direct product, metabelian, supersoluble, monomial
Aliases: C32×C8⋊S3, C12.13C62, C33⋊13M4(2), C24⋊7(C3×S3), C24⋊5(C3×C6), D6.(C3×C12), (C3×C24)⋊17C6, (C3×C24)⋊19S3, C6.2(C6×C12), C8⋊3(S3×C32), (S3×C6).5C12, C6.40(S3×C12), Dic3.(C3×C12), (S3×C12).11C6, C12.122(S3×C6), (C32×C24)⋊12C2, (C3×C12).236D6, (C3×Dic3).5C12, C32⋊8(C3×M4(2)), C3⋊1(C32×M4(2)), (C32×Dic3).7C4, (C32×C12).88C22, C3⋊C8⋊4(C3×C6), (C3×C3⋊C8)⋊11C6, (S3×C3×C6).7C4, C2.3(S3×C3×C12), C4.13(S3×C3×C6), (S3×C3×C12).8C2, (C32×C3⋊C8)⋊18C2, (C4×S3).2(C3×C6), (C3×C6).94(C4×S3), (C3×C12).93(C2×C6), (C3×C6).44(C2×C12), (C32×C6).50(C2×C4), SmallGroup(432,465)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C8⋊S3
G = < a,b,c,d,e | a3=b3=c8=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c5, ede=d-1 >
Subgroups: 280 in 152 conjugacy classes, 78 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, M4(2), 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, C8⋊S3, C3×M4(2), S3×C32, C32×C6, C3×C3⋊C8, C3×C24, C3×C24, C3×C24, S3×C12, C6×C12, C32×Dic3, C32×C12, S3×C3×C6, C3×C8⋊S3, C32×M4(2), C32×C3⋊C8, C32×C24, S3×C3×C12, C32×C8⋊S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C32, C12, D6, C2×C6, M4(2), C3×S3, C3×C6, C4×S3, C2×C12, C3×C12, S3×C6, C62, C8⋊S3, C3×M4(2), S3×C32, S3×C12, C6×C12, S3×C3×C6, C3×C8⋊S3, C32×M4(2), S3×C3×C12, C32×C8⋊S3
►On 144 points
Generators in S144
(1 85 73)(2 86 74)(3 87 75)(4 88 76)(5 81 77)(6 82 78)(7 83 79)(8 84 80)(9 98 95)(10 99 96)(11 100 89)(12 101 90)(13 102 91)(14 103 92)(15 104 93)(16 97 94)(17 113 60)(18 114 61)(19 115 62)(20 116 63)(21 117 64)(22 118 57)(23 119 58)(24 120 59)(25 55 67)(26 56 68)(27 49 69)(28 50 70)(29 51 71)(30 52 72)(31 53 65)(32 54 66)(33 111 46)(34 112 47)(35 105 48)(36 106 41)(37 107 42)(38 108 43)(39 109 44)(40 110 45)(121 144 134)(122 137 135)(123 138 136)(124 139 129)(125 140 130)(126 141 131)(127 142 132)(128 143 133)
(1 21 71)(2 22 72)(3 23 65)(4 24 66)(5 17 67)(6 18 68)(7 19 69)(8 20 70)(9 135 47)(10 136 48)(11 129 41)(12 130 42)(13 131 43)(14 132 44)(15 133 45)(16 134 46)(25 81 113)(26 82 114)(27 83 115)(28 84 116)(29 85 117)(30 86 118)(31 87 119)(32 88 120)(33 97 121)(34 98 122)(35 99 123)(36 100 124)(37 101 125)(38 102 126)(39 103 127)(40 104 128)(49 79 62)(50 80 63)(51 73 64)(52 74 57)(53 75 58)(54 76 59)(55 77 60)(56 78 61)(89 139 106)(90 140 107)(91 141 108)(92 142 109)(93 143 110)(94 144 111)(95 137 112)(96 138 105)
(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 71 21)(2 72 22)(3 65 23)(4 66 24)(5 67 17)(6 68 18)(7 69 19)(8 70 20)(9 135 47)(10 136 48)(11 129 41)(12 130 42)(13 131 43)(14 132 44)(15 133 45)(16 134 46)(25 113 81)(26 114 82)(27 115 83)(28 116 84)(29 117 85)(30 118 86)(31 119 87)(32 120 88)(33 97 121)(34 98 122)(35 99 123)(36 100 124)(37 101 125)(38 102 126)(39 103 127)(40 104 128)(49 62 79)(50 63 80)(51 64 73)(52 57 74)(53 58 75)(54 59 76)(55 60 77)(56 61 78)(89 139 106)(90 140 107)(91 141 108)(92 142 109)(93 143 110)(94 144 111)(95 137 112)(96 138 105)
(1 106)(2 111)(3 108)(4 105)(5 110)(6 107)(7 112)(8 109)(9 115)(10 120)(11 117)(12 114)(13 119)(14 116)(15 113)(16 118)(17 93)(18 90)(19 95)(20 92)(21 89)(22 94)(23 91)(24 96)(25 133)(26 130)(27 135)(28 132)(29 129)(30 134)(31 131)(32 136)(33 74)(34 79)(35 76)(36 73)(37 78)(38 75)(39 80)(40 77)(41 85)(42 82)(43 87)(44 84)(45 81)(46 86)(47 83)(48 88)(49 122)(50 127)(51 124)(52 121)(53 126)(54 123)(55 128)(56 125)(57 97)(58 102)(59 99)(60 104)(61 101)(62 98)(63 103)(64 100)(65 141)(66 138)(67 143)(68 140)(69 137)(70 142)(71 139)(72 144)
G:=sub<Sym(144)| (1,85,73)(2,86,74)(3,87,75)(4,88,76)(5,81,77)(6,82,78)(7,83,79)(8,84,80)(9,98,95)(10,99,96)(11,100,89)(12,101,90)(13,102,91)(14,103,92)(15,104,93)(16,97,94)(17,113,60)(18,114,61)(19,115,62)(20,116,63)(21,117,64)(22,118,57)(23,119,58)(24,120,59)(25,55,67)(26,56,68)(27,49,69)(28,50,70)(29,51,71)(30,52,72)(31,53,65)(32,54,66)(33,111,46)(34,112,47)(35,105,48)(36,106,41)(37,107,42)(38,108,43)(39,109,44)(40,110,45)(121,144,134)(122,137,135)(123,138,136)(124,139,129)(125,140,130)(126,141,131)(127,142,132)(128,143,133), (1,21,71)(2,22,72)(3,23,65)(4,24,66)(5,17,67)(6,18,68)(7,19,69)(8,20,70)(9,135,47)(10,136,48)(11,129,41)(12,130,42)(13,131,43)(14,132,44)(15,133,45)(16,134,46)(25,81,113)(26,82,114)(27,83,115)(28,84,116)(29,85,117)(30,86,118)(31,87,119)(32,88,120)(33,97,121)(34,98,122)(35,99,123)(36,100,124)(37,101,125)(38,102,126)(39,103,127)(40,104,128)(49,79,62)(50,80,63)(51,73,64)(52,74,57)(53,75,58)(54,76,59)(55,77,60)(56,78,61)(89,139,106)(90,140,107)(91,141,108)(92,142,109)(93,143,110)(94,144,111)(95,137,112)(96,138,105), (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), (1,71,21)(2,72,22)(3,65,23)(4,66,24)(5,67,17)(6,68,18)(7,69,19)(8,70,20)(9,135,47)(10,136,48)(11,129,41)(12,130,42)(13,131,43)(14,132,44)(15,133,45)(16,134,46)(25,113,81)(26,114,82)(27,115,83)(28,116,84)(29,117,85)(30,118,86)(31,119,87)(32,120,88)(33,97,121)(34,98,122)(35,99,123)(36,100,124)(37,101,125)(38,102,126)(39,103,127)(40,104,128)(49,62,79)(50,63,80)(51,64,73)(52,57,74)(53,58,75)(54,59,76)(55,60,77)(56,61,78)(89,139,106)(90,140,107)(91,141,108)(92,142,109)(93,143,110)(94,144,111)(95,137,112)(96,138,105), (1,106)(2,111)(3,108)(4,105)(5,110)(6,107)(7,112)(8,109)(9,115)(10,120)(11,117)(12,114)(13,119)(14,116)(15,113)(16,118)(17,93)(18,90)(19,95)(20,92)(21,89)(22,94)(23,91)(24,96)(25,133)(26,130)(27,135)(28,132)(29,129)(30,134)(31,131)(32,136)(33,74)(34,79)(35,76)(36,73)(37,78)(38,75)(39,80)(40,77)(41,85)(42,82)(43,87)(44,84)(45,81)(46,86)(47,83)(48,88)(49,122)(50,127)(51,124)(52,121)(53,126)(54,123)(55,128)(56,125)(57,97)(58,102)(59,99)(60,104)(61,101)(62,98)(63,103)(64,100)(65,141)(66,138)(67,143)(68,140)(69,137)(70,142)(71,139)(72,144)>;
G:=Group( (1,85,73)(2,86,74)(3,87,75)(4,88,76)(5,81,77)(6,82,78)(7,83,79)(8,84,80)(9,98,95)(10,99,96)(11,100,89)(12,101,90)(13,102,91)(14,103,92)(15,104,93)(16,97,94)(17,113,60)(18,114,61)(19,115,62)(20,116,63)(21,117,64)(22,118,57)(23,119,58)(24,120,59)(25,55,67)(26,56,68)(27,49,69)(28,50,70)(29,51,71)(30,52,72)(31,53,65)(32,54,66)(33,111,46)(34,112,47)(35,105,48)(36,106,41)(37,107,42)(38,108,43)(39,109,44)(40,110,45)(121,144,134)(122,137,135)(123,138,136)(124,139,129)(125,140,130)(126,141,131)(127,142,132)(128,143,133), (1,21,71)(2,22,72)(3,23,65)(4,24,66)(5,17,67)(6,18,68)(7,19,69)(8,20,70)(9,135,47)(10,136,48)(11,129,41)(12,130,42)(13,131,43)(14,132,44)(15,133,45)(16,134,46)(25,81,113)(26,82,114)(27,83,115)(28,84,116)(29,85,117)(30,86,118)(31,87,119)(32,88,120)(33,97,121)(34,98,122)(35,99,123)(36,100,124)(37,101,125)(38,102,126)(39,103,127)(40,104,128)(49,79,62)(50,80,63)(51,73,64)(52,74,57)(53,75,58)(54,76,59)(55,77,60)(56,78,61)(89,139,106)(90,140,107)(91,141,108)(92,142,109)(93,143,110)(94,144,111)(95,137,112)(96,138,105), (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), (1,71,21)(2,72,22)(3,65,23)(4,66,24)(5,67,17)(6,68,18)(7,69,19)(8,70,20)(9,135,47)(10,136,48)(11,129,41)(12,130,42)(13,131,43)(14,132,44)(15,133,45)(16,134,46)(25,113,81)(26,114,82)(27,115,83)(28,116,84)(29,117,85)(30,118,86)(31,119,87)(32,120,88)(33,97,121)(34,98,122)(35,99,123)(36,100,124)(37,101,125)(38,102,126)(39,103,127)(40,104,128)(49,62,79)(50,63,80)(51,64,73)(52,57,74)(53,58,75)(54,59,76)(55,60,77)(56,61,78)(89,139,106)(90,140,107)(91,141,108)(92,142,109)(93,143,110)(94,144,111)(95,137,112)(96,138,105), (1,106)(2,111)(3,108)(4,105)(5,110)(6,107)(7,112)(8,109)(9,115)(10,120)(11,117)(12,114)(13,119)(14,116)(15,113)(16,118)(17,93)(18,90)(19,95)(20,92)(21,89)(22,94)(23,91)(24,96)(25,133)(26,130)(27,135)(28,132)(29,129)(30,134)(31,131)(32,136)(33,74)(34,79)(35,76)(36,73)(37,78)(38,75)(39,80)(40,77)(41,85)(42,82)(43,87)(44,84)(45,81)(46,86)(47,83)(48,88)(49,122)(50,127)(51,124)(52,121)(53,126)(54,123)(55,128)(56,125)(57,97)(58,102)(59,99)(60,104)(61,101)(62,98)(63,103)(64,100)(65,141)(66,138)(67,143)(68,140)(69,137)(70,142)(71,139)(72,144) );
G=PermutationGroup([[(1,85,73),(2,86,74),(3,87,75),(4,88,76),(5,81,77),(6,82,78),(7,83,79),(8,84,80),(9,98,95),(10,99,96),(11,100,89),(12,101,90),(13,102,91),(14,103,92),(15,104,93),(16,97,94),(17,113,60),(18,114,61),(19,115,62),(20,116,63),(21,117,64),(22,118,57),(23,119,58),(24,120,59),(25,55,67),(26,56,68),(27,49,69),(28,50,70),(29,51,71),(30,52,72),(31,53,65),(32,54,66),(33,111,46),(34,112,47),(35,105,48),(36,106,41),(37,107,42),(38,108,43),(39,109,44),(40,110,45),(121,144,134),(122,137,135),(123,138,136),(124,139,129),(125,140,130),(126,141,131),(127,142,132),(128,143,133)], [(1,21,71),(2,22,72),(3,23,65),(4,24,66),(5,17,67),(6,18,68),(7,19,69),(8,20,70),(9,135,47),(10,136,48),(11,129,41),(12,130,42),(13,131,43),(14,132,44),(15,133,45),(16,134,46),(25,81,113),(26,82,114),(27,83,115),(28,84,116),(29,85,117),(30,86,118),(31,87,119),(32,88,120),(33,97,121),(34,98,122),(35,99,123),(36,100,124),(37,101,125),(38,102,126),(39,103,127),(40,104