direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C2×C62, C63⋊5C2, C33⋊5C24, C6⋊(C2×C62), (C2×C6)⋊6C62, C3⋊(C22×C62), (C2×C62)⋊13C6, C62⋊20(C2×C6), (C32×C6)⋊5C23, C32⋊4(C23×C6), (C3×C62)⋊17C22, (C3×C6)⋊4(C22×C6), (C22×C6)⋊5(C3×C6), SmallGroup(432,772)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C62 |
Generators and relations for S3×C2×C62
G = < a,b,c,d,e | a2=b6=c6=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1672 in 932 conjugacy classes, 498 normal (10 characteristic)
C1, C2, C2, C3, C3, C3, C22, C22, S3, C6, C6, C23, C23, C32, C32, C32, D6, C2×C6, C2×C6, C24, C3×S3, C3×C6, C3×C6, C22×S3, C22×C6, C22×C6, C22×C6, C33, S3×C6, C62, C62, S3×C23, C23×C6, S3×C32, C32×C6, S3×C2×C6, C2×C62, C2×C62, C2×C62, S3×C3×C6, C3×C62, S3×C22×C6, C22×C62, S3×C62, C63, S3×C2×C62
Quotients: C1, C2, C3, C22, S3, C6, C23, C32, D6, C2×C6, C24, C3×S3, C3×C6, C22×S3, C22×C6, S3×C6, C62, S3×C23, C23×C6, S3×C32, S3×C2×C6, C2×C62, S3×C3×C6, S3×C22×C6, C22×C62, S3×C62, S3×C2×C62
►On 144 points
Generators in S144
(1 53)(2 54)(3 49)(4 50)(5 51)(6 52)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 94)(14 95)(15 96)(16 91)(17 92)(18 93)(19 106)(20 107)(21 108)(22 103)(23 104)(24 105)(25 89)(26 90)(27 85)(28 86)(29 87)(30 88)(31 99)(32 100)(33 101)(34 102)(35 97)(36 98)(37 74)(38 75)(39 76)(40 77)(41 78)(42 73)(43 59)(44 60)(45 55)(46 56)(47 57)(48 58)(61 71)(62 72)(63 67)(64 68)(65 69)(66 70)(109 121)(110 122)(111 123)(112 124)(113 125)(114 126)(115 131)(116 132)(117 127)(118 128)(119 129)(120 130)(133 143)(134 144)(135 139)(136 140)(137 141)(138 142)
(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 57 31 37 15 66)(2 58 32 38 16 61)(3 59 33 39 17 62)(4 60 34 40 18 63)(5 55 35 41 13 64)(6 56 36 42 14 65)(7 141 124 118 21 27)(8 142 125 119 22 28)(9 143 126 120 23 29)(10 144 121 115 24 30)(11 139 122 116 19 25)(12 140 123 117 20 26)(43 101 76 92 72 49)(44 102 77 93 67 50)(45 97 78 94 68 51)(46 98 73 95 69 52)(47 99 74 96 70 53)(48 100 75 91 71 54)(79 137 112 128 108 85)(80 138 113 129 103 86)(81 133 114 130 104 87)(82 134 109 131 105 88)(83 135 110 132 106 89)(84 136 111 127 107 90)
(1 31 15)(2 32 16)(3 33 17)(4 34 18)(5 35 13)(6 36 14)(7 21 124)(8 22 125)(9 23 126)(10 24 121)(11 19 122)(12 20 123)(25 116 139)(26 117 140)(27 118 141)(28 119 142)(29 120 143)(30 115 144)(37 66 57)(38 61 58)(39 62 59)(40 63 60)(41 64 55)(42 65 56)(43 76 72)(44 77 67)(45 78 68)(46 73 69)(47 74 70)(48 75 71)(49 101 92)(50 102 93)(51 97 94)(52 98 95)(53 99 96)(54 100 91)(79 108 112)(80 103 113)(81 104 114)(82 105 109)(83 106 110)(84 107 111)(85 128 137)(86 129 138)(87 130 133)(88 131 134)(89 132 135)(90 127 136)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 59)(8 60)(9 55)(10 56)(11 57)(12 58)(13 120)(14 115)(15 116)(16 117)(17 118)(18 119)(19 66)(20 61)(21 62)(22 63)(23 64)(24 65)(31 139)(32 140)(33 141)(34 142)(35 143)(36 144)(37 122)(38 123)(39 124)(40 125)(41 126)(42 121)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(67 103)(68 104)(69 105)(70 106)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 113)(78 114)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)
G:=sub<Sym(144)| (1,53)(2,54)(3,49)(4,50)(5,51)(6,52)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,94)(14,95)(15,96)(16,91)(17,92)(18,93)(19,106)(20,107)(21,108)(22,103)(23,104)(24,105)(25,89)(26,90)(27,85)(28,86)(29,87)(30,88)(31,99)(32,100)(33,101)(34,102)(35,97)(36,98)(37,74)(38,75)(39,76)(40,77)(41,78)(42,73)(43,59)(44,60)(45,55)(46,56)(47,57)(48,58)(61,71)(62,72)(63,67)(64,68)(65,69)(66,70)(109,121)(110,122)(111,123)(112,124)(113,125)(114,126)(115,131)(116,132)(117,127)(118,128)(119,129)(120,130)(133,143)(134,144)(135,139)(136,140)(137,141)(138,142), (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), (1,57,31,37,15,66)(2,58,32,38,16,61)(3,59,33,39,17,62)(4,60,34,40,18,63)(5,55,35,41,13,64)(6,56,36,42,14,65)(7,141,124,118,21,27)(8,142,125,119,22,28)(9,143,126,120,23,29)(10,144,121,115,24,30)(11,139,122,116,19,25)(12,140,123,117,20,26)(43,101,76,92,72,49)(44,102,77,93,67,50)(45,97,78,94,68,51)(46,98,73,95,69,52)(47,99,74,96,70,53)(48,100,75,91,71,54)(79,137,112,128,108,85)(80,138,113,129,103,86)(81,133,114,130,104,87)(82,134,109,131,105,88)(83,135,110,132,106,89)(84,136,111,127,107,90), (1,31,15)(2,32,16)(3,33,17)(4,34,18)(5,35,13)(6,36,14)(7,21,124)(8,22,125)(9,23,126)(10,24,121)(11,19,122)(12,20,123)(25,116,139)(26,117,140)(27,118,141)(28,119,142)(29,120,143)(30,115,144)(37,66,57)(38,61,58)(39,62,59)(40,63,60)(41,64,55)(42,65,56)(43,76,72)(44,77,67)(45,78,68)(46,73,69)(47,74,70)(48,75,71)(49,101,92)(50,102,93)(51,97,94)(52,98,95)(53,99,96)(54,100,91)(79,108,112)(80,103,113)(81,104,114)(82,105,109)(83,106,110)(84,107,111)(85,128,137)(86,129,138)(87,130,133)(88,131,134)(89,132,135)(90,127,136), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,59)(8,60)(9,55)(10,56)(11,57)(12,58)(13,120)(14,115)(15,116)(16,117)(17,118)(18,119)(19,66)(20,61)(21,62)(22,63)(23,64)(24,65)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,122)(38,123)(39,124)(40,125)(41,126)(42,121)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)>;
G:=Group( (1,53)(2,54)(3,49)(4,50)(5,51)(6,52)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,94)(14,95)(15,96)(16,91)(17,92)(18,93)(19,106)(20,107)(21,108)(22,103)(23,104)(24,105)(25,89)(26,90)(27,85)(28,86)(29,87)(30,88)(31,99)(32,100)(33,101)(34,102)(35,97)(36,98)(37,74)(38,75)(39,76)(40,77)(41,78)(42,73)(43,59)(44,60)(45,55)(46,56)(47,57)(48,58)(61,71)(62,72)(63,67)(64,68)(65,69)(66,70)(109,121)(110,122)(111,123)(112,124)(113,125)(114,126)(115,131)(116,132)(117,127)(118,128)(119,129)(120,130)(133,143)(134,144)(135,139)(136,140)(137,141)(138,142), (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), (1,57,31,37,15,66)(2,58,32,38,16,61)(3,59,33,39,17,62)(4,60,34,40,18,63)(5,55,35,41,13,64)(6,56,36,42,14,65)(7,141,124,118,21,27)(8,142,125,119,22,28)(9,143,126,120,23,29)(10,144,121,115,24,30)(11,139,122,116,19,25)(12,140,123,117,20,26)(43,101,76,92,72,49)(44,102,77,93,67,50)(45,97,78,94,68,51)(46,98,73,95,69,52)(47,99,74,96,70,53)(48,100,75,91,71,54)(79,137,112,128,108,85)(80,138,113,129,103,86)(81,133,114,130,104,87)(82,134,109,131,105,88)(83,135,110,132,106,89)(84,136,111,127,107,90), (1,31,15)(2,32,16)(3,33,17)(4,34,18)(5,35,13)(6,36,14)(7,21,124)(8,22,125)(9,23,126)(10,24,121)(11,19,122)(12,20,123)(25,116,139)(26,117,140)(27,118,141)(28,119,142)(29,120,143)(30,115,144)(37,66,57)(38,61,58)(39,62,59)(40,63,60)(41,64,55)(42,65,56)(43,76,72)(44,77,67)(45,78,68)(46,73,69)(47,74,70)(48,75,71)(49,101,92)(50,102,93)(51,97,94)(52,98,95)(53,99,96)(54,100,91)(79,108,112)(80,103,113)(81,104,114)(82,105,109)(83,106,110)(84,107,111)(85,128,137)(86,129,138)(87,130,133)(88,131,134)(89,132,135)(90,127,136), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,59)(8,60)(9,55)(10,56)(11,57)(12,58)(13,120)(14,115)(15,116)(16,117)(17,118)(18,119)(19,66)(20,61)(21,62)(22,63)(23,64)(24,65)(31,139)(32,140)(33,141)(34,142)(35,143)(36,144)(37,122)(38,123)(39,124)(40,125)(41,126)(42,121)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(67,103)(68,104)(69,105)(70,106)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,113)(78,114)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138) );
G=PermutationGroup([[(1,53),(2,54),(3,49),(4,50),(5,51),(6,52),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,94),(14,95),(15,96),(16,91),(17,92),(18,93),(19,106),(20,107),(21,108),(22,103),(23,104),(24,105),(25,89),(26,90),(27,85),(28,86),(29,87),(30,88),(31,99),(32,100),(33,101),(34,102),(35,97),(36,98),(37,74),(38,75),(39,76),(40,77),(41,78),(42,73),(43,59),(44,60),(45,55),(46,56),(47,57),(48,58),(61,71),(62,72),(63,67),(64,68),(65,69),(66,70),(109,121),(110,122),(111,123),(112,124),(113,125),(114,126),(115,131),(116,132),(117,127),(118,128),(119,129),(120,130),(133,143),(134,144),(135,139),(136,140),(137,141),(138,142)], [(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)], [(1,57,31,37,15,66),(2,58,32,38,16,61),(3,59,33,39,17,62),(4,60,34,40,18,63),(5,55,35,41,13,64),(6,56,36,42,14,65),(7,141,124,118,21,27),(8,142,125,119,22,28),(9,143,126,120,23,29),(10,144,121,115,24,30),(11,139,122,116,19,25),(12,140,123,117,20,26),(43,101,76,92,72,49),(44,102,77,93,67,50),(45,97,78,94,68,51),(46,98,73,95,69,52),(47,99,74,96,70,53),(48,100,75,91,71,54),(79,137,112,128,108,85),(80,138,113,129,103,86),(81,133,114,130,104,87),(82,134,109,131,105,88),(83,135,110,132,106,89),(84,136,111,127,107,90)], [(1,31,15),(2,32,16),(3,33,17),(4,34,18),(5,35,13),(6,36,14),(7,21,124),(8,22,125),(9,23,126),(10,24,121),(11,19,122),(12,20,123),(25,116,139),(26,117,140),(27,118,141),(28,119,142),(29,120,143),(30,115,144),(37,66,57),(38,61,58),(39,62,59),(40,63,60),(41,64,55),(42,65,56),(43,76,72),(44,77,67),(45,78,68),(46,73,69),(47,74,70),(48,75,71),(49,101,92),(50,102,93),(51,97,94),(52,98,95),(53,99,96),(54,100,91),(79,108,112),(80,103,113),(81,104,114),(82,105,109),(83,106,110),(84,107,111),(85,128,137),(86,129,138),(87,130,133),(88,131,134),(89,132,135),(90,127,136)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,59),(8,60),(9,55),(10,56),(11,57),(12,58),(13,120),(14,115),(15,116),(16,117),(17,118),(18,119),(19,66),(20,61),(21,62),(22,63),(23,64),(24,65),(31,139),(32,140),(33,141),(34,142),(35,143),(36,144),(37,122),(38,123),(39,124),(40,125),(41,126),(42,121),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(67,103),(68,104),(69,105),(70,106),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,113),(78,114),(91,127),(92,128),(93,129),(94,130),(95,131),(96,132),(97,133),(98,134),(99,135),(100,136),(101,137),(102,138)]])
216 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | ··· | 3H | 3I | ··· | 3Q | 6A | ··· | 6BD | 6BE | ··· | 6DO | 6DP | ··· | 6GA |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 |
| kernel | S3×C2×C62 | S3×C62 | C63 | S3×C22×C6 | S3×C2×C6 | C2×C62 | C2×C62 | C62 | C22×C6 | C2×C6 |
| # reps | 1 | 14 | 1 | 8 | 112 | 8 | 1 | 7 | 8 | 56 |
Matrix representation of S3×C2×C62 ►in GL4(𝔽7) generated by
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(7))| [1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[3,0,0,0,0,3,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,2,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,4],[1,0,0,0,0,6,0,0,0,0,0,1,0,0,1,0] >;
S3×C2×C62 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_6^2
% in TeX
G:=Group("S3xC2xC6^2"); // GroupNames label
G:=SmallGroup(432,772);
// by ID
G=gap.SmallGroup(432,772);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^6=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations