direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C72, C24⋊4C18, D6.2C36, C36.76D6, Dic3.2C36, C3⋊C8⋊6C18, C3⋊1(C2×C72), (C3×C72)⋊1C2, (S3×C24).C3, (C3×S3).C24, C2.1(S3×C36), C3.4(S3×C24), C6.1(C2×C36), (S3×C6).6C12, (S3×C36).6C2, (S3×C18).4C4, (C4×S3).3C18, C24.27(C3×S3), C6.33(S3×C12), C18.21(C4×S3), C4.12(S3×C18), (C3×C24).20C6, (S3×C12).12C6, C12.116(S3×C6), C12.12(C2×C18), C32.2(C2×C24), (C9×Dic3).4C4, (C3×C36).50C22, (C3×Dic3).6C12, (C3×C9)⋊4(C2×C8), (C9×C3⋊C8)⋊13C2, (C3×C3⋊C8).9C6, (C3×C6).37(C2×C12), (C3×C18).14(C2×C4), (C3×C12).87(C2×C6), SmallGroup(432,109)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C72 |
Generators and relations for S3×C72
G = < a,b,c | a72=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 124 in 74 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, C3×C9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, S3×C8, C2×C24, S3×C9, C3×C18, C72, C72, C2×C36, C3×C3⋊C8, C3×C24, S3×C12, C9×Dic3, C3×C36, S3×C18, C2×C72, S3×C24, C9×C3⋊C8, C3×C72, S3×C36, S3×C72
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C9, C12, D6, C2×C6, C2×C8, C18, C3×S3, C24, C4×S3, C2×C12, C36, C2×C18, S3×C6, S3×C8, C2×C24, S3×C9, C72, C2×C36, S3×C12, S3×C18, C2×C72, S3×C24, S3×C36, S3×C72
►On 144 points
Generators in S144
(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 25 49)(2 26 50)(3 27 51)(4 28 52)(5 29 53)(6 30 54)(7 31 55)(8 32 56)(9 33 57)(10 34 58)(11 35 59)(12 36 60)(13 37 61)(14 38 62)(15 39 63)(16 40 64)(17 41 65)(18 42 66)(19 43 67)(20 44 68)(21 45 69)(22 46 70)(23 47 71)(24 48 72)(73 121 97)(74 122 98)(75 123 99)(76 124 100)(77 125 101)(78 126 102)(79 127 103)(80 128 104)(81 129 105)(82 130 106)(83 131 107)(84 132 108)(85 133 109)(86 134 110)(87 135 111)(88 136 112)(89 137 113)(90 138 114)(91 139 115)(92 140 116)(93 141 117)(94 142 118)(95 143 119)(96 144 120)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 137)(49 138)(50 139)(51 140)(52 141)(53 142)(54 143)(55 144)(56 73)(57 74)(58 75)(59 76)(60 77)(61 78)(62 79)(63 80)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)
G:=sub<Sym(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), (1,25,49)(2,26,50)(3,27,51)(4,28,52)(5,29,53)(6,30,54)(7,31,55)(8,32,56)(9,33,57)(10,34,58)(11,35,59)(12,36,60)(13,37,61)(14,38,62)(15,39,63)(16,40,64)(17,41,65)(18,42,66)(19,43,67)(20,44,68)(21,45,69)(22,46,70)(23,47,71)(24,48,72)(73,121,97)(74,122,98)(75,123,99)(76,124,100)(77,125,101)(78,126,102)(79,127,103)(80,128,104)(81,129,105)(82,130,106)(83,131,107)(84,132,108)(85,133,109)(86,134,110)(87,135,111)(88,136,112)(89,137,113)(90,138,114)(91,139,115)(92,140,116)(93,141,117)(94,142,118)(95,143,119)(96,144,120), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89)>;
G:=Group( (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), (1,25,49)(2,26,50)(3,27,51)(4,28,52)(5,29,53)(6,30,54)(7,31,55)(8,32,56)(9,33,57)(10,34,58)(11,35,59)(12,36,60)(13,37,61)(14,38,62)(15,39,63)(16,40,64)(17,41,65)(18,42,66)(19,43,67)(20,44,68)(21,45,69)(22,46,70)(23,47,71)(24,48,72)(73,121,97)(74,122,98)(75,123,99)(76,124,100)(77,125,101)(78,126,102)(79,127,103)(80,128,104)(81,129,105)(82,130,106)(83,131,107)(84,132,108)(85,133,109)(86,134,110)(87,135,111)(88,136,112)(89,137,113)(90,138,114)(91,139,115)(92,140,116)(93,141,117)(94,142,118)(95,143,119)(96,144,120), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,137)(49,138)(50,139)(51,140)(52,141)(53,142)(54,143)(55,144)(56,73)(57,74)(58,75)(59,76)(60,77)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89) );
G=PermutationGroup([[(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)], [(1,25,49),(2,26,50),(3,27,51),(4,28,52),(5,29,53),(6,30,54),(7,31,55),(8,32,56),(9,33,57),(10,34,58),(11,35,59),(12,36,60),(13,37,61),(14,38,62),(15,39,63),(16,40,64),(17,41,65),(18,42,66),(19,43,67),(20,44,68),(21,45,69),(22,46,70),(23,47,71),(24,48,72),(73,121,97),(74,122,98),(75,123,99),(76,124,100),(77,125,101),(78,126,102),(79,127,103),(80,128,104),(81,129,105),(82,130,106),(83,131,107),(84,132,108),(85,133,109),(86,134,110),(87,135,111),(88,136,112),(89,137,113),(90,138,114),(91,139,115),(92,140,116),(93,141,117),(94,142,118),(95,143,119),(96,144,120)], [(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,109),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,127),(39,128),(40,129),(41,130),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,137),(49,138),(50,139),(51,140),(52,141),(53,142),(54,143),(55,144),(56,73),(57,74),(58,75),(59,76),(60,77),(61,78),(62,79),(63,80),(64,81),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(71,88),(72,89)]])
216 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 18A | ··· | 18F | 18G | ··· | 18L | 18M | ··· | 18X | 24A | ··· | 24H | 24I | ··· | 24T | 24U | ··· | 24AB | 36A | ··· | 36L | 36M | ··· | 36X | 36Y | ··· | 36AJ | 72A | ··· | 72X | 72Y | ··· | 72AV | 72AW | ··· | 72BT |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 | 72 | ··· | 72 | 72 | ··· | 72 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C9 | C12 | C12 | C18 | C18 | C18 | C24 | C36 | C36 | C72 | S3 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C8 | S3×C9 | S3×C12 | S3×C18 | S3×C24 | S3×C36 | S3×C72 |
| kernel | S3×C72 | C9×C3⋊C8 | C3×C72 | S3×C36 | S3×C24 | C9×Dic3 | S3×C18 | C3×C3⋊C8 | C3×C24 | S3×C12 | S3×C9 | S3×C8 | C3×Dic3 | S3×C6 | C3⋊C8 | C24 | C4×S3 | C3×S3 | Dic3 | D6 | S3 | C72 | C36 | C24 | C18 | C12 | C9 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 6 | 4 | 4 | 6 | 6 | 6 | 16 | 12 | 12 | 48 | 1 | 1 | 2 | 2 | 2 | 4 | 6 | 4 | 6 | 8 | 12 | 24 |
Matrix representation of S3×C72 ►in GL2(𝔽73) generated by
| 5 | 0 |
| 0 | 5 |
| 8 | 8 |
| 0 | 64 |
| 1 | 0 |
| 7 | 72 |
G:=sub<GL(2,GF(73))| [5,0,0,5],[8,0,8,64],[1,7,0,72] >;
S3×C72 in GAP, Magma, Sage, TeX
S_3\times C_{72} % in TeX
G:=Group("S3xC72"); // GroupNames label
G:=SmallGroup(432,109);
// by ID
G=gap.SmallGroup(432,109);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,92,142,192,14118]);
// Polycyclic
G:=Group<a,b,c|a^72=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations