metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C16⋊1D14, C56.2D4, C8.3D28, D112⋊2C2, C4.14D56, C28.14D8, C112⋊1C22, D56⋊8C22, M5(2)⋊1D7, C22.5D56, C56.59C23, Dic28⋊7C22, (C2×C14).6D8, (C2×D56)⋊11C2, C112⋊C2⋊1C2, C7⋊1(C16⋊C22), (C2×C8).73D14, (C2×C4).41D28, C14.13(C2×D8), C4.40(C2×D28), C2.15(C2×D56), D56⋊7C2⋊9C2, (C2×C28).128D4, C28.283(C2×D4), (C7×M5(2))⋊1C2, C8.49(C22×D7), (C2×C56).59C22, SmallGroup(448,442)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C16⋊D14
G = < a,b,c | a16=b14=c2=1, bab-1=a9, cac=a7, cbc=b-1 >
Subgroups: 820 in 90 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C16, C2×C8, D8, SD16, Q16, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, M5(2), D16, SD32, C2×D8, C4○D8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C22×D7, C16⋊C22, C112, C56⋊C2, D56, D56, D56, Dic28, C2×C56, C2×D28, C4○D28, D112, C112⋊C2, C7×M5(2), C2×D56, D56⋊7C2, C16⋊D14
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, D28, C22×D7, C16⋊C22, D56, C2×D28, C2×D56, C16⋊D14
►On 112 points
Generators in S112
(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 77 82 47 30 56 99)(2 70 83 40 31 49 100 10 78 91 48 23 57 108)(3 79 84 33 32 58 101)(4 72 85 42 17 51 102 12 80 93 34 25 59 110)(5 65 86 35 18 60 103)(6 74 87 44 19 53 104 14 66 95 36 27 61 112)(7 67 88 37 20 62 105)(8 76 89 46 21 55 106 16 68 81 38 29 63 98)(9 69 90 39 22 64 107)(11 71 92 41 24 50 109)(13 73 94 43 26 52 111)(15 75 96 45 28 54 97)
(1 101)(2 108)(3 99)(4 106)(5 97)(6 104)(7 111)(8 102)(9 109)(10 100)(11 107)(12 98)(13 105)(14 112)(15 103)(16 110)(17 89)(18 96)(19 87)(20 94)(21 85)(22 92)(23 83)(24 90)(25 81)(26 88)(27 95)(28 86)(29 93)(30 84)(31 91)(32 82)(33 47)(34 38)(35 45)(37 43)(39 41)(40 48)(42 46)(49 78)(50 69)(51 76)(52 67)(53 74)(54 65)(55 72)(56 79)(57 70)(58 77)(59 68)(60 75)(61 66)(62 73)(63 80)(64 71)
G:=sub<Sym(112)| (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), (1,77,82,47,30,56,99)(2,70,83,40,31,49,100,10,78,91,48,23,57,108)(3,79,84,33,32,58,101)(4,72,85,42,17,51,102,12,80,93,34,25,59,110)(5,65,86,35,18,60,103)(6,74,87,44,19,53,104,14,66,95,36,27,61,112)(7,67,88,37,20,62,105)(8,76,89,46,21,55,106,16,68,81,38,29,63,98)(9,69,90,39,22,64,107)(11,71,92,41,24,50,109)(13,73,94,43,26,52,111)(15,75,96,45,28,54,97), (1,101)(2,108)(3,99)(4,106)(5,97)(6,104)(7,111)(8,102)(9,109)(10,100)(11,107)(12,98)(13,105)(14,112)(15,103)(16,110)(17,89)(18,96)(19,87)(20,94)(21,85)(22,92)(23,83)(24,90)(25,81)(26,88)(27,95)(28,86)(29,93)(30,84)(31,91)(32,82)(33,47)(34,38)(35,45)(37,43)(39,41)(40,48)(42,46)(49,78)(50,69)(51,76)(52,67)(53,74)(54,65)(55,72)(56,79)(57,70)(58,77)(59,68)(60,75)(61,66)(62,73)(63,80)(64,71)>;
