direct product, metabelian, supersoluble, monomial
Aliases: Q16×C3⋊S3, C24.27D6, C3⋊4(S3×Q16), (C3×Q16)⋊2S3, C6.125(S3×D4), (C3×Q8).38D6, C32⋊12(C2×Q16), C32⋊5Q16⋊9C2, C3⋊Dic3.50D4, (C32×Q16)⋊5C2, C32⋊7Q16⋊7C2, C12.94(C22×S3), (C3×C12).98C23, (C3×C24).30C22, C32⋊4C8.28C22, (Q8×C32).18C22, C32⋊4Q8.19C22, C8.9(C2×C3⋊S3), (C8×C3⋊S3).2C2, C2.22(D4×C3⋊S3), (Q8×C3⋊S3).3C2, Q8.8(C2×C3⋊S3), (C2×C3⋊S3).74D4, C4.8(C22×C3⋊S3), (C3×C6).246(C2×D4), (C4×C3⋊S3).74C22, SmallGroup(288,774)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16×C3⋊S3
G = < a,b,c,d,e | a8=c3=d3=e2=1, b2=a4, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 660 in 180 conjugacy classes, 57 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C8, Q16, Q16, C2×Q8, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, C4×S3, C3×Q8, C2×Q16, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, S3×C8, Dic12, C3⋊Q16, C3×Q16, S3×Q8, C32⋊4C8, C3×C24, C32⋊4Q8, C32⋊4Q8, C4×C3⋊S3, C4×C3⋊S3, Q8×C32, S3×Q16, C8×C3⋊S3, C32⋊5Q16, C32⋊7Q16, C32×Q16, Q8×C3⋊S3, Q16×C3⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊S3, C22×S3, C2×Q16, C2×C3⋊S3, S3×D4, C22×C3⋊S3, S3×Q16, D4×C3⋊S3, Q16×C3⋊S3
►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 61 5 57)(2 60 6 64)(3 59 7 63)(4 58 8 62)(9 66 13 70)(10 65 14 69)(11 72 15 68)(12 71 16 67)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(25 141 29 137)(26 140 30 144)(27 139 31 143)(28 138 32 142)(33 92 37 96)(34 91 38 95)(35 90 39 94)(36 89 40 93)(41 129 45 133)(42 136 46 132)(43 135 47 131)(44 134 48 130)(73 115 77 119)(74 114 78 118)(75 113 79 117)(76 120 80 116)(81 105 85 109)(82 112 86 108)(83 111 87 107)(84 110 88 106)(97 127 101 123)(98 126 102 122)(99 125 103 121)(100 124 104 128)
(1 112 18)(2 105 19)(3 106 20)(4 107 21)(5 108 22)(6 109 23)(7 110 24)(8 111 17)(9 131 77)(10 132 78)(11 133 79)(12 134 80)(13 135 73)(14 136 74)(15 129 75)(16 130 76)(25 122 92)(26 123 93)(27 124 94)(28 125 95)(29 126 96)(30 127 89)(31 128 90)(32 121 91)(33 137 102)(34 138 103)(35 139 104)(36 140 97)(37 141 98)(38 142 99)(39 143 100)(40 144 101)(41 117 72)(42 118 65)(43 119 66)(44 120 67)(45 113 68)(46 114 69)(47 115 70)(48 116 71)(49 62 87)(50 63 88)(51 64 81)(52 57 82)(53 58 83)(54 59 84)(55 60 85)(56 61 86)
(1 119 25)(2 120 26)(3 113 27)(4 114 28)(5 115 29)(6 116 30)(7 117 31)(8 118 32)(9 102 82)(10 103 83)(11 104 84)(12 97 85)(13 98 86)(14 99 87)(15 100 88)(16 101 81)(17 42 91)(18 43 92)(19 44 93)(20 45 94)(21 46 95)(22 47 96)(23 48 89)(24 41 90)(33 52 131)(34 53 132)(35 54 133)(36 55 134)(37 56 135)(38 49 136)(39 50 129)(40 51 130)(57 77 137)(58 78 138)(59 79 139)(60 80 140)(61 73 141)(62 74 142)(63 75 143)(64 76 144)(65 121 111)(66 122 112)(67 123 105)(68 124 106)(69 125 107)(70 126 108)(71 127 109)(72 128 110)
(1 5)(2 6)(3 7)(4 8)(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 105)(24 106)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 113)(32 114)(41 124)(42 125)(43 126)(44 127)(45 128)(46 121)(47 122)(48 123)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 81)(56 82)(57 61)(58 62)(59 63)(60 64)(65 95)(66 96)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)(73 137)(74 138)(75 139)(76 140)(77 141)(78 142)(79 143)(80 144)(97 130)(98 131)(99 132)(100 133)(101 134)(102 135)(103 136)(104 129)
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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,66,13,70)(10,65,14,69)(11,72,15,68)(12,71,16,67)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,141,29,137)(26,140,30,144)(27,139,31,143)(28,138,32,142)(33,92,37,96)(34,91,38,95)(35,90,39,94)(36,89,40,93)(41,129,45,133)(42,136,46,132)(43,135,47,131)(44,134,48,130)(73,115,77,119)(74,114,78,118)(75,113,79,117)(76,120,80,116)(81,105,85,109)(82,112,86,108)(83,111,87,107)(84,110,88,106)(97,127,101,123)(98,126,102,122)(99,125,103,121)(100,124,104,128), (1,112,18)(2,105,19)(3,106,20)(4,107,21)(5,108,22)(6,109,23)(7,110,24)(8,111,17)(9,131,77)(10,132,78)(11,133,79)(12,134,80)(13,135,73)(14,136,74)(15,129,75)(16,130,76)(25,122,92)(26,123,93)(27,124,94)(28,125,95)(29,126,96)(30,127,89)(31,128,90)(32,121,91)(33,137,102)(34,138,103)(35,139,104)(36,140,97)(37,141,98)(38,142,99)(39,143,100)(40,144,101)(41,117,72)(42,118,65)(43,119,66)(44,120,67)(45,113,68)(46,114,69)(47,115,70)(48,116,71)(49,62,87)(50,63,88)(51,64,81)(52,57,82)(53,58,83)(54,59,84)(55,60,85)(56,61,86), (1,119,25)(2,120,26)(3,113,27)(4,114,28)(5,115,29)(6,116,30)(7,117,31)(8,118,32)(9,102,82)(10,103,83)(11,104,84)(12,97,85)(13,98,86)(14,99,87)(15,100,88)(16,101,81)(17,42,91)(18,43,92)(19,44,93)(20,45,94)(21,46,95)(22,47,96)(23,48,89)(24,41,90)(33,52,131)(34,53,132)(35,54,133)(36,55,134)(37,56,135)(38,49,136)(39,50,129)(40,51,130)(57,77,137)(58,78,138)(59,79,139)(60,80,140)(61,73,141)(62,74,142)(63,75,143)(64,76,144)(65,121,111)(66,122,112)(67,123,105)(68,124,106)(69,125,107)(70,126,108)(71,127,109)(72,128,110), (1,5)(2,6)(3,7)(4,8)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,105)(24,106)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,113)(32,114)(41,124)(42,125)(43,126)(44,127)(45,128)(46,121)(47,122)(48,123)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,81)(56,82)(57,61)(58,62)(59,63)(60,64)(65,95)(66,96)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,129)>;
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), (1,61,5,57)(2,60,6,64)(3,59,7,63)(4,58,8,62)(9,66,13,70)(10,65,14,69)(11,72,15,68)(12,71,16,67)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,141,29,137)(26,140,30,144)(27,139,31,143)(28,138,32,142)(33,92,37,96)(34,91,38,95)(35,90,39,94)(36,89,40,93)(41,129,45,133)(42,136,46,132)(43,135,47,131)(44,134,48,130)(73,115,77,119)(74,114,78,118)(75,113,79,117)(76,120,80,116)(81,105,85,109)(82,112,86,108)(83,111,87,107)(84,110,88,106)(97,127,101,123)(98,126,102,122)(99,125,103,121)(100,124,104,128), (1,112,18)(2,105,19)(3,106,20)(4,107,21)(5,108,22)(6,109,23)(7,110,24)(8,111,17)(9,131,77)(10,132,78)(11,133,79)(12,134,80)(13,135,73)(14,136,74)(15,129,75)(16,130,76)(25,122,92)(26,123,93)(27,124,94)(28,125,95)(29,126,96)(30,127,89)(31,128,90)(32,121,91)(33,137,102)(34,138,103)(35,139,104)(36,140,97)(37,141,98)(38,142,99)(39,143,100)(40,144,101)(41,117,72)(42,118,65)(43,119,66)(44,120,67)(45,113,68)(46,114,69)(47,115,70)(48,116,71)(49,62,87)(50,63,88)(51,64,81)(52,57,82)(53,58,83)(54,59,84)(55,60,85)(56,61,86), (1,119,25)(2,120,26)(3,113,27)(4,114,28)(5,115,29)(6,116,30)(7,117,31)(8,118,32)(9,102,82)(10,103,83)(11,104,84)(12,97,85)(13,98,86)(14,99,87)(15,100,88)(16,101,81)(17,42,91)(18,43,92)(19,44,93)(20,45,94)(21,46,95)(22,47,96)(23,48,89)(24,41,90)(33,52,131)(34,53,132)(35,54,133)(36,55,134)(37,56,135)(38,49,136)(39,50,129)(40,51,130)(57,77,137)(58,78,138)(59,79,139)(60,80,140)(61,73,141)(62,74,142)(63,75,143)(64,76,144)(65,121,111)(66,122,112)(67,123,105)(68,124,106)(69,125,107)(70,126,108)(71,127,109)(72,128,110), (1,5)(2,6)(3,7)(4,8)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,105)(24,106)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,113)(32,114)(41,124)(42,125)(43,126)(44,127)(45,128)(46,121)(47,122)(48,123)(49,83)(50,84)(51,85)(52,86)(53,87)(54,88)(55,81)(56,82)(57,61)(58,62)(59,63)(60,64)(65,95)(66,96)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,129) );
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)], [(1,61,5,57),(2,60,6,64),(3,59,7,63),(4,58,8,62),(9,66,13,70),(10,65,14,69),(11,72,15,68),(12,71,16,67),(17,49,21,53),(18,56,22,52),(19,55,23,51),(20,54,24,50),(25,141,29,137),(26,140,30,144),(27,139,31,143),(28,138,32,142),(33,92,37,96),(34,91,38,95),(35,90,39,94),(36,89,40,93),(41,129,45,133),(42,136,46,132),(43,135,47,131),(44,134,48,130),(73,115,77,119),(74,114,78,118),(75,113,79,117),(76,120,80,116),(81,105,85,109),(82,112,86,108),(83,111,87,107),(84,110,88,106),(97,127,101,123),(98,126,102,122),(99,125,103,121),(100,124,104,128)], [(1,112,18),(2,105,19),(3,106,20),(4,107,21),(5,108,22),(6,109,23),(7,110,24),(8,111,17),(9,131,77),(10,132,78),(11,133,79),(12,134,80),(13,135,73),(14,136,74),(15,129,75),(16,130,76),(25,122,92),(26,123,93),(27,124,94),(28,125,95),(29,126,96),(30,127,89),(31,128,90),(32,121,91),(33,137,102),(34,138,103),(35,139,104),(36,140,97),(37,141,98),(38,142,99),(39,143,100),(40,144,101),(41,117,72),(42,118,65),(43,119,66),(44,120,67),(45,113,68),(46,114,69),(47,115,70),(48,116,71),(49,62,87),(50,63,88),(51,64,81),(52,57,82),(53,58,83),(54,59,84),(55,60,85),(56,61,86)], [(1,119,25),(2,120,26),(3,113,27),(4,114,28),(5,115,29),(6,116,30),(7,117,31),(8,118,32),(9,102,82),(10,103,83),(11,104,84),(12,97,85),(13,98,86),(14,99,87),(15,100,88),(16,101,81),(17,42,91),(18,43,92),(19,44,93),(20,45,94),(21,46,95),(22,47,96),(23,48,89),(24,41,90),(33,52,131),(34,53,132),(35,54,133),(36,55,134),(37,56,135),(38,49,136),(39,50,129),(40,51,130),(57,77,137),(58,78,138),(59,79,139),(60,80,140),(61,73,141),(62,74,142),(63,75,143),(64,76,144),(65,121,111),(66,122,112),(67,123,105),(68,124,106),(69,125,107),(70,126,108),(71,127,109),(72,128,110)], [(1,5),(2,6),(3,7),(4,8),(9,37),(10,38),(11,39),(12,40),(13,33),(14,34),(15,35),(16,36),(17,107),(18,108),(19,109),(20,110),(21,111),(22,112),(23,105),(24,106),(25,115),(26,116),(27,117),(28,118),(29,119),(30,120),(31,113),(32,114),(41,124),(42,125),(43,126),(44,127),(45,128),(46,121),(47,122),(48,123),(49,83),(50,84),(51,85),(52,86),(53,87),(54,88),(55,81),(56,82),(57,61),(58,62),(59,63),(60,64),(65,95),(66,96),(67,89),(68,90),(69,91),(70,92),(71,93),(72,94),(73,137),(74,138),(75,139),(76,140),(77,141),(78,142),(79,143),(80,144),(97,130),(98,131),(99,132),(100,133),(101,134),(102,135),(103,136),(104,129)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 18 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | Q16 | S3×D4 | S3×Q16 |
| kernel | Q16×C3⋊S3 | C8×C3⋊S3 | C32⋊5Q16 | C32⋊7Q16 | C32×Q16 | Q8×C3⋊S3 | C3×Q16 | C3⋊Dic3 | C2×C3⋊S3 | C24 | C3×Q8 | C3⋊S3 | C6 | C3 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 1 | 1 | 4 | 8 | 4 | 4 | 8 |
Matrix representation of Q16×C3⋊S3 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 |
| 0 | 0 | 0 | 0 | 25 | 32 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 68 | 43 |
| 0 | 0 | 0 | 0 | 69 | 5 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,25,0,0,0,0,35,32],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,68,69,0,0,0,0,43,5],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
Q16×C3⋊S3 in GAP, Magma, Sage, TeX
Q_{16}\times C_3\rtimes S_3 % in TeX
G:=Group("Q16xC3:S3"); // GroupNames label
G:=SmallGroup(288,774);
// by ID
G=gap.SmallGroup(288,774);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,135,100,346,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^8=c^3=d^3=e^2=1,b^2=a^4,b*a*b^-1=a^-1,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^-1,e*d*e=d^-1>;
// generators/relations