,128),(49,79,62),(50,80,63),(51,73,64),(52,74,57),(53,75,58),(54,76,59),(55,77,60),(56,78,61),(89,139,106),(90,140,107),(91,141,108),(92,142,109),(93,143,110),(94,144,111),(95,137,112),(96,138,105)], [(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)], [(1,71,21),(2,72,22),(3,65,23),(4,66,24),(5,67,17),(6,68,18),(7,69,19),(8,70,20),(9,135,47),(10,136,48),(11,129,41),(12,130,42),(13,131,43),(14,132,44),(15,133,45),(16,134,46),(25,113,81),(26,114,82),(27,115,83),(28,116,84),(29,117,85),(30,118,86),(31,119,87),(32,120,88),(33,97,121),(34,98,122),(35,99,123),(36,100,124),(37,101,125),(38,102,126),(39,103,127),(40,104,128),(49,62,79),(50,63,80),(51,64,73),(52,57,74),(53,58,75),(54,59,76),(55,60,77),(56,61,78),(89,139,106),(90,140,107),(91,141,108),(92,142,109),(93,143,110),(94,144,111),(95,137,112),(96,138,105)], [(1,106),(2,111),(3,108),(4,105),(5,110),(6,107),(7,112),(8,109),(9,115),(10,120),(11,117),(12,114),(13,119),(14,116),(15,113),(16,118),(17,93),(18,90),(19,95),(20,92),(21,89),(22,94),(23,91),(24,96),(25,133),(26,130),(27,135),(28,132),(29,129),(30,134),(31,131),(32,136),(33,74),(34,79),(35,76),(36,73),(37,78),(38,75),(39,80),(40,77),(41,85),(42,82),(43,87),(44,84),(45,81),(46,86),(47,83),(48,88),(49,122),(50,127),(51,124),(52,121),(53,126),(54,123),(55,128),(56,125),(57,97),(58,102),(59,99),(60,104),(61,101),(62,98),(63,103),(64,100),(65,141),(66,138),(67,143),(68,140),(69,137),(70,142),(71,139),(72,144)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 6A | ··· | 6H | 6I | ··· | 6Q | 6R | ··· | 6Y | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12AH | 12AI | ··· | 12AP | 24A | ··· | 24AZ | 24BA | ··· | 24BP |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | S3 | D6 | M4(2) | C3×S3 | C4×S3 | S3×C6 | C8⋊S3 | C3×M4(2) | S3×C12 | C3×C8⋊S3 |
| kernel | C32×C8⋊S3 | C32×C3⋊C8 | C32×C24 | S3×C3×C12 | C3×C8⋊S3 | C32×Dic3 | S3×C3×C6 | C3×C3⋊C8 | C3×C24 | S3×C12 | C3×Dic3 | S3×C6 | C3×C24 | C3×C12 | C33 | C24 | C3×C6 | C12 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 8 | 8 | 8 | 16 | 16 | 1 | 1 | 2 | 8 | 2 | 8 | 4 | 16 | 16 | 32 |
Matrix representation of C32×C8⋊S3 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 0 | 1 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 63 | 0 |
| 0 | 0 | 5 | 10 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 14 | 64 |
| 70 | 21 | 0 | 0 |
| 17 | 3 | 0 | 0 |
| 0 | 0 | 55 | 1 |
| 0 | 0 | 42 | 18 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[0,27,0,0,1,0,0,0,0,0,63,5,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,8,14,0,0,0,64],[70,17,0,0,21,3,0,0,0,0,55,42,0,0,1,18] >;
C32×C8⋊S3 in GAP, Magma, Sage, TeX
C_3^2\times C_8\rtimes S_3
% in TeX
G:=Group("C3^2xC8:S3"); // GroupNames label
G:=SmallGroup(432,465);
// by ID
G=gap.SmallGroup(432,465);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,1037,260,102,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^5,e*d*e=d^-1>;
// generators/relations