direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×D16, D8⋊1D14, C16⋊4D14, D112⋊4C2, C112⋊2C22, D14.12D8, D56⋊5C22, Dic7.3D8, C56.13C23, C7⋊2(C2×D16), (D7×D8)⋊3C2, C7⋊C8.11D4, C4.1(D4×D7), (C7×D16)⋊2C2, (D7×C16)⋊1C2, C7⋊D16⋊1C2, C7⋊C16⋊5C22, C28.7(C2×D4), C2.16(D7×D8), (C4×D7).18D4, C14.32(C2×D8), (C7×D8)⋊5C22, (C8×D7).9C22, C8.19(C22×D7), SmallGroup(448,444)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×D16
G = < a,b,c,d | a7=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 880 in 98 conjugacy classes, 33 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, C23, D7, D7, C14, C14, C16, C16, C2×C8, D8, D8, C2×D4, Dic7, C28, D14, D14, C2×C14, C2×C16, D16, D16, C2×D8, C7⋊C8, C56, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, C2×D16, C7⋊C16, C112, C8×D7, D56, D4⋊D7, C7×D8, D4×D7, D7×C16, D112, C7⋊D16, C7×D16, D7×D8, D7×D16
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, D16, C2×D8, C22×D7, C2×D16, D4×D7, D7×D8, D7×D16
►On 112 points
Generators in S112
(1 98 61 70 24 89 34)(2 99 62 71 25 90 35)(3 100 63 72 26 91 36)(4 101 64 73 27 92 37)(5 102 49 74 28 93 38)(6 103 50 75 29 94 39)(7 104 51 76 30 95 40)(8 105 52 77 31 96 41)(9 106 53 78 32 81 42)(10 107 54 79 17 82 43)(11 108 55 80 18 83 44)(12 109 56 65 19 84 45)(13 110 57 66 20 85 46)(14 111 58 67 21 86 47)(15 112 59 68 22 87 48)(16 97 60 69 23 88 33)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 33)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 49)(29 50)(30 51)(31 52)(32 53)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 97)(89 98)(90 99)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)
(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 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(17 22)(18 21)(19 20)(23 32)(24 31)(25 30)(26 29)(27 28)(33 42)(34 41)(35 40)(36 39)(37 38)(43 48)(44 47)(45 46)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(65 66)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 88)(82 87)(83 86)(84 85)(89 96)(90 95)(91 94)(92 93)(97 106)(98 105)(99 104)(100 103)(101 102)(107 112)(108 111)(109 110)
G:=sub<Sym(112)| (1,98,61,70,24,89,34)(2,99,62,71,25,90,35)(3,100,63,72,26,91,36)(4,101,64,73,27,92,37)(5,102,49,74,28,93,38)(6,103,50,75,29,94,39)(7,104,51,76,30,95,40)(8,105,52,77,31,96,41)(9,106,53,78,32,81,42)(10,107,54,79,17,82,43)(11,108,55,80,18,83,44)(12,109,56,65,19,84,45)(13,110,57,66,20,85,46)(14,111,58,67,21,86,47)(15,112,59,68,22,87,48)(16,97,60,69,23,88,33), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,33)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,49)(29,50)(30,51)(31,52)(32,53)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,97)(89,98)(90,99)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,22)(18,21)(19,20)(23,32)(24,31)(25,30)(26,29)(27,28)(33,42)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(65,66)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,88)(82,87)(83,86)(84,85)(89,96)(90,95)(91,94)(92,93)(97,106)(98,105)(99,104)(100,103)(101,102)(107,112)(108,111)(109,110)>;
G:=Group( (1,98,61,70,24,89,34)(2,99,62,71,25,90,35)(3,100,63,72,26,91,36)(4,101,64,73,27,92,37)(5,102,49,74,28,93,38)(6,103,50,75,29,94,39)(7,104,51,76,30,95,40)(8,105,52,77,31,96,41)(9,106,53,78,32,81,42)(10,107,54,79,17,82,43)(11,108,55,80,18,83,44)(12,109,56,65,19,84,45)(13,110,57,66,20,85,46)(14,111,58,67,21,86,47)(15,112,59,68,22,87,48)(16,97,60,69,23,88,33), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,33)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,49)(29,50)(30,51)(31,52)(32,53)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,97)(89,98)(90,99)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105), (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), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(17,22)(18,21)(19,20)(23,32)(24,31)(25,30)(26,29)(27,28)(33,42)(34,41)(35,40)(36,39)(37,38)(43,48)(44,47)(45,46)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(65,66)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,88)(82,87)(83,86)(84,85)(89,96)(90,95)(91,94)(92,93)(97,106)(98,105)(99,104)(100,103)(101,102)(107,112)(108,111)(109,110) );
G=PermutationGroup([[(1,98,61,70,24,89,34),(2,99,62,71,25,90,35),(3,100,63,72,26,91,36),(4,101,64,73,27,92,37),(5,102,49,74,28,93,38),(6,103,50,75,29,94,39),(7,104,51,76,30,95,40),(8,105,52,77,31,96,41),(9,106,53,78,32,81,42),(10,107,54,79,17,82,43),(11,108,55,80,18,83,44),(12,109,56,65,19,84,45),(13,110,57,66,20,85,46),(14,111,58,67,21,86,47),(15,112,59,68,22,87,48),(16,97,60,69,23,88,33)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,33),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,49),(29,50),(30,51),(31,52),(32,53),(81,106),(82,107),(83,108),(84,109),(85,110),(86,111),(87,112),(88,97),(89,98),(90,99),(91,100),(92,101),(93,102),(94,103),(95,104),(96,105)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(17,22),(18,21),(19,20),(23,32),(24,31),(25,30),(26,29),(27,28),(33,42),(34,41),(35,40),(36,39),(37,38),(43,48),(44,47),(45,46),(49,64),(50,63),(51,62),(52,61),(53,60),(54,59),(55,58),(56,57),(65,66),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(81,88),(82,87),(83,86),(84,85),(89,96),(90,95),(91,94),(92,93),(97,106),(98,105),(99,104),(100,103),(101,102),(107,112),(108,111),(109,110)]])
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 14D | ··· | 14I | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 28A | 28B | 28C | 56A | ··· | 56F | 112A | ··· | 112L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 28 | 28 | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 7 | 7 | 8 | 8 | 56 | 56 | 2 | 14 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 16 | ··· | 16 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | D16 | D4×D7 | D7×D8 | D7×D16 |
| kernel | D7×D16 | D7×C16 | D112 | C7⋊D16 | C7×D16 | D7×D8 | C7⋊C8 | C4×D7 | D16 | Dic7 | D14 | C16 | D8 | D7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 8 | 3 | 6 | 12 |
Matrix representation of D7×D16 ►in GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 112 | 79 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 14 | 57 |
| 0 | 0 | 60 | 91 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 99 | 56 |
| 0 | 0 | 51 | 14 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,79,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,14,60,0,0,57,91],[112,0,0,0,0,112,0,0,0,0,99,51,0,0,56,14] >;
D7×D16 in GAP, Magma, Sage, TeX
D_7\times D_{16} % in TeX
G:=Group("D7xD16"); // GroupNames label
G:=SmallGroup(448,444);
// by ID
G=gap.SmallGroup(448,444);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,135,346,185,192,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations