metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊4D14, SD16⋊2D14, C56.1C23, M4(2)⋊8D14, C28.20C24, Dic28⋊1C22, D28.13C23, Dic14.13C23, C8⋊C22⋊6D7, D8⋊3D7⋊1C2, D8⋊D7⋊2C2, (C2×D4)⋊30D14, D4⋊D7⋊6C22, (C4×D7).43D4, C4.190(D4×D7), (D4×D7)⋊9C22, (C7×D8)⋊2C22, (C8×D7)⋊2C22, C7⋊C8.10C23, C8.1(C22×D7), C4○D4.28D14, D14.54(C2×D4), SD16⋊D7⋊1C2, C8.D14⋊1C2, C28.241(C2×D4), C8⋊D7⋊2C22, C56⋊C2⋊2C22, (D7×M4(2))⋊2C2, D4.D7⋊5C22, (Q8×D7)⋊10C22, C7⋊Q16⋊3C22, C22.47(D4×D7), C4.20(C23×D7), SD16⋊3D7⋊1C2, D4.9D14⋊9C2, (D4×C14)⋊22C22, C7⋊3(D8⋊C22), Dic7.60(C2×D4), (C7×SD16)⋊2C22, (C7×D4).13C23, D4.13(C22×D7), (C22×D7).43D4, (C4×D7).30C23, D4.D14⋊10C2, Q8.13(C22×D7), (C7×Q8).13C23, D4⋊2D7⋊10C22, (C2×C28).111C23, (C2×Dic7).195D4, Q8⋊2D7⋊10C22, C4○D28.28C22, C14.121(C22×D4), (C7×M4(2))⋊2C22, C4.Dic7⋊13C22, (C2×Dic14)⋊39C22, C2.94(C2×D4×D7), (D7×C4○D4)⋊4C2, (C7×C8⋊C22)⋊2C2, (C2×C14).66(C2×D4), (C2×D4⋊2D7)⋊26C2, (C2×C4×D7).161C22, (C2×C4).95(C22×D7), (C7×C4○D4).24C22, SmallGroup(448,1226)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16⋊D14
G = < a,b,c,d | a8=b2=c14=d2=1, bab=dad=a3, cac-1=a-1, cbc-1=a6b, dbd=a2b, dcd=c-1 >
Subgroups: 1260 in 262 conjugacy classes, 99 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×M4(2), C4○D8, C8⋊C22, C8⋊C22, C8.C22, C2×C4○D4, C7⋊C8, C56, Dic14, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×D4, C7×Q8, C22×D7, C22×D7, C22×C14, D8⋊C22, C8×D7, C8⋊D7, C56⋊C2, Dic28, C4.Dic7, D4⋊D7, D4.D7, C7⋊Q16, C7×M4(2), C7×D8, C7×SD16, C2×Dic14, C2×C4×D7, C2×C4×D7, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C22×Dic7, C2×C7⋊D4, D4×C14, C7×C4○D4, D7×M4(2), C8.D14, D8⋊D7, D8⋊3D7, SD16⋊D7, SD16⋊3D7, D4.D14, D4.9D14, C7×C8⋊C22, C2×D4⋊2D7, D7×C4○D4, SD16⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D8⋊C22, D4×D7, C23×D7, C2×D4×D7, SD16⋊D14
►On 112 points
(1 43 36 80 99 86 16 65)(2 66 17 87 100 81 37 44)(3 45 38 82 101 88 18 67)(4 68 19 89 102 83 39 46)(5 47 40 84 103 90 20 69)(6 70 21 91 104 71 41 48)(7 49 42 72 105 92 22 57)(8 58 23 93 106 73 29 50)(9 51 30 74 107 94 24 59)(10 60 25 95 108 75 31 52)(11 53 32 76 109 96 26 61)(12 62 27 97 110 77 33 54)(13 55 34 78 111 98 28 63)(14 64 15 85 112 79 35 56)
(1 58)(2 94)(3 60)(4 96)(5 62)(6 98)(7 64)(8 86)(9 66)(10 88)(11 68)(12 90)(13 70)(14 92)(15 72)(16 93)(17 74)(18 95)(19 76)(20 97)(21 78)(22 85)(23 80)(24 87)(25 82)(26 89)(27 84)(28 91)(29 65)(30 44)(31 67)(32 46)(33 69)(34 48)(35 57)(36 50)(37 59)(38 52)(39 61)(40 54)(41 63)(42 56)(43 106)(45 108)(47 110)(49 112)(51 100)(53 102)(55 104)(71 111)(73 99)(75 101)(77 103)(79 105)(81 107)(83 109)
(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)
(1 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 16)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(43 85)(44 98)(45 97)(46 96)(47 95)(48 94)(49 93)(50 92)(51 91)(52 90)(53 89)(54 88)(55 87)(56 86)(57 73)(58 72)(59 71)(60 84)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)
G:=sub<Sym(112)| (1,43,36,80,99,86,16,65)(2,66,17,87,100,81,37,44)(3,45,38,82,101,88,18,67)(4,68,19,89,102,83,39,46)(5,47,40,84,103,90,20,69)(6,70,21,91,104,71,41,48)(7,49,42,72,105,92,22,57)(8,58,23,93,106,73,29,50)(9,51,30,74,107,94,24,59)(10,60,25,95,108,75,31,52)(11,53,32,76,109,96,26,61)(12,62,27,97,110,77,33,54)(13,55,34,78,111,98,28,63)(14,64,15,85,112,79,35,56), (1,58)(2,94)(3,60)(4,96)(5,62)(6,98)(7,64)(8,86)(9,66)(10,88)(11,68)(12,90)(13,70)(14,92)(15,72)(16,93)(17,74)(18,95)(19,76)(20,97)(21,78)(22,85)(23,80)(24,87)(25,82)(26,89)(27,84)(28,91)(29,65)(30,44)(31,67)(32,46)(33,69)(34,48)(35,57)(36,50)(37,59)(38,52)(39,61)(40,54)(41,63)(42,56)(43,106)(45,108)(47,110)(49,112)(51,100)(53,102)(55,104)(71,111)(73,99)(75,101)(77,103)(79,105)(81,107)(83,109), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,16)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,85)(44,98)(45,97)(46,96)(47,95)(48,94)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,73)(58,72)(59,71)(60,84)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)>;
G:=Group( (1,43,36,80,99,86,16,65)(2,66,17,87,100,81,37,44)(3,45,38,82,101,88,18,67)(4,68,19,89,102,83,39,46)(5,47,40,84,103,90,20,69)(6,70,21,91,104,71,41,48)(7,49,42,72,105,92,22,57)(8,58,23,93,106,73,29,50)(9,51,30,74,107,94,24,59)(10,60,25,95,108,75,31,52)(11,53,32,76,109,96,26,61)(12,62,27,97,110,77,33,54)(13,55,34,78,111,98,28,63)(14,64,15,85,112,79,35,56), (1,58)(2,94)(3,60)(4,96)(5,62)(6,98)(7,64)(8,86)(9,66)(10,88)(11,68)(12,90)(13,70)(14,92)(15,72)(16,93)(17,74)(18,95)(19,76)(20,97)(21,78)(22,85)(23,80)(24,87)(25,82)(26,89)(27,84)(28,91)(29,65)(30,44)(31,67)(32,46)(33,69)(34,48)(35,57)(36,50)(37,59)(38,52)(39,61)(40,54)(41,63)(42,56)(43,106)(45,108)(47,110)(49,112)(51,100)(53,102)(55,104)(71,111)(73,99)(75,101)(77,103)(79,105)(81,107)(83,109), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,16)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,85)(44,98)(45,97)(46,96)(47,95)(48,94)(49,93)(50,92)(51,91)(52,90)(53,89)(54,88)(55,87)(56,86)(57,73)(58,72)(59,71)(60,84)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106) );
G=PermutationGroup([[(1,43,36,80,99,86,16,65),(2,66,17,87,100,81,37,44),(3,45,38,82,101,88,18,67),(4,68,19,89,102,83,39,46),(5,47,40,84,103,90,20,69),(6,70,21,91,104,71,41,48),(7,49,42,72,105,92,22,57),(8,58,23,93,106,73,29,50),(9,51,30,74,107,94,24,59),(10,60,25,95,108,75,31,52),(11,53,32,76,109,96,26,61),(12,62,27,97,110,77,33,54),(13,55,34,78,111,98,28,63),(14,64,15,85,112,79,35,56)], [(1,58),(2,94),(3,60),(4,96),(5,62),(6,98),(7,64),(8,86),(9,66),(10,88),(11,68),(12,90),(13,70),(14,92),(15,72),(16,93),(17,74),(18,95),(19,76),(20,97),(21,78),(22,85),(23,80),(24,87),(25,82),(26,89),(27,84),(28,91),(29,65),(30,44),(31,67),(32,46),(33,69),(34,48),(35,57),(36,50),(37,59),(38,52),(39,61),(40,54),(41,63),(42,56),(43,106),(45,108),(47,110),(49,112),(51,100),(53,102),(55,104),(71,111),(73,99),(75,101),(77,103),(79,105),(81,107),(83,109)], [(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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16),(17,28),(18,27),(19,26),(20,25),(21,24),(22,23),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(43,85),(44,98),(45,97),(46,96),(47,95),(48,94),(49,93),(50,92),(51,91),(52,90),(53,89),(54,88),(55,87),(56,86),(57,73),(58,72),(59,71),(60,84),(61,83),(62,82),(63,81),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14O | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | 7 | 7 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | D8⋊C22 | D4×D7 | D4×D7 | SD16⋊D14 |
| kernel | SD16⋊D14 | D7×M4(2) | C8.D14 | D8⋊D7 | D8⋊3D7 | SD16⋊D7 | SD16⋊3D7 | D4.D14 | D4.9D14 | C7×C8⋊C22 | C2×D4⋊2D7 | D7×C4○D4 | C4×D7 | C2×Dic7 | C22×D7 | C8⋊C22 | M4(2) | D8 | SD16 | C2×D4 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 6 | 6 | 3 | 3 | 2 | 3 | 3 | 3 |
Matrix representation of SD16⋊D14 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 38 | 2 | 44 |
| 0 | 0 | 45 | 94 | 15 | 91 |
| 0 | 0 | 80 | 12 | 7 | 41 |
| 0 | 0 | 86 | 103 | 37 | 101 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 111 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 70 | 67 | 105 | 50 |
| 0 | 0 | 103 | 28 | 1 | 8 |
| 103 | 103 | 0 | 0 | 0 | 0 |
| 10 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 81 | 0 |
| 0 | 0 | 45 | 94 | 15 | 91 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 31 | 78 | 25 | 19 |
| 10 | 10 | 0 | 0 | 0 | 0 |
| 24 | 103 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 81 | 0 |
| 0 | 0 | 69 | 19 | 17 | 22 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 36 | 35 | 13 | 94 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,24,45,80,86,0,0,38,94,12,103,0,0,2,15,7,37,0,0,44,91,41,101],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,70,103,0,0,111,1,67,28,0,0,0,0,105,1,0,0,0,0,50,8],[103,10,0,0,0,0,103,89,0,0,0,0,0,0,112,45,0,31,0,0,0,94,0,78,0,0,81,15,1,25,0,0,0,91,0,19],[10,24,0,0,0,0,10,103,0,0,0,0,0,0,112,69,0,36,0,0,0,19,0,35,0,0,81,17,1,13,0,0,0,22,0,94] >;
SD16⋊D14 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\rtimes D_{14} % in TeX
G:=Group("SD16:D14"); // GroupNames label
G:=SmallGroup(448,1226);
// by ID
G=gap.SmallGroup(448,1226);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,1123,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=d*a*d=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^6*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations