metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊2D14, D16⋊2D7, C16⋊2D14, D14.6D8, C112⋊4C22, Dic7.8D8, C56.14C23, D56.1C22, Dic28⋊4C22, C7⋊C8.2D4, (D7×D8)⋊4C2, C4.2(D4×D7), (C7×D16)⋊4C2, C7⋊D16⋊2C2, C7⋊C16⋊1C22, D8.D7⋊1C2, (C4×D7).7D4, C28.8(C2×D4), C2.17(D7×D8), D8⋊3D7⋊3C2, C16⋊D7⋊3C2, C112⋊C2⋊3C2, C7⋊2(C16⋊C22), C14.33(C2×D8), (C7×D8)⋊6C22, (C8×D7).3C22, C8.20(C22×D7), SmallGroup(448,445)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊D14
G = < a,b,c,d | a8=b2=c14=d2=1, bab=cac-1=dad=a-1, cbc-1=a5b, dbd=ab, dcd=c-1 >
Subgroups: 688 in 90 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, Q8, C23, D7, C14, C14, C16, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, Dic7, Dic7, C28, D14, D14, C2×C14, M5(2), D16, D16, SD32, C2×D8, C4○D8, C7⋊C8, C56, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C7×D4, C22×D7, C16⋊C22, C7⋊C16, C112, C8×D7, D56, Dic28, D4⋊D7, D4.D7, C7×D8, D4×D7, D4⋊2D7, C16⋊D7, C112⋊C2, C7⋊D16, D8.D7, C7×D16, D7×D8, D8⋊3D7, D8⋊D14
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C22×D7, C16⋊C22, D4×D7, D7×D8, D8⋊D14
►On 112 points
Generators in S112
(1 39 16 46 14 53 23 32)(2 33 24 54 8 47 17 40)(3 41 18 48 9 55 25 34)(4 35 26 56 10 49 19 42)(5 29 20 50 11 43 27 36)(6 37 28 44 12 51 21 30)(7 31 22 52 13 45 15 38)(57 80 85 104 111 92 73 64)(58 65 74 93 112 105 86 81)(59 82 87 106 99 94 75 66)(60 67 76 95 100 107 88 83)(61 84 89 108 101 96 77 68)(62 69 78 97 102 109 90 71)(63 72 91 110 103 98 79 70)
(1 75)(2 83)(3 77)(4 71)(5 79)(6 73)(7 81)(8 95)(9 89)(10 97)(11 91)(12 85)(13 93)(14 87)(15 65)(16 99)(17 67)(18 101)(19 69)(20 103)(21 57)(22 105)(23 59)(24 107)(25 61)(26 109)(27 63)(28 111)(29 98)(30 64)(31 86)(32 66)(33 88)(34 68)(35 90)(36 70)(37 92)(38 58)(39 94)(40 60)(41 96)(42 62)(43 72)(44 104)(45 74)(46 106)(47 76)(48 108)(49 78)(50 110)(51 80)(52 112)(53 82)(54 100)(55 84)(56 102)
(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 17)(18 28)(19 27)(20 26)(21 25)(22 24)(29 56)(30 55)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(69 70)(71 72)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(97 98)(99 106)(100 105)(101 104)(102 103)(107 112)(108 111)(109 110)
G:=sub<Sym(112)| (1,39,16,46,14,53,23,32)(2,33,24,54,8,47,17,40)(3,41,18,48,9,55,25,34)(4,35,26,56,10,49,19,42)(5,29,20,50,11,43,27,36)(6,37,28,44,12,51,21,30)(7,31,22,52,13,45,15,38)(57,80,85,104,111,92,73,64)(58,65,74,93,112,105,86,81)(59,82,87,106,99,94,75,66)(60,67,76,95,100,107,88,83)(61,84,89,108,101,96,77,68)(62,69,78,97,102,109,90,71)(63,72,91,110,103,98,79,70), (1,75)(2,83)(3,77)(4,71)(5,79)(6,73)(7,81)(8,95)(9,89)(10,97)(11,91)(12,85)(13,93)(14,87)(15,65)(16,99)(17,67)(18,101)(19,69)(20,103)(21,57)(22,105)(23,59)(24,107)(25,61)(26,109)(27,63)(28,111)(29,98)(30,64)(31,86)(32,66)(33,88)(34,68)(35,90)(36,70)(37,92)(38,58)(39,94)(40,60)(41,96)(42,62)(43,72)(44,104)(45,74)(46,106)(47,76)(48,108)(49,78)(50,110)(51,80)(52,112)(53,82)(54,100)(55,84)(56,102), (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)(15,17)(18,28)(19,27)(20,26)(21,25)(22,24)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(69,70)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,98)(99,106)(100,105)(101,104)(102,103)(107,112)(108,111)(109,110)>;
G:=Group( (1,39,16,46,14,53,23,32)(2,33,24,54,8,47,17,40)(3,41,18,48,9,55,25,34)(4,35,26,56,10,49,19,42)(5,29,20,50,11,43,27,36)(6,37,28,44,12,51,21,30)(7,31,22,52,13,45,15,38)(57,80,85,104,111,92,73,64)(58,65,74,93,112,105,86,81)(59,82,87,106,99,94,75,66)(60,67,76,95,100,107,88,83)(61,84,89,108,101,96,77,68)(62,69,78,97,102,109,90,71)(63,72,91,110,103,98,79,70), (1,75)(2,83)(3,77)(4,71)(5,79)(6,73)(7,81)(8,95)(9,89)(10,97)(11,91)(12,85)(13,93)(14,87)(15,65)(16,99)(17,67)(18,101)(19,69)(20,103)(21,57)(22,105)(23,59)(24,107)(25,61)(26,109)(27,63)(28,111)(29,98)(30,64)(31,86)(32,66)(33,88)(34,68)(35,90)(36,70)(37,92)(38,58)(39,94)(40,60)(41,96)(42,62)(43,72)(44,104)(45,74)(46,106)(47,76)(48,108)(49,78)(50,110)(51,80)(52,112)(53,82)(54,100)(55,84)(56,102), (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)(15,17)(18,28)(19,27)(20,26)(21,25)(22,24)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(69,70)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,98)(99,106)(100,105)(101,104)(102,103)(107,112)(108,111)(109,110) );
G=PermutationGroup([[(1,39,16,46,14,53,23,32),(2,33,24,54,8,47,17,40),(3,41,18,48,9,55,25,34),(4,35,26,56,10,49,19,42),(5,29,20,50,11,43,27,36),(6,37,28,44,12,51,21,30),(7,31,22,52,13,45,15,38),(57,80,85,104,111,92,73,64),(58,65,74,93,112,105,86,81),(59,82,87,106,99,94,75,66),(60,67,76,95,100,107,88,83),(61,84,89,108,101,96,77,68),(62,69,78,97,102,109,90,71),(63,72,91,110,103,98,79,70)], [(1,75),(2,83),(3,77),(4,71),(5,79),(6,73),(7,81),(8,95),(9,89),(10,97),(11,91),(12,85),(13,93),(14,87),(15,65),(16,99),(17,67),(18,101),(19,69),(20,103),(21,57),(22,105),(23,59),(24,107),(25,61),(26,109),(27,63),(28,111),(29,98),(30,64),(31,86),(32,66),(33,88),(34,68),(35,90),(36,70),(37,92),(38,58),(39,94),(40,60),(41,96),(42,62),(43,72),(44,104),(45,74),(46,106),(47,76),(48,108),(49,78),(50,110),(51,80),(52,112),(53,82),(54,100),(55,84),(56,102)], [(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),(15,17),(18,28),(19,27),(20,26),(21,25),(22,24),(29,56),(30,55),(31,54),(32,53),(33,52),(34,51),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63),(69,70),(71,72),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(97,98),(99,106),(100,105),(101,104),(102,103),(107,112),(108,111),(109,110)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 14A | 14B | 14C | 14D | ··· | 14I | 16A | 16B | 16C | 16D | 28A | 28B | 28C | 56A | ··· | 56F | 112A | ··· | 112L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 16 | 16 | 16 | 16 | 28 | 28 | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 8 | 8 | 14 | 56 | 2 | 14 | 56 | 2 | 2 | 2 | 2 | 2 | 28 | 2 | 2 | 2 | 16 | ··· | 16 | 4 | 4 | 28 | 28 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | C16⋊C22 | D4×D7 | D7×D8 | D8⋊D14 |
| kernel | D8⋊D14 | C16⋊D7 | C112⋊C2 | C7⋊D16 | D8.D7 | C7×D16 | D7×D8 | D8⋊3D7 | C7⋊C8 | C4×D7 | D16 | Dic7 | D14 | C16 | D8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 2 | 3 | 6 | 12 |
Matrix representation of D8⋊D14 ►in GL8(𝔽113)
| 0 | 0 | 112 | 1 | 0 | 0 | 0 | 0 |
| 34 | 1 | 111 | 79 | 0 | 0 | 0 | 0 |
| 27 | 44 | 112 | 0 | 0 | 0 | 0 | 0 |
| 26 | 44 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 109 | 0 | 0 |
| 0 | 0 | 0 | 0 | 85 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 89 | 62 | 82 | 82 |
| 0 | 0 | 0 | 0 | 81 | 25 | 31 | 82 |
| 31 | 0 | 82 | 31 | 0 | 0 | 0 | 0 |
| 37 | 62 | 51 | 76 | 0 | 0 | 0 | 0 |
| 46 | 8 | 51 | 0 | 0 | 0 | 0 | 0 |
| 77 | 8 | 82 | 82 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 | 105 | 105 |
| 0 | 0 | 0 | 0 | 12 | 51 | 0 | 11 |
| 0 | 0 | 0 | 0 | 73 | 51 | 93 | 93 |
| 0 | 0 | 0 | 0 | 46 | 88 | 19 | 81 |
| 54 | 88 | 50 | 59 | 0 | 0 | 0 | 0 |
| 80 | 9 | 104 | 24 | 0 | 0 | 0 | 0 |
| 88 | 0 | 25 | 25 | 0 | 0 | 0 | 0 |
| 88 | 88 | 25 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 109 | 0 | 0 |
| 0 | 0 | 0 | 0 | 85 | 62 | 0 | 0 |
| 0 | 0 | 0 | 0 | 111 | 47 | 0 | 112 |
| 0 | 0 | 0 | 0 | 51 | 97 | 112 | 0 |
| 54 | 88 | 59 | 50 | 0 | 0 | 0 | 0 |
| 80 | 9 | 24 | 104 | 0 | 0 | 0 | 0 |
| 88 | 0 | 25 | 25 | 0 | 0 | 0 | 0 |
| 79 | 88 | 25 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 109 | 0 | 0 |
| 0 | 0 | 0 | 0 | 85 | 62 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 46 | 0 | 1 |
| 0 | 0 | 0 | 0 | 46 | 97 | 1 | 0 |
G:=sub<GL(8,GF(113))| [0,34,27,26,0,0,0,0,0,1,44,44,0,0,0,0,112,111,112,112,0,0,0,0,1,79,0,0,0,0,0,0,0,0,0,0,51,85,89,81,0,0,0,0,109,0,62,25,0,0,0,0,0,0,82,31,0,0,0,0,0,0,82,82],[31,37,46,77,0,0,0,0,0,62,8,8,0,0,0,0,82,51,51,82,0,0,0,0,31,76,0,82,0,0,0,0,0,0,0,0,1,12,73,46,0,0,0,0,4,51,51,88,0,0,0,0,105,0,93,19,0,0,0,0,105,11,93,81],[54,80,88,88,0,0,0,0,88,9,0,88,0,0,0,0,50,104,25,25,0,0,0,0,59,24,25,25,0,0,0,0,0,0,0,0,51,85,111,51,0,0,0,0,109,62,47,97,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0],[54,80,88,79,0,0,0,0,88,9,0,88,0,0,0,0,59,24,25,25,0,0,0,0,50,104,25,25,0,0,0,0,0,0,0,0,51,85,31,46,0,0,0,0,109,62,46,97,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
D8⋊D14 in GAP, Magma, Sage, TeX
D_8\rtimes D_{14} % in TeX
G:=Group("D8:D14"); // GroupNames label
G:=SmallGroup(448,445);
// by ID
G=gap.SmallGroup(448,445);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,135,346,185,192,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^5*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations