direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D7×C3×C6, C42⋊6C6, C7⋊3C62, C21⋊8(C2×C6), C14⋊3(C3×C6), (C3×C42)⋊3C2, (C3×C21)⋊8C22, SmallGroup(252,41)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C3×C6 |
Generators and relations for D7×C3×C6
G = < a,b,c,d | a3=b6=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 168 in 60 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C6, C6, C7, C32, C2×C6, D7, C14, C3×C6, C3×C6, C21, D14, C62, C3×D7, C42, C3×C21, C6×D7, C32×D7, C3×C42, D7×C3×C6
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, D7, C3×C6, D14, C62, C3×D7, C6×D7, C32×D7, D7×C3×C6
►On 126 points
Generators in S126
(1 48 27)(2 49 28)(3 43 22)(4 44 23)(5 45 24)(6 46 25)(7 47 26)(8 50 29)(9 51 30)(10 52 31)(11 53 32)(12 54 33)(13 55 34)(14 56 35)(15 57 36)(16 58 37)(17 59 38)(18 60 39)(19 61 40)(20 62 41)(21 63 42)(64 106 85)(65 107 86)(66 108 87)(67 109 88)(68 110 89)(69 111 90)(70 112 91)(71 113 92)(72 114 93)(73 115 94)(74 116 95)(75 117 96)(76 118 97)(77 119 98)(78 120 99)(79 121 100)(80 122 101)(81 123 102)(82 124 103)(83 125 104)(84 126 105)
(1 76 20 69 13 83)(2 77 21 70 14 84)(3 71 15 64 8 78)(4 72 16 65 9 79)(5 73 17 66 10 80)(6 74 18 67 11 81)(7 75 19 68 12 82)(22 92 36 85 29 99)(23 93 37 86 30 100)(24 94 38 87 31 101)(25 95 39 88 32 102)(26 96 40 89 33 103)(27 97 41 90 34 104)(28 98 42 91 35 105)(43 113 57 106 50 120)(44 114 58 107 51 121)(45 115 59 108 52 122)(46 116 60 109 53 123)(47 117 61 110 54 124)(48 118 62 111 55 125)(49 119 63 112 56 126)
(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)
(1 68)(2 67)(3 66)(4 65)(5 64)(6 70)(7 69)(8 73)(9 72)(10 71)(11 77)(12 76)(13 75)(14 74)(15 80)(16 79)(17 78)(18 84)(19 83)(20 82)(21 81)(22 87)(23 86)(24 85)(25 91)(26 90)(27 89)(28 88)(29 94)(30 93)(31 92)(32 98)(33 97)(34 96)(35 95)(36 101)(37 100)(38 99)(39 105)(40 104)(41 103)(42 102)(43 108)(44 107)(45 106)(46 112)(47 111)(48 110)(49 109)(50 115)(51 114)(52 113)(53 119)(54 118)(55 117)(56 116)(57 122)(58 121)(59 120)(60 126)(61 125)(62 124)(63 123)
G:=sub<Sym(126)| (1,48,27)(2,49,28)(3,43,22)(4,44,23)(5,45,24)(6,46,25)(7,47,26)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42)(64,106,85)(65,107,86)(66,108,87)(67,109,88)(68,110,89)(69,111,90)(70,112,91)(71,113,92)(72,114,93)(73,115,94)(74,116,95)(75,117,96)(76,118,97)(77,119,98)(78,120,99)(79,121,100)(80,122,101)(81,123,102)(82,124,103)(83,125,104)(84,126,105), (1,76,20,69,13,83)(2,77,21,70,14,84)(3,71,15,64,8,78)(4,72,16,65,9,79)(5,73,17,66,10,80)(6,74,18,67,11,81)(7,75,19,68,12,82)(22,92,36,85,29,99)(23,93,37,86,30,100)(24,94,38,87,31,101)(25,95,39,88,32,102)(26,96,40,89,33,103)(27,97,41,90,34,104)(28,98,42,91,35,105)(43,113,57,106,50,120)(44,114,58,107,51,121)(45,115,59,108,52,122)(46,116,60,109,53,123)(47,117,61,110,54,124)(48,118,62,111,55,125)(49,119,63,112,56,126), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,70)(7,69)(8,73)(9,72)(10,71)(11,77)(12,76)(13,75)(14,74)(15,80)(16,79)(17,78)(18,84)(19,83)(20,82)(21,81)(22,87)(23,86)(24,85)(25,91)(26,90)(27,89)(28,88)(29,94)(30,93)(31,92)(32,98)(33,97)(34,96)(35,95)(36,101)(37,100)(38,99)(39,105)(40,104)(41,103)(42,102)(43,108)(44,107)(45,106)(46,112)(47,111)(48,110)(49,109)(50,115)(51,114)(52,113)(53,119)(54,118)(55,117)(56,116)(57,122)(58,121)(59,120)(60,126)(61,125)(62,124)(63,123)>;
G:=Group( (1,48,27)(2,49,28)(3,43,22)(4,44,23)(5,45,24)(6,46,25)(7,47,26)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42)(64,106,85)(65,107,86)(66,108,87)(67,109,88)(68,110,89)(69,111,90)(70,112,91)(71,113,92)(72,114,93)(73,115,94)(74,116,95)(75,117,96)(76,118,97)(77,119,98)(78,120,99)(79,121,100)(80,122,101)(81,123,102)(82,124,103)(83,125,104)(84,126,105), (1,76,20,69,13,83)(2,77,21,70,14,84)(3,71,15,64,8,78)(4,72,16,65,9,79)(5,73,17,66,10,80)(6,74,18,67,11,81)(7,75,19,68,12,82)(22,92,36,85,29,99)(23,93,37,86,30,100)(24,94,38,87,31,101)(25,95,39,88,32,102)(26,96,40,89,33,103)(27,97,41,90,34,104)(28,98,42,91,35,105)(43,113,57,106,50,120)(44,114,58,107,51,121)(45,115,59,108,52,122)(46,116,60,109,53,123)(47,117,61,110,54,124)(48,118,62,111,55,125)(49,119,63,112,56,126), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,70)(7,69)(8,73)(9,72)(10,71)(11,77)(12,76)(13,75)(14,74)(15,80)(16,79)(17,78)(18,84)(19,83)(20,82)(21,81)(22,87)(23,86)(24,85)(25,91)(26,90)(27,89)(28,88)(29,94)(30,93)(31,92)(32,98)(33,97)(34,96)(35,95)(36,101)(37,100)(38,99)(39,105)(40,104)(41,103)(42,102)(43,108)(44,107)(45,106)(46,112)(47,111)(48,110)(49,109)(50,115)(51,114)(52,113)(53,119)(54,118)(55,117)(56,116)(57,122)(58,121)(59,120)(60,126)(61,125)(62,124)(63,123) );
G=PermutationGroup([[(1,48,27),(2,49,28),(3,43,22),(4,44,23),(5,45,24),(6,46,25),(7,47,26),(8,50,29),(9,51,30),(10,52,31),(11,53,32),(12,54,33),(13,55,34),(14,56,35),(15,57,36),(16,58,37),(17,59,38),(18,60,39),(19,61,40),(20,62,41),(21,63,42),(64,106,85),(65,107,86),(66,108,87),(67,109,88),(68,110,89),(69,111,90),(70,112,91),(71,113,92),(72,114,93),(73,115,94),(74,116,95),(75,117,96),(76,118,97),(77,119,98),(78,120,99),(79,121,100),(80,122,101),(81,123,102),(82,124,103),(83,125,104),(84,126,105)], [(1,76,20,69,13,83),(2,77,21,70,14,84),(3,71,15,64,8,78),(4,72,16,65,9,79),(5,73,17,66,10,80),(6,74,18,67,11,81),(7,75,19,68,12,82),(22,92,36,85,29,99),(23,93,37,86,30,100),(24,94,38,87,31,101),(25,95,39,88,32,102),(26,96,40,89,33,103),(27,97,41,90,34,104),(28,98,42,91,35,105),(43,113,57,106,50,120),(44,114,58,107,51,121),(45,115,59,108,52,122),(46,116,60,109,53,123),(47,117,61,110,54,124),(48,118,62,111,55,125),(49,119,63,112,56,126)], [(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)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,70),(7,69),(8,73),(9,72),(10,71),(11,77),(12,76),(13,75),(14,74),(15,80),(16,79),(17,78),(18,84),(19,83),(20,82),(21,81),(22,87),(23,86),(24,85),(25,91),(26,90),(27,89),(28,88),(29,94),(30,93),(31,92),(32,98),(33,97),(34,96),(35,95),(36,101),(37,100),(38,99),(39,105),(40,104),(41,103),(42,102),(43,108),(44,107),(45,106),(46,112),(47,111),(48,110),(49,109),(50,115),(51,114),(52,113),(53,119),(54,118),(55,117),(56,116),(57,122),(58,121),(59,120),(60,126),(61,125),(62,124),(63,123)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 6A | ··· | 6H | 6I | ··· | 6X | 7A | 7B | 7C | 14A | 14B | 14C | 21A | ··· | 21X | 42A | ··· | 42X |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 7 | 7 | 7 | 14 | 14 | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 7 | 7 | 1 | ··· | 1 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D7 | D14 | C3×D7 | C6×D7 |
| kernel | D7×C3×C6 | C32×D7 | C3×C42 | C6×D7 | C3×D7 | C42 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 2 | 1 | 8 | 16 | 8 | 3 | 3 | 24 | 24 |
Matrix representation of D7×C3×C6 ►in GL4(𝔽43) generated by
| 6 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 42 | 19 |
| 1 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(43))| [6,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,42,0,0,1,19],[1,0,0,0,0,42,0,0,0,0,0,1,0,0,1,0] >;
D7×C3×C6 in GAP, Magma, Sage, TeX
D_7\times C_3\times C_6
% in TeX
G:=Group("D7xC3xC6"); // GroupNames label
G:=SmallGroup(252,41);
// by ID
G=gap.SmallGroup(252,41);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-7,5404]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations