direct product, metabelian, supersoluble, monomial
Aliases: C32×Dic12, C33⋊10Q16, C12.10C62, C8.(S3×C32), C24.1(C3×C6), C24.13(C3×S3), (C3×C24).19S3, (C3×C24).14C6, C6.39(C3×D12), (C3×C6).86D12, C3⋊1(C32×Q16), C32⋊7(C3×Q16), C12.114(S3×C6), C6.3(D4×C32), (C3×C12).233D6, (C32×C24).2C2, Dic6.1(C3×C6), (C32×C6).54D4, C2.5(C32×D12), (C3×Dic6).10C6, (C32×Dic6).6C2, (C32×C12).81C22, C4.10(S3×C3×C6), (C3×C6).46(C3×D4), (C3×C12).85(C2×C6), SmallGroup(432,468)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×Dic12
G = < a,b,c,d | a3=b3=c24=1, d2=c12, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 280 in 140 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, C8, Q8, C32, C32, C32, Dic3, C12, C12, C12, Q16, C3×C6, C3×C6, C3×C6, C24, C24, C24, Dic6, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, Dic12, C3×Q16, C32×C6, C3×C24, C3×C24, C3×C24, C3×Dic6, Q8×C32, C32×Dic3, C32×C12, C3×Dic12, C32×Q16, C32×C24, C32×Dic6, C32×Dic12
Quotients: C1, C2, C3, C22, S3, C6, D4, C32, D6, C2×C6, Q16, C3×S3, C3×C6, D12, C3×D4, S3×C6, C62, Dic12, C3×Q16, S3×C32, C3×D12, D4×C32, S3×C3×C6, C3×Dic12, C32×Q16, C32×D12, C32×Dic12
►On 144 points
Generators in S144
(1 29 60)(2 30 61)(3 31 62)(4 32 63)(5 33 64)(6 34 65)(7 35 66)(8 36 67)(9 37 68)(10 38 69)(11 39 70)(12 40 71)(13 41 72)(14 42 49)(15 43 50)(16 44 51)(17 45 52)(18 46 53)(19 47 54)(20 48 55)(21 25 56)(22 26 57)(23 27 58)(24 28 59)(73 144 108)(74 121 109)(75 122 110)(76 123 111)(77 124 112)(78 125 113)(79 126 114)(80 127 115)(81 128 116)(82 129 117)(83 130 118)(84 131 119)(85 132 120)(86 133 97)(87 134 98)(88 135 99)(89 136 100)(90 137 101)(91 138 102)(92 139 103)(93 140 104)(94 141 105)(95 142 106)(96 143 107)
(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 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)(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)(97 113 105)(98 114 106)(99 115 107)(100 116 108)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(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 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 97 13 109)(2 120 14 108)(3 119 15 107)(4 118 16 106)(5 117 17 105)(6 116 18 104)(7 115 19 103)(8 114 20 102)(9 113 21 101)(10 112 22 100)(11 111 23 99)(12 110 24 98)(25 90 37 78)(26 89 38 77)(27 88 39 76)(28 87 40 75)(29 86 41 74)(30 85 42 73)(31 84 43 96)(32 83 44 95)(33 82 45 94)(34 81 46 93)(35 80 47 92)(36 79 48 91)(49 144 61 132)(50 143 62 131)(51 142 63 130)(52 141 64 129)(53 140 65 128)(54 139 66 127)(55 138 67 126)(56 137 68 125)(57 136 69 124)(58 135 70 123)(59 134 71 122)(60 133 72 121)
G:=sub<Sym(144)| (1,29,60)(2,30,61)(3,31,62)(4,32,63)(5,33,64)(6,34,65)(7,35,66)(8,36,67)(9,37,68)(10,38,69)(11,39,70)(12,40,71)(13,41,72)(14,42,49)(15,43,50)(16,44,51)(17,45,52)(18,46,53)(19,47,54)(20,48,55)(21,25,56)(22,26,57)(23,27,58)(24,28,59)(73,144,108)(74,121,109)(75,122,110)(76,123,111)(77,124,112)(78,125,113)(79,126,114)(80,127,115)(81,128,116)(82,129,117)(83,130,118)(84,131,119)(85,132,120)(86,133,97)(87,134,98)(88,135,99)(89,136,100)(90,137,101)(91,138,102)(92,139,103)(93,140,104)(94,141,105)(95,142,106)(96,143,107), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(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)(97,113,105)(98,114,106)(99,115,107)(100,116,108)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(121,137,129)(122,138,130)(123,139,131)(124,140,132)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(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), (1,97,13,109)(2,120,14,108)(3,119,15,107)(4,118,16,106)(5,117,17,105)(6,116,18,104)(7,115,19,103)(8,114,20,102)(9,113,21,101)(10,112,22,100)(11,111,23,99)(12,110,24,98)(25,90,37,78)(26,89,38,77)(27,88,39,76)(28,87,40,75)(29,86,41,74)(30,85,42,73)(31,84,43,96)(32,83,44,95)(33,82,45,94)(34,81,46,93)(35,80,47,92)(36,79,48,91)(49,144,61,132)(50,143,62,131)(51,142,63,130)(52,141,64,129)(53,140,65,128)(54,139,66,127)(55,138,67,126)(56,137,68,125)(57,136,69,124)(58,135,70,123)(59,134,71,122)(60,133,72,121)>;
G:=Group( (1,29,60)(2,30,61)(3,31,62)(4,32,63)(5,33,64)(6,34,65)(7,35,66)(8,36,67)(9,37,68)(10,38,69)(11,39,70)(12,40,71)(13,41,72)(14,42,49)(15,43,50)(16,44,51)(17,45,52)(18,46,53)(19,47,54)(20,48,55)(21,25,56)(22,26,57)(23,27,58)(24,28,59)(73,144,108)(74,121,109)(75,122,110)(76,123,111)(77,124,112)(78,125,113)(79,126,114)(80,127,115)(81,128,116)(82,129,117)(83,130,118)(84,131,119)(85,132,120)(86,133,97)(87,134,98)(88,135,99)(89,136,100)(90,137,101)(91,138,102)(92,139,103)(93,140,104)(94,141,105)(95,142,106)(96,143,107), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(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)(97,113,105)(98,114,106)(99,115,107)(100,116,108)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(121,137,129)(122,138,130)(123,139,131)(124,140,132)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(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), (1,97,13,109)(2,120,14,108)(3,119,15,107)(4,118,16,106)(5,117,17,105)(6,116,18,104)(7,115,19,103)(8,114,20,102)(9,113,21,101)(10,112,22,100)(11,111,23,99)(12,110,24,98)(25,90,37,78)(26,89,38,77)(27,88,39,76)(28,87,40,75)(29,86,41,74)(30,85,42,73)(31,84,43,96)(32,83,44,95)(33,82,45,94)(34,81,46,93)(35,80,47,92)(36,79,48,91)(49,144,61,132)(50,143,62,131)(51,142,63,130)(52,141,64,129)(53,140,65,128)(54,139,66,127)(55,138,67,126)(56,137,68,125)(57,136,69,124)(58,135,70,123)(59,134,71,122)(60,133,72,121) );
G=PermutationGroup([[(1,29,60),(2,30,61),(3,31,62),(4,32,63),(5,33,64),(6,34,65),(7,35,66),(8,36,67),(9,37,68),(10,38,69),(11,39,70),(12,40,71),(13,41,72),(14,42,49),(15,43,50),(16,44,51),(17,45,52),(18,46,53),(19,47,54),(20,48,55),(21,25,56),(22,26,57),(23,27,58),(24,28,59),(73,144,108),(74,121,109),(75,122,110),(76,123,111),(77,124,112),(78,125,113),(79,126,114),(80,127,115),(81,128,116),(82,129,117),(83,130,118),(84,131,119),(85,132,120),(86,133,97),(87,134,98),(88,135,99),(89,136,100),(90,137,101),(91,138,102),(92,139,103),(93,140,104),(94,141,105),(95,142,106),(96,143,107)], [(1,9,17),(2,10,18),(3,11,19),(4,12,20),(5,13,21),(6,14,22),(7,15,23),(8,16,24),(25,33,41),(26,34,42),(27,35,43),(28,36,44),(29,37,45),(30,38,46),(31,39,47),(32,40,48),(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),(97,113,105),(98,114,106),(99,115,107),(100,116,108),(101,117,109),(102,118,110),(103,119,111),(104,120,112),(121,137,129),(122,138,130),(123,139,131),(124,140,132),(125,141,133),(126,142,134),(127,143,135),(128,144,136)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(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)], [(1,97,13,109),(2,120,14,108),(3,119,15,107),(4,118,16,106),(5,117,17,105),(6,116,18,104),(7,115,19,103),(8,114,20,102),(9,113,21,101),(10,112,22,100),(11,111,23,99),(12,110,24,98),(25,90,37,78),(26,89,38,77),(27,88,39,76),(28,87,40,75),(29,86,41,74),(30,85,42,73),(31,84,43,96),(32,83,44,95),(33,82,45,94),(34,81,46,93),(35,80,47,92),(36,79,48,91),(49,144,61,132),(50,143,62,131),(51,142,63,130),(52,141,64,129),(53,140,65,128),(54,139,66,127),(55,138,67,126),(56,137,68,125),(57,136,69,124),(58,135,70,123),(59,134,71,122),(60,133,72,121)]])
135 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 6A | ··· | 6H | 6I | ··· | 6Q | 8A | 8B | 12A | ··· | 12Z | 12AA | ··· | 12AP | 24A | ··· | 24AZ |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D6 | Q16 | C3×S3 | D12 | C3×D4 | S3×C6 | Dic12 | C3×Q16 | C3×D12 | C3×Dic12 |
| kernel | C32×Dic12 | C32×C24 | C32×Dic6 | C3×Dic12 | C3×C24 | C3×Dic6 | C3×C24 | C32×C6 | C3×C12 | C33 | C24 | C3×C6 | C3×C6 | C12 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 8 | 8 | 16 | 1 | 1 | 1 | 2 | 8 | 2 | 8 | 8 | 4 | 16 | 16 | 32 |
Matrix representation of C32×Dic12 ►in GL3(𝔽73) generated by
| 8 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 8 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 72 | 0 | 0 |
| 0 | 30 | 0 |
| 0 | 22 | 56 |
| 72 | 0 | 0 |
| 0 | 34 | 7 |
| 0 | 12 | 39 |
G:=sub<GL(3,GF(73))| [8,0,0,0,1,0,0,0,1],[8,0,0,0,64,0,0,0,64],[72,0,0,0,30,22,0,0,56],[72,0,0,0,34,12,0,7,39] >;
C32×Dic12 in GAP, Magma, Sage, TeX
C_3^2\times {\rm Dic}_{12} % in TeX
G:=Group("C3^2xDic12"); // GroupNames label
G:=SmallGroup(432,468);
// by ID
G=gap.SmallGroup(432,468);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,533,764,3784,102,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^24=1,d^2=c^12,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations