direct product, metabelian, nilpotent (class 3), monomial
Aliases: SD16×C3×C6, C8⋊3C62, Q8⋊2C62, D4.1C62, C62.145D4, (C2×C24)⋊13C6, (C6×C24)⋊21C2, C24⋊13(C2×C6), (C6×Q8)⋊14C6, C6.92(C6×D4), (C6×D4).25C6, C12.51(C3×D4), C4.2(C2×C62), C4.7(D4×C32), (C3×C24)⋊33C22, (C3×C12).148D4, (C2×C4).26C62, C12.56(C22×C6), (C6×C12).375C22, (C3×C12).186C23, (Q8×C32)⋊25C22, C22.15(D4×C32), (D4×C32).32C22, (C2×C8)⋊5(C3×C6), (Q8×C3×C6)⋊17C2, C2.12(D4×C3×C6), (C2×Q8)⋊5(C3×C6), (D4×C3×C6).20C2, (C3×Q8)⋊11(C2×C6), (C2×D4).6(C3×C6), (C2×C6).73(C3×D4), (C3×D4).16(C2×C6), (C3×C6).309(C2×D4), (C2×C12).162(C2×C6), SmallGroup(288,830)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C3×C6
G = < a,b,c,d | a3=b6=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
Subgroups: 324 in 204 conjugacy classes, 132 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C32, C12, C12, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3×C6, C3×C6, C3×C6, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C2×SD16, C3×C12, C3×C12, C62, C62, C2×C24, C3×SD16, C6×D4, C6×Q8, C3×C24, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8×C32, C2×C62, C6×SD16, C6×C24, C32×SD16, D4×C3×C6, Q8×C3×C6, SD16×C3×C6
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, SD16, C2×D4, C3×C6, C3×D4, C22×C6, C2×SD16, C62, C3×SD16, C6×D4, D4×C32, C2×C62, C6×SD16, C32×SD16, D4×C3×C6, SD16×C3×C6
►On 144 points
Generators in S144
(1 99 91)(2 100 92)(3 101 93)(4 102 94)(5 103 95)(6 104 96)(7 97 89)(8 98 90)(9 111 135)(10 112 136)(11 105 129)(12 106 130)(13 107 131)(14 108 132)(15 109 133)(16 110 134)(17 122 114)(18 123 115)(19 124 116)(20 125 117)(21 126 118)(22 127 119)(23 128 120)(24 121 113)(25 49 41)(26 50 42)(27 51 43)(28 52 44)(29 53 45)(30 54 46)(31 55 47)(32 56 48)(33 138 82)(34 139 83)(35 140 84)(36 141 85)(37 142 86)(38 143 87)(39 144 88)(40 137 81)(57 77 65)(58 78 66)(59 79 67)(60 80 68)(61 73 69)(62 74 70)(63 75 71)(64 76 72)
(1 38 18 62 42 135)(2 39 19 63 43 136)(3 40 20 64 44 129)(4 33 21 57 45 130)(5 34 22 58 46 131)(6 35 23 59 47 132)(7 36 24 60 48 133)(8 37 17 61 41 134)(9 99 143 123 74 26)(10 100 144 124 75 27)(11 101 137 125 76 28)(12 102 138 126 77 29)(13 103 139 127 78 30)(14 104 140 128 79 31)(15 97 141 121 80 32)(16 98 142 122 73 25)(49 110 90 86 114 69)(50 111 91 87 115 70)(51 112 92 88 116 71)(52 105 93 81 117 72)(53 106 94 82 118 65)(54 107 95 83 119 66)(55 108 96 84 120 67)(56 109 89 85 113 68)
(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 58)(2 61)(3 64)(4 59)(5 62)(6 57)(7 60)(8 63)(9 127)(10 122)(11 125)(12 128)(13 123)(14 126)(15 121)(16 124)(17 136)(18 131)(19 134)(20 129)(21 132)(22 135)(23 130)(24 133)(25 144)(26 139)(27 142)(28 137)(29 140)(30 143)(31 138)(32 141)(33 47)(34 42)(35 45)(36 48)(37 43)(38 46)(39 41)(40 44)(49 88)(50 83)(51 86)(52 81)(53 84)(54 87)(55 82)(56 85)(65 96)(66 91)(67 94)(68 89)(69 92)(70 95)(71 90)(72 93)(73 100)(74 103)(75 98)(76 101)(77 104)(78 99)(79 102)(80 97)(105 117)(106 120)(107 115)(108 118)(109 113)(110 116)(111 119)(112 114)
