direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C2×C36, C6⋊1(C2×C36), (C6×C36)⋊5C2, (C2×C12)⋊5C18, C12⋊3(C2×C18), (S3×C6).7C12, C6.38(S3×C12), (C6×C12).41C6, (C3×C36)⋊8C22, C3⋊1(C22×C36), D6.4(C2×C18), (C2×C18).52D6, (S3×C12).13C6, C12.120(S3×C6), (C2×Dic3)⋊5C18, Dic3⋊3(C2×C18), C6.2(C22×C18), C22.9(S3×C18), C62.54(C2×C6), (Dic3×C18)⋊11C2, (C22×S3).2C18, (C6×C18).27C22, (C3×C18).29C23, C18.50(C22×S3), (C6×Dic3).16C6, (S3×C18).15C22, (C9×Dic3)⋊10C22, C32.2(C22×C12), (S3×C2×C12).C3, (S3×C2×C6).9C6, C3.4(S3×C2×C12), C2.1(S3×C2×C18), C6.63(S3×C2×C6), (C3×C18)⋊4(C2×C4), (C3×S3).(C2×C12), (S3×C2×C18).4C2, (C3×C9)⋊5(C22×C4), (C2×C6).86(S3×C6), (S3×C6).19(C2×C6), (C3×C6).42(C2×C12), (C2×C6).12(C2×C18), (C2×C12).50(C3×S3), (C3×C12).91(C2×C6), (C3×C6).39(C22×C6), (C3×Dic3).19(C2×C6), SmallGroup(432,345)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C36 |
Generators and relations for S3×C2×C36
G = < a,b,c,d | a2=b36=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 308 in 178 conjugacy classes, 105 normal (33 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22×C4, C18, C18, C18, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C3×C9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, S3×C2×C4, C22×C12, S3×C9, C3×C18, C3×C18, C2×C36, C2×C36, C22×C18, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, C9×Dic3, C3×C36, S3×C18, C6×C18, C22×C36, S3×C2×C12, S3×C36, Dic3×C18, C6×C36, S3×C2×C18, S3×C2×C36
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C9, C12, D6, C2×C6, C22×C4, C18, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, C36, C2×C18, S3×C6, S3×C2×C4, C22×C12, S3×C9, C2×C36, C22×C18, S3×C12, S3×C2×C6, S3×C18, C22×C36, S3×C2×C12, S3×C36, S3×C2×C18, S3×C2×C36
►On 144 points
Generators in S144
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 144)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)
(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 13 25)(2 14 26)(3 15 27)(4 16 28)(5 17 29)(6 18 30)(7 19 31)(8 20 32)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(37 49 61)(38 50 62)(39 51 63)(40 52 64)(41 53 65)(42 54 66)(43 55 67)(44 56 68)(45 57 69)(46 58 70)(47 59 71)(48 60 72)(73 97 85)(74 98 86)(75 99 87)(76 100 88)(77 101 89)(78 102 90)(79 103 91)(80 104 92)(81 105 93)(82 106 94)(83 107 95)(84 108 96)(109 133 121)(110 134 122)(111 135 123)(112 136 124)(113 137 125)(114 138 126)(115 139 127)(116 140 128)(117 141 129)(118 142 130)(119 143 131)(120 144 132)
(1 108)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 143)(58 144)(59 109)(60 110)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 121)(72 122)
G:=sub<Sym(144)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,109)(83,110)(84,111)(85,112)(86,113)(87,114)(88,115)(89,116)(90,117)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135), (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), (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,31)(8,20,32)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72)(73,97,85)(74,98,86)(75,99,87)(76,100,88)(77,101,89)(78,102,90)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96)(109,133,121)(110,134,122)(111,135,123)(112,136,124)(113,137,125)(114,138,126)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,108)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,143)(58,144)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122)>;
G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,144)(82,109)(83,110)(84,111)(85,112)(86,113)(87,114)(88,115)(89,116)(90,117)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135), (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), (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,31)(8,20,32)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(37,49,61)(38,50,62)(39,51,63)(40,52,64)(41,53,65)(42,54,66)(43,55,67)(44,56,68)(45,57,69)(46,58,70)(47,59,71)(48,60,72)(73,97,85)(74,98,86)(75,99,87)(76,100,88)(77,101,89)(78,102,90)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96)(109,133,121)(110,134,122)(111,135,123)(112,136,124)(113,137,125)(114,138,126)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,108)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,143)(58,144)(59,109)(60,110)(61,111)(62,112)(63,113)(64,114)(65,115)(66,116)(67,117)(68,118)(69,119)(70,120)(71,121)(72,122) );
G=PermutationGroup([[(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(73,136),(74,137),(75,138),(76,139),(77,140),(78,141),(79,142),(80,143),(81,144),(82,109),(83,110),(84,111),(85,112),(86,113),(87,114),(88,115),(89,116),(90,117),(91,118),(92,119),(93,120),(94,121),(95,122),(96,123),(97,124),(98,125),(99,126),(100,127),(101,128),(102,129),(103,130),(104,131),(105,132),(106,133),(107,134),(108,135)], [(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)], [(1,13,25),(2,14,26),(3,15,27),(4,16,28),(5,17,29),(6,18,30),(7,19,31),(8,20,32),(9,21,33),(10,22,34),(11,23,35),(12,24,36),(37,49,61),(38,50,62),(39,51,63),(40,52,64),(41,53,65),(42,54,66),(43,55,67),(44,56,68),(45,57,69),(46,58,70),(47,59,71),(48,60,72),(73,97,85),(74,98,86),(75,99,87),(76,100,88),(77,101,89),(78,102,90),(79,103,91),(80,104,92),(81,105,93),(82,106,94),(83,107,95),(84,108,96),(109,133,121),(110,134,122),(111,135,123),(112,136,124),(113,137,125),(114,138,126),(115,139,127),(116,140,128),(117,141,129),(118,142,130),(119,143,131),(120,144,132)], [(1,108),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,84),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,101),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(37,123),(38,124),(39,125),(40,126),(41,127),(42,128),(43,129),(44,130),(45,131),(46,132),(47,133),(48,134),(49,135),(50,136),(51,137),(52,138),(53,139),(54,140),(55,141),(56,142),(57,143),(58,144),(59,109),(60,110),(61,111),(62,112),(63,113),(64,114),(65,115),(66,116),(67,117),(68,118),(69,119),(70,120),(71,121),(72,122)]])
216 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12H | 12I | ··· | 12T | 12U | ··· | 12AB | 18A | ··· | 18R | 18S | ··· | 18AJ | 18AK | ··· | 18BH | 36A | ··· | 36X | 36Y | ··· | 36AV | 36AW | ··· | 36BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C9 | C12 | C18 | C18 | C18 | C18 | C36 | S3 | D6 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C6 | S3×C9 | S3×C12 | S3×C18 | S3×C18 | S3×C36 |
| kernel | S3×C2×C36 | S3×C36 | Dic3×C18 | C6×C36 | S3×C2×C18 | S3×C2×C12 | S3×C18 | S3×C12 | C6×Dic3 | C6×C12 | S3×C2×C6 | S3×C2×C4 | S3×C6 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C2×C36 | C36 | C2×C18 | C2×C12 | C18 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 2 | 6 | 16 | 24 | 6 | 6 | 6 | 48 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 6 | 8 | 12 | 6 | 24 |
Matrix representation of S3×C2×C36 ►in GL3(𝔽37) generated by
| 36 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 33 | 0 | 0 |
| 0 | 19 | 0 |
| 0 | 0 | 19 |
| 1 | 0 | 0 |
| 0 | 10 | 10 |
| 0 | 0 | 26 |
| 36 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 9 | 36 |
G:=sub<GL(3,GF(37))| [36,0,0,0,1,0,0,0,1],[33,0,0,0,19,0,0,0,19],[1,0,0,0,10,0,0,10,26],[36,0,0,0,1,9,0,0,36] >;
S3×C2×C36 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_{36} % in TeX
G:=Group("S3xC2xC36"); // GroupNames label
G:=SmallGroup(432,345);
// by ID
G=gap.SmallGroup(432,345);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,142,192,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^36=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