direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C3×C24, C12.12C62, C24⋊4(C3×C6), C3⋊1(C6×C24), (C3×C24)⋊14C6, C6.1(C6×C12), C33⋊16(C2×C8), (S3×C6).8C12, C6.39(S3×C12), C32⋊9(C2×C24), (C32×C24)⋊8C2, D6.2(C3×C12), (S3×C12).14C6, C12.121(S3×C6), (C3×C12).235D6, (C3×Dic3).8C12, Dic3.2(C3×C12), (C32×Dic3).8C4, (C32×C12).87C22, C3⋊C8⋊6(C3×C6), (C3×C3⋊C8)⋊13C6, (S3×C3×C6).8C4, C2.1(S3×C3×C12), C4.12(S3×C3×C6), (S3×C3×C12).9C2, (C32×C3⋊C8)⋊20C2, (C4×S3).3(C3×C6), (C3×C6).93(C4×S3), (C3×C12).92(C2×C6), (C3×C6).43(C2×C12), (C32×C6).49(C2×C4), SmallGroup(432,464)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C3×C24 |
Generators and relations for S3×C3×C24
G = < a,b,c,d | a3=b24=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 280 in 164 conjugacy classes, 90 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, C8, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C24, C24, C4×S3, C2×C12, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C62, S3×C8, C2×C24, S3×C32, C32×C6, C3×C3⋊C8, C3×C24, C3×C24, C3×C24, S3×C12, C6×C12, C32×Dic3, C32×C12, S3×C3×C6, S3×C24, C6×C24, C32×C3⋊C8, C32×C24, S3×C3×C12, S3×C3×C24
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C32, C12, D6, C2×C6, C2×C8, C3×S3, C3×C6, C24, C4×S3, C2×C12, C3×C12, S3×C6, C62, S3×C8, C2×C24, S3×C32, C3×C24, S3×C12, C6×C12, S3×C3×C6, S3×C24, C6×C24, S3×C3×C12, S3×C3×C24
►On 144 points
Generators in S144
(1 107 72)(2 108 49)(3 109 50)(4 110 51)(5 111 52)(6 112 53)(7 113 54)(8 114 55)(9 115 56)(10 116 57)(11 117 58)(12 118 59)(13 119 60)(14 120 61)(15 97 62)(16 98 63)(17 99 64)(18 100 65)(19 101 66)(20 102 67)(21 103 68)(22 104 69)(23 105 70)(24 106 71)(25 141 87)(26 142 88)(27 143 89)(28 144 90)(29 121 91)(30 122 92)(31 123 93)(32 124 94)(33 125 95)(34 126 96)(35 127 73)(36 128 74)(37 129 75)(38 130 76)(39 131 77)(40 132 78)(41 133 79)(42 134 80)(43 135 81)(44 136 82)(45 137 83)(46 138 84)(47 139 85)(48 140 86)
(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 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(49 57 65)(50 58 66)(51 59 67)(52 60 68)(53 61 69)(54 62 70)(55 63 71)(56 64 72)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 88)(97 105 113)(98 106 114)(99 107 115)(100 108 116)(101 109 117)(102 110 118)(103 111 119)(104 112 120)(121 137 129)(122 138 130)(123 139 131)(124 140 132)(125 141 133)(126 142 134)(127 143 135)(128 144 136)
(1 140)(2 141)(3 142)(4 143)(5 144)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 136)(22 137)(23 138)(24 139)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(73 118)(74 119)(75 120)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)
G:=sub<Sym(144)| (1,107,72)(2,108,49)(3,109,50)(4,110,51)(5,111,52)(6,112,53)(7,113,54)(8,114,55)(9,115,56)(10,116,57)(11,117,58)(12,118,59)(13,119,60)(14,120,61)(15,97,62)(16,98,63)(17,99,64)(18,100,65)(19,101,66)(20,102,67)(21,103,68)(22,104,69)(23,105,70)(24,106,71)(25,141,87)(26,142,88)(27,143,89)(28,144,90)(29,121,91)(30,122,92)(31,123,93)(32,124,94)(33,125,95)(34,126,96)(35,127,73)(36,128,74)(37,129,75)(38,130,76)(39,131,77)(40,132,78)(41,133,79)(42,134,80)(43,135,81)(44,136,82)(45,137,83)(46,138,84)(47,139,85)(48,140,86), (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), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(97,105,113)(98,106,114)(99,107,115)(100,108,116)(101,109,117)(102,110,118)(103,111,119)(104,112,120)(121,137,129)(122,138,130)(123,139,131)(124,140,132)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (1,140)(2,141)(3,142)(4,143)(5,144)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,136)(22,137)(23,138)(24,139)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(73,118)(74,119)(75,120)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117)>;
G:=Group( (1,107,72)(2,108,49)(3,109,50)(4,110,51)(5,111,52)(6,112,53)(7,113,54)(8,114,55)(9,115,56)(10,116,57)(11,117,58)(12,118,59)(13,119,60)(14,120,61)(15,97,62)(16,98,63)(17,99,64)(18,100,65)(19,101,66)(20,102,67)(21,103,68)(22,104,69)(23,105,70)(24,106,71)(25,141,87)(26,142,88)(27,143,89)(28,144,90)(29,121,91)(30,122,92)(31,123,93)(32,124,94)(33,125,95)(34,126,96)(35,127,73)(36,128,74)(37,129,75)(38,130,76)(39,131,77)(40,132,78)(41,133,79)(42,134,80)(43,135,81)(44,136,82)(45,137,83)(46,138,84)(47,139,85)(48,140,86), (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), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(97,105,113)(98,106,114)(99,107,115)(100,108,116)(101,109,117)(102,110,118)(103,111,119)(104,112,120)(121,137,129)(122,138,130)(123,139,131)(124,140,132)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (1,140)(2,141)(3,142)(4,143)(5,144)(6,121)(7,122)(8,123)(9,124)(10,125)(11,126)(12,127)(13,128)(14,129)(15,130)(16,131)(17,132)(18,133)(19,134)(20,135)(21,136)(22,137)(23,138)(24,139)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(73,118)(74,119)(75,120)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111)(91,112)(92,113)(93,114)(94,115)(95,116)(96,117) );
G=PermutationGroup([[(1,107,72),(2,108,49),(3,109,50),(4,110,51),(5,111,52),(6,112,53),(7,113,54),(8,114,55),(9,115,56),(10,116,57),(11,117,58),(12,118,59),(13,119,60),(14,120,61),(15,97,62),(16,98,63),(17,99,64),(18,100,65),(19,101,66),(20,102,67),(21,103,68),(22,104,69),(23,105,70),(24,106,71),(25,141,87),(26,142,88),(27,143,89),(28,144,90),(29,121,91),(30,122,92),(31,123,93),(32,124,94),(33,125,95),(34,126,96),(35,127,73),(36,128,74),(37,129,75),(38,130,76),(39,131,77),(40,132,78),(41,133,79),(42,134,80),(43,135,81),(44,136,82),(45,137,83),(46,138,84),(47,139,85),(48,140,86)], [(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)], [(1,9,17),(2,10,18),(3,11,19),(4,12,20),(5,13,21),(6,14,22),(7,15,23),(8,16,24),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(49,57,65),(50,58,66),(51,59,67),(52,60,68),(53,61,69),(54,62,70),(55,63,71),(56,64,72),(73,89,81),(74,90,82),(75,91,83),(76,92,84),(77,93,85),(78,94,86),(79,95,87),(80,96,88),(97,105,113),(98,106,114),(99,107,115),(100,108,116),(101,109,117),(102,110,118),(103,111,119),(104,112,120),(121,137,129),(122,138,130),(123,139,131),(124,140,132),(125,141,133),(126,142,134),(127,143,135),(128,144,136)], [(1,140),(2,141),(3,142),(4,143),(5,144),(6,121),(7,122),(8,123),(9,124),(10,125),(11,126),(12,127),(13,128),(14,129),(15,130),(16,131),(17,132),(18,133),(19,134),(20,135),(21,136),(22,137),(23,138),(24,139),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(73,118),(74,119),(75,120),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111),(91,112),(92,113),(93,114),(94,115),(95,116),(96,117)]])
216 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 4D | 6A | ··· | 6H | 6I | ··· | 6Q | 6R | ··· | 6AG | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12P | 12Q | ··· | 12AH | 12AI | ··· | 12AX | 24A | ··· | 24AF | 24AG | ··· | 24BP | 24BQ | ··· | 24CV |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C8 | S3×C12 | S3×C24 |
| kernel | S3×C3×C24 | C32×C3⋊C8 | C32×C24 | S3×C3×C12 | S3×C24 | C32×Dic3 | S3×C3×C6 | C3×C3⋊C8 | C3×C24 | S3×C12 | S3×C32 | C3×Dic3 | S3×C6 | C3×S3 | C3×C24 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 8 | 8 | 8 | 8 | 16 | 16 | 64 | 1 | 1 | 8 | 2 | 8 | 4 | 16 | 32 |
Matrix representation of S3×C3×C24 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 52 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 24 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 64 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[52,0,0,0,0,8,0,0,0,0,24,0,0,0,0,24],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,64],[72,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
S3×C3×C24 in GAP, Magma, Sage, TeX
S_3\times C_3\times C_{24} % in TeX
G:=Group("S3xC3xC24"); // GroupNames label
G:=SmallGroup(432,464);
// by ID
G=gap.SmallGroup(432,464);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,260,102,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^24=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