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), (1,77,82,47,30,56,99)(2,70,83,40,31,49,100,10,78,91,48,23,57,108)(3,79,84,33,32,58,101)(4,72,85,42,17,51,102,12,80,93,34,25,59,110)(5,65,86,35,18,60,103)(6,74,87,44,19,53,104,14,66,95,36,27,61,112)(7,67,88,37,20,62,105)(8,76,89,46,21,55,106,16,68,81,38,29,63,98)(9,69,90,39,22,64,107)(11,71,92,41,24,50,109)(13,73,94,43,26,52,111)(15,75,96,45,28,54,97), (1,101)(2,108)(3,99)(4,106)(5,97)(6,104)(7,111)(8,102)(9,109)(10,100)(11,107)(12,98)(13,105)(14,112)(15,103)(16,110)(17,89)(18,96)(19,87)(20,94)(21,85)(22,92)(23,83)(24,90)(25,81)(26,88)(27,95)(28,86)(29,93)(30,84)(31,91)(32,82)(33,47)(34,38)(35,45)(37,43)(39,41)(40,48)(42,46)(49,78)(50,69)(51,76)(52,67)(53,74)(54,65)(55,72)(56,79)(57,70)(58,77)(59,68)(60,75)(61,66)(62,73)(63,80)(64,71) );
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)], [(1,77,82,47,30,56,99),(2,70,83,40,31,49,100,10,78,91,48,23,57,108),(3,79,84,33,32,58,101),(4,72,85,42,17,51,102,12,80,93,34,25,59,110),(5,65,86,35,18,60,103),(6,74,87,44,19,53,104,14,66,95,36,27,61,112),(7,67,88,37,20,62,105),(8,76,89,46,21,55,106,16,68,81,38,29,63,98),(9,69,90,39,22,64,107),(11,71,92,41,24,50,109),(13,73,94,43,26,52,111),(15,75,96,45,28,54,97)], [(1,101),(2,108),(3,99),(4,106),(5,97),(6,104),(7,111),(8,102),(9,109),(10,100),(11,107),(12,98),(13,105),(14,112),(15,103),(16,110),(17,89),(18,96),(19,87),(20,94),(21,85),(22,92),(23,83),(24,90),(25,81),(26,88),(27,95),(28,86),(29,93),(30,84),(31,91),(32,82),(33,47),(34,38),(35,45),(37,43),(39,41),(40,48),(42,46),(49,78),(50,69),(51,76),(52,67),(53,74),(54,65),(55,72),(56,79),(57,70),(58,77),(59,68),(60,75),(61,66),(62,73),(63,80),(64,71)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 14A | 14B | 14C | 14D | 14E | 14F | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56R | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 56 | 56 | 56 | 2 | 2 | 56 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | D28 | D28 | D56 | D56 | C16⋊C22 | C16⋊D14 |
| kernel | C16⋊D14 | D112 | C112⋊C2 | C7×M5(2) | C2×D56 | D56⋊7C2 | C56 | C2×C28 | M5(2) | C28 | C2×C14 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 6 | 3 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of C16⋊D14 ►in GL4(𝔽113) generated by
| 1 | 0 | 111 | 0 |
| 0 | 1 | 0 | 111 |
| 68 | 78 | 112 | 0 |
| 35 | 15 | 0 | 112 |
| 88 | 88 | 0 | 0 |
| 25 | 34 | 0 | 0 |
| 88 | 88 | 25 | 25 |
| 25 | 34 | 88 | 79 |
| 105 | 40 | 0 | 0 |
| 86 | 8 | 0 | 0 |
| 43 | 86 | 19 | 94 |
| 93 | 70 | 13 | 94 |
G:=sub<GL(4,GF(113))| [1,0,68,35,0,1,78,15,111,0,112,0,0,111,0,112],[88,25,88,25,88,34,88,34,0,0,25,88,0,0,25,79],[105,86,43,93,40,8,86,70,0,0,19,13,0,0,94,94] >;
C16⋊D14 in GAP, Magma, Sage, TeX
C_{16}\rtimes D_{14} % in TeX
G:=Group("C16:D14"); // GroupNames label
G:=SmallGroup(448,442);
// by ID
G=gap.SmallGroup(448,442);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,387,142,675,192,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^16=b^14=c^2=1,b*a*b^-1=a^9,c*a*c=a^7,c*b*c=b^-1>;
// generators/relations