120,95,114)(90,113,96,119)(121,140,127,134)(122,133,128,139)(123,138,129,144)(124,143,130,137)(125,136,131,142)(126,141,132,135) );
G=PermutationGroup([[(1,31),(2,32),(3,33),(4,34),(5,35),(6,36),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,73),(37,50),(38,51),(39,52),(40,53),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,49),(61,106),(62,107),(63,108),(64,97),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(85,134),(86,135),(87,136),(88,137),(89,138),(90,139),(91,140),(92,141),(93,142),(94,143),(95,144),(96,133),(109,130),(110,131),(111,132),(112,121),(113,122),(114,123),(115,124),(116,125),(117,126),(118,127),(119,128),(120,129)], [(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)], [(1,58,94,121,14,67,7,52,88,127,20,61),(2,53,95,128,15,62,8,59,89,122,21,68),(3,60,96,123,16,69,9,54,90,129,22,63),(4,55,85,130,17,64,10,49,91,124,23,70),(5,50,86,125,18,71,11,56,92,131,24,65),(6,57,87,132,19,66,12,51,93,126,13,72),(25,39,137,118,81,106,31,45,143,112,75,100),(26,46,138,113,82,101,32,40,144,119,76,107),(27,41,139,120,83,108,33,47,133,114,77,102),(28,48,140,115,84,103,34,42,134,109,78,97),(29,43,141,110,73,98,35,37,135,116,79,104),(30,38,142,117,74,105,36,44,136,111,80,99)], [(1,97,7,103),(2,102,8,108),(3,107,9,101),(4,100,10,106),(5,105,11,99),(6,98,12,104),(13,37,19,43),(14,42,20,48),(15,47,21,41),(16,40,22,46),(17,45,23,39),(18,38,24,44),(25,70,31,64),(26,63,32,69),(27,68,33,62),(28,61,34,67),(29,66,35,72),(30,71,36,65),(49,75,55,81),(50,80,56,74),(51,73,57,79),(52,78,58,84),(53,83,59,77),(54,76,60,82),(85,112,91,118),(86,117,92,111),(87,110,93,116),(88,115,94,109),(89,120,95,114),(90,113,96,119),(121,140,127,134),(122,133,128,139),(123,138,129,144),(124,143,130,137),(125,136,131,142),(126,141,132,135)]])
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