G:=sub<Sym(144)| (1,99,91)(2,100,92)(3,101,93)(4,102,94)(5,103,95)(6,104,96)(7,97,89)(8,98,90)(9,111,135)(10,112,136)(11,105,129)(12,106,130)(13,107,131)(14,108,132)(15,109,133)(16,110,134)(17,122,114)(18,123,115)(19,124,116)(20,125,117)(21,126,118)(22,127,119)(23,128,120)(24,121,113)(25,49,41)(26,50,42)(27,51,43)(28,52,44)(29,53,45)(30,54,46)(31,55,47)(32,56,48)(33,138,82)(34,139,83)(35,140,84)(36,141,85)(37,142,86)(38,143,87)(39,144,88)(40,137,81)(57,77,65)(58,78,66)(59,79,67)(60,80,68)(61,73,69)(62,74,70)(63,75,71)(64,76,72), (1,38,18,62,42,135)(2,39,19,63,43,136)(3,40,20,64,44,129)(4,33,21,57,45,130)(5,34,22,58,46,131)(6,35,23,59,47,132)(7,36,24,60,48,133)(8,37,17,61,41,134)(9,99,143,123,74,26)(10,100,144,124,75,27)(11,101,137,125,76,28)(12,102,138,126,77,29)(13,103,139,127,78,30)(14,104,140,128,79,31)(15,97,141,121,80,32)(16,98,142,122,73,25)(49,110,90,86,114,69)(50,111,91,87,115,70)(51,112,92,88,116,71)(52,105,93,81,117,72)(53,106,94,82,118,65)(54,107,95,83,119,66)(55,108,96,84,120,67)(56,109,89,85,113,68), (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), (1,58)(2,61)(3,64)(4,59)(5,62)(6,57)(7,60)(8,63)(9,127)(10,122)(11,125)(12,128)(13,123)(14,126)(15,121)(16,124)(17,136)(18,131)(19,134)(20,129)(21,132)(22,135)(23,130)(24,133)(25,144)(26,139)(27,142)(28,137)(29,140)(30,143)(31,138)(32,141)(33,47)(34,42)(35,45)(36,48)(37,43)(38,46)(39,41)(40,44)(49,88)(50,83)(51,86)(52,81)(53,84)(54,87)(55,82)(56,85)(65,96)(66,91)(67,94)(68,89)(69,92)(70,95)(71,90)(72,93)(73,100)(74,103)(75,98)(76,101)(77,104)(78,99)(79,102)(80,97)(105,117)(106,120)(107,115)(108,118)(109,113)(110,116)(111,119)(112,114)>;
G:=Group( (1,99,91)(2,100,92)(3,101,93)(4,102,94)(5,103,95)(6,104,96)(7,97,89)(8,98,90)(9,111,135)(10,112,136)(11,105,129)(12,106,130)(13,107,131)(14,108,132)(15,109,133)(16,110,134)(17,122,114)(18,123,115)(19,124,116)(20,125,117)(21,126,118)(22,127,119)(23,128,120)(24,121,113)(25,49,41)(26,50,42)(27,51,43)(28,52,44)(29,53,45)(30,54,46)(31,55,47)(32,56,48)(33,138,82)(34,139,83)(35,140,84)(36,141,85)(37,142,86)(38,143,87)(39,144,88)(40,137,81)(57,77,65)(58,78,66)(59,79,67)(60,80,68)(61,73,69)(62,74,70)(63,75,71)(64,76,72), (1,38,18,62,42,135)(2,39,19,63,43,136)(3,40,20,64,44,129)(4,33,21,57,45,130)(5,34,22,58,46,131)(6,35,23,59,47,132)(7,36,24,60,48,133)(8,37,17,61,41,134)(9,99,143,123,74,26)(10,100,144,124,75,27)(11,101,137,125,76,28)(12,102,138,126,77,29)(13,103,139,127,78,30)(14,104,140,128,79,31)(15,97,141,121,80,32)(16,98,142,122,73,25)(49,110,90,86,114,69)(50,111,91,87,115,70)(51,112,92,88,116,71)(52,105,93,81,117,72)(53,106,94,82,118,65)(54,107,95,83,119,66)(55,108,96,84,120,67)(56,109,89,85,113,68), (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), (1,58)(2,61)(3,64)(4,59)(5,62)(6,57)(7,60)(8,63)(9,127)(10,122)(11,125)(12,128)(13,123)(14,126)(15,121)(16,124)(17,136)(18,131)(19,134)(20,129)(21,132)(22,135)(23,130)(24,133)(25,144)(26,139)(27,142)(28,137)(29,140)(30,143)(31,138)(32,141)(33,47)(34,42)(35,45)(36,48)(37,43)(38,46)(39,41)(40,44)(49,88)(50,83)(51,86)(52,81)(53,84)(54,87)(55,82)(56,85)(65,96)(66,91)(67,94)(68,89)(69,92)(70,95)(71,90)(72,93)(73,100)(74,103)(75,98)(76,101)(77,104)(78,99)(79,102)(80,97)(105,117)(106,120)(107,115)(108,118)(109,113)(110,116)(111,119)(112,114) );
G=PermutationGroup([[(1,99,91),(2,100,92),(3,101,93),(4,102,94),(5,103,95),(6,104,96),(7,97,89),(8,98,90),(9,111,135),(10,112,136),(11,105,129),(12,106,130),(13,107,131),(14,108,132),(15,109,133),(16,110,134),(17,122,114),(18,123,115),(19,124,116),(20,125,117),(21,126,118),(22,127,119),(23,128,120),(24,121,113),(25,49,41),(26,50,42),(27,51,43),(28,52,44),(29,53,45),(30,54,46),(31,55,47),(32,56,48),(33,138,82),(34,139,83),(35,140,84),(36,141,85),(37,142,86),(38,143,87),(39,144,88),(40,137,81),(57,77,65),(58,78,66),(59,79,67),(60,80,68),(61,73,69),(62,74,70),(63,75,71),(64,76,72)], [(1,38,18,62,42,135),(2,39,19,63,43,136),(3,40,20,64,44,129),(4,33,21,57,45,130),(5,34,22,58,46,131),(6,35,23,59,47,132),(7,36,24,60,48,133),(8,37,17,61,41,134),(9,99,143,123,74,26),(10,100,144,124,75,27),(11,101,137,125,76,28),(12,102,138,126,77,29),(13,103,139,127,78,30),(14,104,140,128,79,31),(15,97,141,121,80,32),(16,98,142,122,73,25),(49,110,90,86,114,69),(50,111,91,87,115,70),(51,112,92,88,116,71),(52,105,93,81,117,72),(53,106,94,82,118,65),(54,107,95,83,119,66),(55,108,96,84,120,67),(56,109,89,85,113,68)], [(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)], [(1,58),(2,61),(3,64),(4,59),(5,62),(6,57),(7,60),(8,63),(9,127),(10,122),(11,125),(12,128),(13,123),(14,126),(15,121),(16,124),(17,136),(18,131),(19,134),(20,129),(21,132),(22,135),(23,130),(24,133),(25,144),(26,139),(27,142),(28,137),(29,140),(30,143),(31,138),(32,141),(33,47),(34,42),(35,45),(36,48),(37,43),(38,46),(39,41),(40,44),(49,88),(50,83),(51,86),(52,81),(53,84),(54,87),(55,82),(56,85),(65,96),(66,91),(67,94),(68,89),(69,92),(70,95),(71,90),(72,93),(73,100),(74,103),(75,98),(76,101),(77,104),(78,99),(79,102),(80,97),(105,117),(106,120),(107,115),(108,118),(109,113),(110,116),(111,119),(112,114)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 6A | ··· | 6X | 6Y | ··· | 6AN | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12AF | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | SD16 | C3×D4 | C3×D4 | C3×SD16 |
| kernel | SD16×C3×C6 | C6×C24 | C32×SD16 | D4×C3×C6 | Q8×C3×C6 | C6×SD16 | C2×C24 | C3×SD16 | C6×D4 | C6×Q8 | C3×C12 | C62 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 1 | 4 | 1 | 1 | 8 | 8 | 32 | 8 | 8 | 1 | 1 | 4 | 8 | 8 | 32 |
Matrix representation of SD16×C3×C6 ►in GL5(𝔽73)
| 8 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 67 |
| 0 | 0 | 0 | 6 | 6 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(5,GF(73))| [8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,67,6],[72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72] >;
SD16×C3×C6 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_3\times C_6 % in TeX
G:=Group("SD16xC3xC6"); // GroupNames label
G:=SmallGroup(288,830);
// by ID
G=gap.SmallGroup(288,830);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1008,1037,9077,4548,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^8=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^3>;
// generators/relations