direct product, metabelian, supersoluble, monomial, A-group
Aliases: C16×C3⋊S3, C48⋊5S3, C24.91D6, C3⋊2(S3×C16), (C3×C48)⋊8C2, C6.12(S3×C8), C32⋊8(C2×C16), C12.75(C4×S3), C3⋊Dic3.8C8, C24.S3⋊10C2, (C3×C24).68C22, C32⋊4C8.11C4, C2.1(C8×C3⋊S3), (C2×C3⋊S3).8C8, C4.16(C4×C3⋊S3), C8.18(C2×C3⋊S3), (C4×C3⋊S3).18C4, (C8×C3⋊S3).14C2, (C3×C6).32(C2×C8), (C3×C12).107(C2×C4), SmallGroup(288,272)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C16×C3⋊S3 |
Generators and relations for C16×C3⋊S3
G = < a,b,c,d | a16=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 228 in 84 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2 [×2], C3 [×4], C4, C4, C22, S3 [×8], C6 [×4], C8, C8, C2×C4, C32, Dic3 [×4], C12 [×4], D6 [×4], C16, C16, C2×C8, C3⋊S3 [×2], C3×C6, C3⋊C8 [×4], C24 [×4], C4×S3 [×4], C2×C16, C3⋊Dic3, C3×C12, C2×C3⋊S3, C3⋊C16 [×4], C48 [×4], S3×C8 [×4], C32⋊4C8, C3×C24, C4×C3⋊S3, S3×C16 [×4], C24.S3, C3×C48, C8×C3⋊S3, C16×C3⋊S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, D6 [×4], C16 [×2], C2×C8, C3⋊S3, C4×S3 [×4], C2×C16, C2×C3⋊S3, S3×C8 [×4], C4×C3⋊S3, S3×C16 [×4], C8×C3⋊S3, C16×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 42 110)(2 43 111)(3 44 112)(4 45 97)(5 46 98)(6 47 99)(7 48 100)(8 33 101)(9 34 102)(10 35 103)(11 36 104)(12 37 105)(13 38 106)(14 39 107)(15 40 108)(16 41 109)(17 144 56)(18 129 57)(19 130 58)(20 131 59)(21 132 60)(22 133 61)(23 134 62)(24 135 63)(25 136 64)(26 137 49)(27 138 50)(28 139 51)(29 140 52)(30 141 53)(31 142 54)(32 143 55)(65 122 89)(66 123 90)(67 124 91)(68 125 92)(69 126 93)(70 127 94)(71 128 95)(72 113 96)(73 114 81)(74 115 82)(75 116 83)(76 117 84)(77 118 85)(78 119 86)(79 120 87)(80 121 88)
(1 116 135)(2 117 136)(3 118 137)(4 119 138)(5 120 139)(6 121 140)(7 122 141)(8 123 142)(9 124 143)(10 125 144)(11 126 129)(12 127 130)(13 128 131)(14 113 132)(15 114 133)(16 115 134)(17 103 68)(18 104 69)(19 105 70)(20 106 71)(21 107 72)(22 108 73)(23 109 74)(24 110 75)(25 111 76)(26 112 77)(27 97 78)(28 98 79)(29 99 80)(30 100 65)(31 101 66)(32 102 67)(33 90 54)(34 91 55)(35 92 56)(36 93 57)(37 94 58)(38 95 59)(39 96 60)(40 81 61)(41 82 62)(42 83 63)(43 84 64)(44 85 49)(45 86 50)(46 87 51)(47 88 52)(48 89 53)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 81)(31 82)(32 83)(33 109)(34 110)(35 111)(36 112)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 65)(62 66)(63 67)(64 68)(113 140)(114 141)(115 142)(116 143)(117 144)(118 129)(119 130)(120 131)(121 132)(122 133)(123 134)(124 135)(125 136)(126 137)(127 138)(128 139)
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), (1,42,110)(2,43,111)(3,44,112)(4,45,97)(5,46,98)(6,47,99)(7,48,100)(8,33,101)(9,34,102)(10,35,103)(11,36,104)(12,37,105)(13,38,106)(14,39,107)(15,40,108)(16,41,109)(17,144,56)(18,129,57)(19,130,58)(20,131,59)(21,132,60)(22,133,61)(23,134,62)(24,135,63)(25,136,64)(26,137,49)(27,138,50)(28,139,51)(29,140,52)(30,141,53)(31,142,54)(32,143,55)(65,122,89)(66,123,90)(67,124,91)(68,125,92)(69,126,93)(70,127,94)(71,128,95)(72,113,96)(73,114,81)(74,115,82)(75,116,83)(76,117,84)(77,118,85)(78,119,86)(79,120,87)(80,121,88), (1,116,135)(2,117,136)(3,118,137)(4,119,138)(5,120,139)(6,121,140)(7,122,141)(8,123,142)(9,124,143)(10,125,144)(11,126,129)(12,127,130)(13,128,131)(14,113,132)(15,114,133)(16,115,134)(17,103,68)(18,104,69)(19,105,70)(20,106,71)(21,107,72)(22,108,73)(23,109,74)(24,110,75)(25,111,76)(26,112,77)(27,97,78)(28,98,79)(29,99,80)(30,100,65)(31,101,66)(32,102,67)(33,90,54)(34,91,55)(35,92,56)(36,93,57)(37,94,58)(38,95,59)(39,96,60)(40,81,61)(41,82,62)(42,83,63)(43,84,64)(44,85,49)(45,86,50)(46,87,51)(47,88,52)(48,89,53), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,81)(31,82)(32,83)(33,109)(34,110)(35,111)(36,112)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,65)(62,66)(63,67)(64,68)(113,140)(114,141)(115,142)(116,143)(117,144)(118,129)(119,130)(120,131)(121,132)(122,133)(123,134)(124,135)(125,136)(126,137)(127,138)(128,139)>;
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), (1,42,110)(2,43,111)(3,44,112)(4,45,97)(5,46,98)(6,47,99)(7,48,100)(8,33,101)(9,34,102)(10,35,103)(11,36,104)(12,37,105)(13,38,106)(14,39,107)(15,40,108)(16,41,109)(17,144,56)(18,129,57)(19,130,58)(20,131,59)(21,132,60)(22,133,61)(23,134,62)(24,135,63)(25,136,64)(26,137,49)(27,138,50)(28,139,51)(29,140,52)(30,141,53)(31,142,54)(32,143,55)(65,122,89)(66,123,90)(67,124,91)(68,125,92)(69,126,93)(70,127,94)(71,128,95)(72,113,96)(73,114,81)(74,115,82)(75,116,83)(76,117,84)(77,118,85)(78,119,86)(79,120,87)(80,121,88), (1,116,135)(2,117,136)(3,118,137)(4,119,138)(5,120,139)(6,121,140)(7,122,141)(8,123,142)(9,124,143)(10,125,144)(11,126,129)(12,127,130)(13,128,131)(14,113,132)(15,114,133)(16,115,134)(17,103,68)(18,104,69)(19,105,70)(20,106,71)(21,107,72)(22,108,73)(23,109,74)(24,110,75)(25,111,76)(26,112,77)(27,97,78)(28,98,79)(29,99,80)(30,100,65)(31,101,66)(32,102,67)(33,90,54)(34,91,55)(35,92,56)(36,93,57)(37,94,58)(38,95,59)(39,96,60)(40,81,61)(41,82,62)(42,83,63)(43,84,64)(44,85,49)(45,86,50)(46,87,51)(47,88,52)(48,89,53), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,84)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(25,92)(26,93)(27,94)(28,95)(29,96)(30,81)(31,82)(32,83)(33,109)(34,110)(35,111)(36,112)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,65)(62,66)(63,67)(64,68)(113,140)(114,141)(115,142)(116,143)(117,144)(118,129)(119,130)(120,131)(121,132)(122,133)(123,134)(124,135)(125,136)(126,137)(127,138)(128,139) );
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)], [(1,42,110),(2,43,111),(3,44,112),(4,45,97),(5,46,98),(6,47,99),(7,48,100),(8,33,101),(9,34,102),(10,35,103),(11,36,104),(12,37,105),(13,38,106),(14,39,107),(15,40,108),(16,41,109),(17,144,56),(18,129,57),(19,130,58),(20,131,59),(21,132,60),(22,133,61),(23,134,62),(24,135,63),(25,136,64),(26,137,49),(27,138,50),(28,139,51),(29,140,52),(30,141,53),(31,142,54),(32,143,55),(65,122,89),(66,123,90),(67,124,91),(68,125,92),(69,126,93),(70,127,94),(71,128,95),(72,113,96),(73,114,81),(74,115,82),(75,116,83),(76,117,84),(77,118,85),(78,119,86),(79,120,87),(80,121,88)], [(1,116,135),(2,117,136),(3,118,137),(4,119,138),(5,120,139),(6,121,140),(7,122,141),(8,123,142),(9,124,143),(10,125,144),(11,126,129),(12,127,130),(13,128,131),(14,113,132),(15,114,133),(16,115,134),(17,103,68),(18,104,69),(19,105,70),(20,106,71),(21,107,72),(22,108,73),(23,109,74),(24,110,75),(25,111,76),(26,112,77),(27,97,78),(28,98,79),(29,99,80),(30,100,65),(31,101,66),(32,102,67),(33,90,54),(34,91,55),(35,92,56),(36,93,57),(37,94,58),(38,95,59),(39,96,60),(40,81,61),(41,82,62),(42,83,63),(43,84,64),(44,85,49),(45,86,50),(46,87,51),(47,88,52),(48,89,53)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,84),(18,85),(19,86),(20,87),(21,88),(22,89),(23,90),(24,91),(25,92),(26,93),(27,94),(28,95),(29,96),(30,81),(31,82),(32,83),(33,109),(34,110),(35,111),(36,112),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,65),(62,66),(63,67),(64,68),(113,140),(114,141),(115,142),(116,143),(117,144),(118,129),(119,130),(120,131),(121,132),(122,133),(123,134),(124,135),(125,136),(126,137),(127,138),(128,139)])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | ··· | 12H | 16A | ··· | 16H | 16I | ··· | 16P | 24A | ··· | 24P | 48A | ··· | 48AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 1 | 1 | 9 | 9 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | D6 | C4×S3 | S3×C8 | S3×C16 |
| kernel | C16×C3⋊S3 | C24.S3 | C3×C48 | C8×C3⋊S3 | C32⋊4C8 | C4×C3⋊S3 | C3⋊Dic3 | C2×C3⋊S3 | C3⋊S3 | C48 | C24 | C12 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 4 | 4 | 8 | 16 | 32 |
Matrix representation of C16×C3⋊S3 ►in GL5(𝔽97)
| 8 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 0 | 33 | 0 |
| 0 | 0 | 0 | 0 | 33 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 96 | 1 | 0 | 0 |
| 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 95 | 94 |
| 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 96 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 96 | 96 |
G:=sub<GL(5,GF(97))| [8,0,0,0,0,0,22,0,0,0,0,0,22,0,0,0,0,0,33,0,0,0,0,0,33],[1,0,0,0,0,0,96,96,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,95,1,0,0,0,94,1],[1,0,0,0,0,0,1,0,0,0,0,96,96,0,0,0,0,0,1,96,0,0,0,0,96] >;
C16×C3⋊S3 in GAP, Magma, Sage, TeX
C_{16}\times C_3\rtimes S_3 % in TeX
G:=Group("C16xC3:S3"); // GroupNames label
G:=SmallGroup(288,272);
// by ID
G=gap.SmallGroup(288,272);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,36,58,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations