direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×Q16, C8.9D14, Q8.3D14, Dic28⋊5C2, D14.14D4, C28.8C23, C56.7C22, Dic7.5D4, Dic14.4C22, C7⋊2(C2×Q16), (Q8×D7).C2, (C7×Q16)⋊2C2, (C8×D7).1C2, C7⋊Q16⋊3C2, C2.22(D4×D7), C7⋊C8.7C22, C14.34(C2×D4), C4.8(C22×D7), (C7×Q8).3C22, (C4×D7).11C22, SmallGroup(224,112)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×Q16
G = < a,b,c,d | a7=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 238 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, Q8, Q8, D7, C14, C2×C8, Q16, Q16, C2×Q8, Dic7, Dic7, C28, C28, D14, C2×Q16, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, C7×Q8, C8×D7, Dic28, C7⋊Q16, C7×Q16, Q8×D7, D7×Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, D14, C2×Q16, C22×D7, D4×D7, D7×Q16
►On 112 points
Generators in S112
(1 102 21 12 86 40 50)(2 103 22 13 87 33 51)(3 104 23 14 88 34 52)(4 97 24 15 81 35 53)(5 98 17 16 82 36 54)(6 99 18 9 83 37 55)(7 100 19 10 84 38 56)(8 101 20 11 85 39 49)(25 74 43 59 70 91 109)(26 75 44 60 71 92 110)(27 76 45 61 72 93 111)(28 77 46 62 65 94 112)(29 78 47 63 66 95 105)(30 79 48 64 67 96 106)(31 80 41 57 68 89 107)(32 73 42 58 69 90 108)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 13)(10 14)(11 15)(12 16)(17 86)(18 87)(19 88)(20 81)(21 82)(22 83)(23 84)(24 85)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 97)(40 98)(41 72)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(57 61)(58 62)(59 63)(60 64)(73 94)(74 95)(75 96)(76 89)(77 90)(78 91)(79 92)(80 93)
(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 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 62 13 58)(10 61 14 57)(11 60 15 64)(12 59 16 63)(17 47 21 43)(18 46 22 42)(19 45 23 41)(20 44 24 48)(33 90 37 94)(34 89 38 93)(35 96 39 92)(36 95 40 91)(49 110 53 106)(50 109 54 105)(51 108 55 112)(52 107 56 111)(65 87 69 83)(66 86 70 82)(67 85 71 81)(68 84 72 88)(73 99 77 103)(74 98 78 102)(75 97 79 101)(76 104 80 100)
G:=sub<Sym(112)| (1,102,21,12,86,40,50)(2,103,22,13,87,33,51)(3,104,23,14,88,34,52)(4,97,24,15,81,35,53)(5,98,17,16,82,36,54)(6,99,18,9,83,37,55)(7,100,19,10,84,38,56)(8,101,20,11,85,39,49)(25,74,43,59,70,91,109)(26,75,44,60,71,92,110)(27,76,45,61,72,93,111)(28,77,46,62,65,94,112)(29,78,47,63,66,95,105)(30,79,48,64,67,96,106)(31,80,41,57,68,89,107)(32,73,42,58,69,90,108), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,13)(10,14)(11,15)(12,16)(17,86)(18,87)(19,88)(20,81)(21,82)(22,83)(23,84)(24,85)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,97)(40,98)(41,72)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(57,61)(58,62)(59,63)(60,64)(73,94)(74,95)(75,96)(76,89)(77,90)(78,91)(79,92)(80,93), (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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,62,13,58)(10,61,14,57)(11,60,15,64)(12,59,16,63)(17,47,21,43)(18,46,22,42)(19,45,23,41)(20,44,24,48)(33,90,37,94)(34,89,38,93)(35,96,39,92)(36,95,40,91)(49,110,53,106)(50,109,54,105)(51,108,55,112)(52,107,56,111)(65,87,69,83)(66,86,70,82)(67,85,71,81)(68,84,72,88)(73,99,77,103)(74,98,78,102)(75,97,79,101)(76,104,80,100)>;
G:=Group( (1,102,21,12,86,40,50)(2,103,22,13,87,33,51)(3,104,23,14,88,34,52)(4,97,24,15,81,35,53)(5,98,17,16,82,36,54)(6,99,18,9,83,37,55)(7,100,19,10,84,38,56)(8,101,20,11,85,39,49)(25,74,43,59,70,91,109)(26,75,44,60,71,92,110)(27,76,45,61,72,93,111)(28,77,46,62,65,94,112)(29,78,47,63,66,95,105)(30,79,48,64,67,96,106)(31,80,41,57,68,89,107)(32,73,42,58,69,90,108), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,13)(10,14)(11,15)(12,16)(17,86)(18,87)(19,88)(20,81)(21,82)(22,83)(23,84)(24,85)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,97)(40,98)(41,72)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(57,61)(58,62)(59,63)(60,64)(73,94)(74,95)(75,96)(76,89)(77,90)(78,91)(79,92)(80,93), (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), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,62,13,58)(10,61,14,57)(11,60,15,64)(12,59,16,63)(17,47,21,43)(18,46,22,42)(19,45,23,41)(20,44,24,48)(33,90,37,94)(34,89,38,93)(35,96,39,92)(36,95,40,91)(49,110,53,106)(50,109,54,105)(51,108,55,112)(52,107,56,111)(65,87,69,83)(66,86,70,82)(67,85,71,81)(68,84,72,88)(73,99,77,103)(74,98,78,102)(75,97,79,101)(76,104,80,100) );
G=PermutationGroup([[(1,102,21,12,86,40,50),(2,103,22,13,87,33,51),(3,104,23,14,88,34,52),(4,97,24,15,81,35,53),(5,98,17,16,82,36,54),(6,99,18,9,83,37,55),(7,100,19,10,84,38,56),(8,101,20,11,85,39,49),(25,74,43,59,70,91,109),(26,75,44,60,71,92,110),(27,76,45,61,72,93,111),(28,77,46,62,65,94,112),(29,78,47,63,66,95,105),(30,79,48,64,67,96,106),(31,80,41,57,68,89,107),(32,73,42,58,69,90,108)], [(1,54),(2,55),(3,56),(4,49),(5,50),(6,51),(7,52),(8,53),(9,13),(10,14),(11,15),(12,16),(17,86),(18,87),(19,88),(20,81),(21,82),(22,83),(23,84),(24,85),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,99),(34,100),(35,101),(36,102),(37,103),(38,104),(39,97),(40,98),(41,72),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(57,61),(58,62),(59,63),(60,64),(73,94),(74,95),(75,96),(76,89),(77,90),(78,91),(79,92),(80,93)], [(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)], [(1,25,5,29),(2,32,6,28),(3,31,7,27),(4,30,8,26),(9,62,13,58),(10,61,14,57),(11,60,15,64),(12,59,16,63),(17,47,21,43),(18,46,22,42),(19,45,23,41),(20,44,24,48),(33,90,37,94),(34,89,38,93),(35,96,39,92),(36,95,40,91),(49,110,53,106),(50,109,54,105),(51,108,55,112),(52,107,56,111),(65,87,69,83),(66,86,70,82),(67,85,71,81),(68,84,72,88),(73,99,77,103),(74,98,78,102),(75,97,79,101),(76,104,80,100)]])
D7×Q16 is a maximal subgroup of
SD32⋊D7 Q32⋊D7 D28.30D4 D8.10D14 D28.44D4
D7×Q16 is a maximal quotient of
Dic7⋊4Q16 Dic7.1Q16 Dic7⋊Q16 Dic7.Q16 D14⋊4Q16 D14.Q16 D14⋊Q16 Dic28⋊6C4 C56⋊2Q8 Dic14⋊2Q8 D14.2Q16 D14⋊2Q16 C56.26D4 Dic7⋊3Q16 D14⋊5Q16 D14⋊3Q16
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 28A | 28B | 28C | 28D | ··· | 28I | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 7 | 7 | 2 | 4 | 4 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | Q16 | D14 | D14 | D4×D7 | D7×Q16 |
| kernel | D7×Q16 | C8×D7 | Dic28 | C7⋊Q16 | C7×Q16 | Q8×D7 | Dic7 | D14 | Q16 | D7 | C8 | Q8 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 4 | 3 | 6 | 3 | 6 |
Matrix representation of D7×Q16 ►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 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 14 |
| 0 | 0 | 8 | 62 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 49 | 65 |
| 0 | 0 | 83 | 64 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,79,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,0,8,0,0,14,62],[112,0,0,0,0,112,0,0,0,0,49,83,0,0,65,64] >;
D7×Q16 in GAP, Magma, Sage, TeX
D_7\times Q_{16} % in TeX
G:=Group("D7xQ16"); // GroupNames label
G:=SmallGroup(224,112);
// by ID
G=gap.SmallGroup(224,112);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,116,86,297,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations