metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊8D14, Q16⋊7D14, C28.51D8, C56.32D4, C56.29C23, D56.11C22, C4○D8⋊2D7, C7⋊D16⋊6C2, C7⋊C16⋊4C22, (C2×C14).9D8, (C2×D56)⋊22C2, C7⋊5(C16⋊C22), (C2×C8).98D14, C14.68(C2×D8), C7⋊SD32⋊6C2, (C7×D8)⋊8C22, C8.7(C7⋊D4), C28.C8⋊6C2, C4.24(D4⋊D7), (C2×C28).185D4, C28.191(C2×D4), (C7×Q16)⋊7C22, C8.35(C22×D7), C22.5(D4⋊D7), (C2×C56).104C22, (C7×C4○D8)⋊4C2, C2.23(C2×D4⋊D7), C4.17(C2×C7⋊D4), (C2×C4).81(C7⋊D4), SmallGroup(448,727)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q16⋊D14
G = < a,b,c,d | a8=c14=d2=1, b2=a4, bab-1=dad=a-1, ac=ca, cbc-1=a4b, dbd=a3b, dcd=c-1 >
Subgroups: 628 in 90 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, C28, C28, D14, C2×C14, C2×C14, M5(2), D16, SD32, C2×D8, C4○D8, C56, D28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C16⋊C22, C7⋊C16, D56, D56, C2×C56, C7×D8, C7×SD16, C7×Q16, C2×D28, C7×C4○D4, C28.C8, C7⋊D16, C7⋊SD32, C2×D56, C7×C4○D8, Q16⋊D14
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C7⋊D4, C22×D7, C16⋊C22, D4⋊D7, C2×C7⋊D4, C2×D4⋊D7, Q16⋊D14
►On 112 points
Generators in S112
(1 47 19 40 13 54 26 33)(2 48 20 41 14 55 27 34)(3 49 21 42 8 56 28 35)(4 43 15 36 9 50 22 29)(5 44 16 37 10 51 23 30)(6 45 17 38 11 52 24 31)(7 46 18 39 12 53 25 32)(57 96 79 112 64 89 72 105)(58 97 80 99 65 90 73 106)(59 98 81 100 66 91 74 107)(60 85 82 101 67 92 75 108)(61 86 83 102 68 93 76 109)(62 87 84 103 69 94 77 110)(63 88 71 104 70 95 78 111)
(1 60 13 67)(2 68 14 61)(3 62 8 69)(4 70 9 63)(5 64 10 57)(6 58 11 65)(7 66 12 59)(15 71 22 78)(16 79 23 72)(17 73 24 80)(18 81 25 74)(19 75 26 82)(20 83 27 76)(21 77 28 84)(29 95 36 88)(30 89 37 96)(31 97 38 90)(32 91 39 98)(33 85 40 92)(34 93 41 86)(35 87 42 94)(43 104 50 111)(44 112 51 105)(45 106 52 99)(46 100 53 107)(47 108 54 101)(48 102 55 109)(49 110 56 103)
(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 32)(2 31)(3 30)(4 29)(5 35)(6 34)(7 33)(8 37)(9 36)(10 42)(11 41)(12 40)(13 39)(14 38)(15 50)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 43)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(99 109)(100 108)(101 107)(102 106)(103 105)(110 112)
G:=sub<Sym(112)| (1,47,19,40,13,54,26,33)(2,48,20,41,14,55,27,34)(3,49,21,42,8,56,28,35)(4,43,15,36,9,50,22,29)(5,44,16,37,10,51,23,30)(6,45,17,38,11,52,24,31)(7,46,18,39,12,53,25,32)(57,96,79,112,64,89,72,105)(58,97,80,99,65,90,73,106)(59,98,81,100,66,91,74,107)(60,85,82,101,67,92,75,108)(61,86,83,102,68,93,76,109)(62,87,84,103,69,94,77,110)(63,88,71,104,70,95,78,111), (1,60,13,67)(2,68,14,61)(3,62,8,69)(4,70,9,63)(5,64,10,57)(6,58,11,65)(7,66,12,59)(15,71,22,78)(16,79,23,72)(17,73,24,80)(18,81,25,74)(19,75,26,82)(20,83,27,76)(21,77,28,84)(29,95,36,88)(30,89,37,96)(31,97,38,90)(32,91,39,98)(33,85,40,92)(34,93,41,86)(35,87,42,94)(43,104,50,111)(44,112,51,105)(45,106,52,99)(46,100,53,107)(47,108,54,101)(48,102,55,109)(49,110,56,103), (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), (1,32)(2,31)(3,30)(4,29)(5,35)(6,34)(7,33)(8,37)(9,36)(10,42)(11,41)(12,40)(13,39)(14,38)(15,50)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,43)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112)>;
G:=Group( (1,47,19,40,13,54,26,33)(2,48,20,41,14,55,27,34)(3,49,21,42,8,56,28,35)(4,43,15,36,9,50,22,29)(5,44,16,37,10,51,23,30)(6,45,17,38,11,52,24,31)(7,46,18,39,12,53,25,32)(57,96,79,112,64,89,72,105)(58,97,80,99,65,90,73,106)(59,98,81,100,66,91,74,107)(60,85,82,101,67,92,75,108)(61,86,83,102,68,93,76,109)(62,87,84,103,69,94,77,110)(63,88,71,104,70,95,78,111), (1,60,13,67)(2,68,14,61)(3,62,8,69)(4,70,9,63)(5,64,10,57)(6,58,11,65)(7,66,12,59)(15,71,22,78)(16,79,23,72)(17,73,24,80)(18,81,25,74)(19,75,26,82)(20,83,27,76)(21,77,28,84)(29,95,36,88)(30,89,37,96)(31,97,38,90)(32,91,39,98)(33,85,40,92)(34,93,41,86)(35,87,42,94)(43,104,50,111)(44,112,51,105)(45,106,52,99)(46,100,53,107)(47,108,54,101)(48,102,55,109)(49,110,56,103), (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), (1,32)(2,31)(3,30)(4,29)(5,35)(6,34)(7,33)(8,37)(9,36)(10,42)(11,41)(12,40)(13,39)(14,38)(15,50)(16,56)(17,55)(18,54)(19,53)(20,52)(21,51)(22,43)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112) );
G=PermutationGroup([[(1,47,19,40,13,54,26,33),(2,48,20,41,14,55,27,34),(3,49,21,42,8,56,28,35),(4,43,15,36,9,50,22,29),(5,44,16,37,10,51,23,30),(6,45,17,38,11,52,24,31),(7,46,18,39,12,53,25,32),(57,96,79,112,64,89,72,105),(58,97,80,99,65,90,73,106),(59,98,81,100,66,91,74,107),(60,85,82,101,67,92,75,108),(61,86,83,102,68,93,76,109),(62,87,84,103,69,94,77,110),(63,88,71,104,70,95,78,111)], [(1,60,13,67),(2,68,14,61),(3,62,8,69),(4,70,9,63),(5,64,10,57),(6,58,11,65),(7,66,12,59),(15,71,22,78),(16,79,23,72),(17,73,24,80),(18,81,25,74),(19,75,26,82),(20,83,27,76),(21,77,28,84),(29,95,36,88),(30,89,37,96),(31,97,38,90),(32,91,39,98),(33,85,40,92),(34,93,41,86),(35,87,42,94),(43,104,50,111),(44,112,51,105),(45,106,52,99),(46,100,53,107),(47,108,54,101),(48,102,55,109),(49,110,56,103)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,35),(6,34),(7,33),(8,37),(9,36),(10,42),(11,41),(12,40),(13,39),(14,38),(15,50),(16,56),(17,55),(18,54),(19,53),(20,52),(21,51),(22,43),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(57,84),(58,83),(59,82),(60,81),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(99,109),(100,108),(101,107),(102,106),(103,105),(110,112)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14L | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | 28H | 28I | 28J | ··· | 28O | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 8 | 56 | 56 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C16⋊C22 | D4⋊D7 | D4⋊D7 | Q16⋊D14 |
| kernel | Q16⋊D14 | C28.C8 | C7⋊D16 | C7⋊SD32 | C2×D56 | C7×C4○D8 | C56 | C2×C28 | C4○D8 | C28 | C2×C14 | C2×C8 | D8 | Q16 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of Q16⋊D14 ►in GL4(𝔽113) generated by
| 41 | 48 | 0 | 0 |
| 65 | 21 | 0 | 0 |
| 11 | 61 | 21 | 65 |
| 52 | 108 | 48 | 41 |
| 14 | 62 | 2 | 0 |
| 51 | 7 | 0 | 2 |
| 72 | 27 | 99 | 51 |
| 86 | 89 | 62 | 106 |
| 33 | 33 | 0 | 0 |
| 80 | 104 | 0 | 0 |
| 2 | 96 | 80 | 80 |
| 17 | 75 | 33 | 9 |
| 108 | 17 | 0 | 0 |
| 85 | 5 | 0 | 0 |
| 38 | 62 | 4 | 109 |
| 65 | 75 | 32 | 109 |
G:=sub<GL(4,GF(113))| [41,65,11,52,48,21,61,108,0,0,21,48,0,0,65,41],[14,51,72,86,62,7,27,89,2,0,99,62,0,2,51,106],[33,80,2,17,33,104,96,75,0,0,80,33,0,0,80,9],[108,85,38,65,17,5,62,75,0,0,4,32,0,0,109,109] >;
Q16⋊D14 in GAP, Magma, Sage, TeX
Q_{16}\rtimes D_{14} % in TeX
G:=Group("Q16:D14"); // GroupNames label
G:=SmallGroup(448,727);
// by ID
G=gap.SmallGroup(448,727);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,387,675,185,192,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^14=d^2=1,b^2=a^4,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations