direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C2×C22, C33⋊4C23, C66⋊4C22, C6⋊(C2×C22), C3⋊(C22×C22), (C2×C66)⋊7C2, (C2×C6)⋊3C22, SmallGroup(264,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C22 |
Generators and relations for S3×C2×C22
G = < a,b,c,d | a2=b22=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 108 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, C11, D6, C2×C6, C22, C22, C22×S3, C33, C2×C22, C2×C22, S3×C11, C66, C22×C22, S3×C22, C2×C66, S3×C2×C22
Quotients: C1, C2, C22, S3, C23, C11, D6, C22, C22×S3, C2×C22, S3×C11, C22×C22, S3×C22, S3×C2×C22
►On 132 points
Generators in S132
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 121)(24 122)(25 123)(26 124)(27 125)(28 126)(29 127)(30 128)(31 129)(32 130)(33 131)(34 132)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 117)(42 118)(43 119)(44 120)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 101)(77 102)(78 103)(79 104)(80 105)(81 106)(82 107)(83 108)(84 109)(85 110)(86 89)(87 90)(88 91)
(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)
(1 29 84)(2 30 85)(3 31 86)(4 32 87)(5 33 88)(6 34 67)(7 35 68)(8 36 69)(9 37 70)(10 38 71)(11 39 72)(12 40 73)(13 41 74)(14 42 75)(15 43 76)(16 44 77)(17 23 78)(18 24 79)(19 25 80)(20 26 81)(21 27 82)(22 28 83)(45 116 98)(46 117 99)(47 118 100)(48 119 101)(49 120 102)(50 121 103)(51 122 104)(52 123 105)(53 124 106)(54 125 107)(55 126 108)(56 127 109)(57 128 110)(58 129 89)(59 130 90)(60 131 91)(61 132 92)(62 111 93)(63 112 94)(64 113 95)(65 114 96)(66 115 97)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 99)(42 100)(43 101)(44 102)(67 132)(68 111)(69 112)(70 113)(71 114)(72 115)(73 116)(74 117)(75 118)(76 119)(77 120)(78 121)(79 122)(80 123)(81 124)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 131)
G:=sub<Sym(132)| (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,121)(24,122)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,129)(32,130)(33,131)(34,132)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(85,110)(86,89)(87,90)(88,91), (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), (1,29,84)(2,30,85)(3,31,86)(4,32,87)(5,33,88)(6,34,67)(7,35,68)(8,36,69)(9,37,70)(10,38,71)(11,39,72)(12,40,73)(13,41,74)(14,42,75)(15,43,76)(16,44,77)(17,23,78)(18,24,79)(19,25,80)(20,26,81)(21,27,82)(22,28,83)(45,116,98)(46,117,99)(47,118,100)(48,119,101)(49,120,102)(50,121,103)(51,122,104)(52,123,105)(53,124,106)(54,125,107)(55,126,108)(56,127,109)(57,128,110)(58,129,89)(59,130,90)(60,131,91)(61,132,92)(62,111,93)(63,112,94)(64,113,95)(65,114,96)(66,115,97), (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,101)(44,102)(67,132)(68,111)(69,112)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)>;
G:=Group( (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,121)(24,122)(25,123)(26,124)(27,125)(28,126)(29,127)(30,128)(31,129)(32,130)(33,131)(34,132)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,117)(42,118)(43,119)(44,120)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(85,110)(86,89)(87,90)(88,91), (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), (1,29,84)(2,30,85)(3,31,86)(4,32,87)(5,33,88)(6,34,67)(7,35,68)(8,36,69)(9,37,70)(10,38,71)(11,39,72)(12,40,73)(13,41,74)(14,42,75)(15,43,76)(16,44,77)(17,23,78)(18,24,79)(19,25,80)(20,26,81)(21,27,82)(22,28,83)(45,116,98)(46,117,99)(47,118,100)(48,119,101)(49,120,102)(50,121,103)(51,122,104)(52,123,105)(53,124,106)(54,125,107)(55,126,108)(56,127,109)(57,128,110)(58,129,89)(59,130,90)(60,131,91)(61,132,92)(62,111,93)(63,112,94)(64,113,95)(65,114,96)(66,115,97), (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,99)(42,100)(43,101)(44,102)(67,132)(68,111)(69,112)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131) );
G=PermutationGroup([[(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,121),(24,122),(25,123),(26,124),(27,125),(28,126),(29,127),(30,128),(31,129),(32,130),(33,131),(34,132),(35,111),(36,112),(37,113),(38,114),(39,115),(40,116),(41,117),(42,118),(43,119),(44,120),(67,92),(68,93),(69,94),(70,95),(71,96),(72,97),(73,98),(74,99),(75,100),(76,101),(77,102),(78,103),(79,104),(80,105),(81,106),(82,107),(83,108),(84,109),(85,110),(86,89),(87,90),(88,91)], [(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)], [(1,29,84),(2,30,85),(3,31,86),(4,32,87),(5,33,88),(6,34,67),(7,35,68),(8,36,69),(9,37,70),(10,38,71),(11,39,72),(12,40,73),(13,41,74),(14,42,75),(15,43,76),(16,44,77),(17,23,78),(18,24,79),(19,25,80),(20,26,81),(21,27,82),(22,28,83),(45,116,98),(46,117,99),(47,118,100),(48,119,101),(49,120,102),(50,121,103),(51,122,104),(52,123,105),(53,124,106),(54,125,107),(55,126,108),(56,127,109),(57,128,110),(58,129,89),(59,130,90),(60,131,91),(61,132,92),(62,111,93),(63,112,94),(64,113,95),(65,114,96),(66,115,97)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,97),(40,98),(41,99),(42,100),(43,101),(44,102),(67,132),(68,111),(69,112),(70,113),(71,114),(72,115),(73,116),(74,117),(75,118),(76,119),(77,120),(78,121),(79,122),(80,123),(81,124),(82,125),(83,126),(84,127),(85,128),(86,129),(87,130),(88,131)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 6A | 6B | 6C | 11A | ··· | 11J | 22A | ··· | 22AD | 22AE | ··· | 22BR | 33A | ··· | 33J | 66A | ··· | 66AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | S3 | D6 | S3×C11 | S3×C22 |
| kernel | S3×C2×C22 | S3×C22 | C2×C66 | C22×S3 | D6 | C2×C6 | C2×C22 | C22 | C22 | C2 |
| # reps | 1 | 6 | 1 | 10 | 60 | 10 | 1 | 3 | 10 | 30 |
Matrix representation of S3×C2×C22 ►in GL4(𝔽67) generated by
| 66 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 66 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 66 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 66 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 66 | 66 |
G:=sub<GL(4,GF(67))| [66,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,66,0,0,0,0,15,0,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,66,1,0,0,66,0],[1,0,0,0,0,66,0,0,0,0,1,66,0,0,0,66] >;
S3×C2×C22 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_{22} % in TeX
G:=Group("S3xC2xC22"); // GroupNames label
G:=SmallGroup(264,37);
// by ID
G=gap.SmallGroup(264,37);
# by ID
G:=PCGroup([5,-2,-2,-2,-11,-3,4404]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^22=c^3=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