metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊13D14, D28.28D4, C28.4C24, D56⋊16C22, C56.40C23, D28.2C23, Dic14.28D4, Dic28⋊14C22, Dic14.2C23, (D7×D8)⋊6C2, (C2×C8)⋊9D14, C7⋊2(D4○D8), (C2×D8)⋊12D7, (C14×D8)⋊3C2, C4.75(D4×D7), C7⋊D4.8D4, C7⋊C8.1C23, D8⋊3D7⋊6C2, D8⋊D7⋊5C2, (C2×D4)⋊14D14, (C2×C56)⋊3C22, D4⋊D7⋊1C22, C28.79(C2×D4), (D4×D7)⋊1C22, (C8×D7)⋊7C22, D56⋊7C2⋊3C2, D4⋊6D14⋊5C2, C4.4(C23×D7), D14.26(C2×D4), C4○D28⋊3C22, (C7×D8)⋊11C22, D4.D7⋊1C22, (C7×D4).2C23, D4.2(C22×D7), (C4×D7).2C23, C8.10(C22×D7), C22.20(D4×D7), D28.2C4⋊2C2, D4.D14⋊7C2, D4⋊2D7⋊1C22, (D4×C14)⋊20C22, C8⋊D7⋊13C22, C56⋊C2⋊14C22, Dic7.31(C2×D4), (C2×C28).521C23, C14.105(C22×D4), C4.Dic7⋊28C22, C2.78(C2×D4×D7), (C2×C14).394(C2×D4), (C2×C4).229(C22×D7), SmallGroup(448,1210)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊13D14
G = < a,b,c,d | a8=b2=c14=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a6b, dbd=a4b, dcd=c-1 >
Subgroups: 1476 in 268 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, D8, SD16, Q16, C2×D4, C2×D4, C4○D4, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×D8, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C7×D4, C7×D4, C22×D7, C22×C14, D4○D8, C8×D7, C8⋊D7, C56⋊C2, D56, Dic28, C4.Dic7, D4⋊D7, D4.D7, C2×C56, C7×D8, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7, C2×C7⋊D4, D4×C14, D28.2C4, D56⋊7C2, D7×D8, D8⋊D7, D8⋊3D7, D4.D14, C14×D8, D4⋊6D14, D8⋊13D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4○D8, D4×D7, C23×D7, C2×D4×D7, D8⋊13D14
►On 112 points
(1 89 39 108 14 101 32 96)(2 97 33 102 8 109 40 90)(3 91 41 110 9 103 34 98)(4 85 35 104 10 111 42 92)(5 93 29 112 11 105 36 86)(6 87 37 106 12 99 30 94)(7 95 31 100 13 107 38 88)(15 58 53 80 25 73 46 65)(16 66 47 74 26 81 54 59)(17 60 55 82 27 75 48 67)(18 68 49 76 28 83 56 61)(19 62 43 84 22 77 50 69)(20 70 51 78 23 71 44 63)(21 64 45 72 24 79 52 57)
(1 59)(2 67)(3 61)(4 69)(5 63)(6 57)(7 65)(8 82)(9 76)(10 84)(11 78)(12 72)(13 80)(14 74)(15 88)(16 96)(17 90)(18 98)(19 92)(20 86)(21 94)(22 104)(23 112)(24 106)(25 100)(26 108)(27 102)(28 110)(29 71)(30 64)(31 73)(32 66)(33 75)(34 68)(35 77)(36 70)(37 79)(38 58)(39 81)(40 60)(41 83)(42 62)(43 111)(44 93)(45 99)(46 95)(47 101)(48 97)(49 103)(50 85)(51 105)(52 87)(53 107)(54 89)(55 109)(56 91)
(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 21)(19 20)(22 23)(24 28)(25 27)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(43 51)(44 50)(45 49)(46 48)(52 56)(53 55)(57 61)(58 60)(62 70)(63 69)(64 68)(65 67)(71 77)(72 76)(73 75)(78 84)(79 83)(80 82)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)
G:=sub<Sym(112)| (1,89,39,108,14,101,32,96)(2,97,33,102,8,109,40,90)(3,91,41,110,9,103,34,98)(4,85,35,104,10,111,42,92)(5,93,29,112,11,105,36,86)(6,87,37,106,12,99,30,94)(7,95,31,100,13,107,38,88)(15,58,53,80,25,73,46,65)(16,66,47,74,26,81,54,59)(17,60,55,82,27,75,48,67)(18,68,49,76,28,83,56,61)(19,62,43,84,22,77,50,69)(20,70,51,78,23,71,44,63)(21,64,45,72,24,79,52,57), (1,59)(2,67)(3,61)(4,69)(5,63)(6,57)(7,65)(8,82)(9,76)(10,84)(11,78)(12,72)(13,80)(14,74)(15,88)(16,96)(17,90)(18,98)(19,92)(20,86)(21,94)(22,104)(23,112)(24,106)(25,100)(26,108)(27,102)(28,110)(29,71)(30,64)(31,73)(32,66)(33,75)(34,68)(35,77)(36,70)(37,79)(38,58)(39,81)(40,60)(41,83)(42,62)(43,111)(44,93)(45,99)(46,95)(47,101)(48,97)(49,103)(50,85)(51,105)(52,87)(53,107)(54,89)(55,109)(56,91), (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)(18,21)(19,20)(22,23)(24,28)(25,27)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,51)(44,50)(45,49)(46,48)(52,56)(53,55)(57,61)(58,60)(62,70)(63,69)(64,68)(65,67)(71,77)(72,76)(73,75)(78,84)(79,83)(80,82)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)>;
G:=Group( (1,89,39,108,14,101,32,96)(2,97,33,102,8,109,40,90)(3,91,41,110,9,103,34,98)(4,85,35,104,10,111,42,92)(5,93,29,112,11,105,36,86)(6,87,37,106,12,99,30,94)(7,95,31,100,13,107,38,88)(15,58,53,80,25,73,46,65)(16,66,47,74,26,81,54,59)(17,60,55,82,27,75,48,67)(18,68,49,76,28,83,56,61)(19,62,43,84,22,77,50,69)(20,70,51,78,23,71,44,63)(21,64,45,72,24,79,52,57), (1,59)(2,67)(3,61)(4,69)(5,63)(6,57)(7,65)(8,82)(9,76)(10,84)(11,78)(12,72)(13,80)(14,74)(15,88)(16,96)(17,90)(18,98)(19,92)(20,86)(21,94)(22,104)(23,112)(24,106)(25,100)(26,108)(27,102)(28,110)(29,71)(30,64)(31,73)(32,66)(33,75)(34,68)(35,77)(36,70)(37,79)(38,58)(39,81)(40,60)(41,83)(42,62)(43,111)(44,93)(45,99)(46,95)(47,101)(48,97)(49,103)(50,85)(51,105)(52,87)(53,107)(54,89)(55,109)(56,91), (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)(18,21)(19,20)(22,23)(24,28)(25,27)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,51)(44,50)(45,49)(46,48)(52,56)(53,55)(57,61)(58,60)(62,70)(63,69)(64,68)(65,67)(71,77)(72,76)(73,75)(78,84)(79,83)(80,82)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106) );
G=PermutationGroup([[(1,89,39,108,14,101,32,96),(2,97,33,102,8,109,40,90),(3,91,41,110,9,103,34,98),(4,85,35,104,10,111,42,92),(5,93,29,112,11,105,36,86),(6,87,37,106,12,99,30,94),(7,95,31,100,13,107,38,88),(15,58,53,80,25,73,46,65),(16,66,47,74,26,81,54,59),(17,60,55,82,27,75,48,67),(18,68,49,76,28,83,56,61),(19,62,43,84,22,77,50,69),(20,70,51,78,23,71,44,63),(21,64,45,72,24,79,52,57)], [(1,59),(2,67),(3,61),(4,69),(5,63),(6,57),(7,65),(8,82),(9,76),(10,84),(11,78),(12,72),(13,80),(14,74),(15,88),(16,96),(17,90),(18,98),(19,92),(20,86),(21,94),(22,104),(23,112),(24,106),(25,100),(26,108),(27,102),(28,110),(29,71),(30,64),(31,73),(32,66),(33,75),(34,68),(35,77),(36,70),(37,79),(38,58),(39,81),(40,60),(41,83),(42,62),(43,111),(44,93),(45,99),(46,95),(47,101),(48,97),(49,103),(50,85),(51,105),(52,87),(53,107),(54,89),(55,109),(56,91)], [(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),(18,21),(19,20),(22,23),(24,28),(25,27),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(43,51),(44,50),(45,49),(46,48),(52,56),(53,55),(57,61),(58,60),(62,70),(63,69),(64,68),(65,67),(71,77),(72,76),(73,75),(78,84),(79,83),(80,82),(85,105),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99),(92,112),(93,111),(94,110),(95,109),(96,108),(97,107),(98,106)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 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 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D4○D8 | D4×D7 | D4×D7 | D8⋊13D14 |
| kernel | D8⋊13D14 | D28.2C4 | D56⋊7C2 | D7×D8 | D8⋊D7 | D8⋊3D7 | D4.D14 | C14×D8 | D4⋊6D14 | Dic14 | D28 | C7⋊D4 | C2×D8 | C2×C8 | D8 | C2×D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 3 | 3 | 12 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of D8⋊13D14 ►in GL4(𝔽113) generated by
| 0 | 0 | 46 | 26 |
| 0 | 0 | 87 | 0 |
| 0 | 100 | 51 | 0 |
| 13 | 23 | 0 | 51 |
| 74 | 84 | 78 | 93 |
| 29 | 39 | 89 | 96 |
| 65 | 103 | 21 | 29 |
| 101 | 74 | 55 | 92 |
| 33 | 33 | 14 | 86 |
| 80 | 104 | 57 | 27 |
| 0 | 0 | 81 | 80 |
| 0 | 0 | 66 | 8 |
| 80 | 80 | 0 | 0 |
| 9 | 33 | 0 | 0 |
| 0 | 0 | 104 | 104 |
| 0 | 0 | 34 | 9 |
G:=sub<GL(4,GF(113))| [0,0,0,13,0,0,100,23,46,87,51,0,26,0,0,51],[74,29,65,101,84,39,103,74,78,89,21,55,93,96,29,92],[33,80,0,0,33,104,0,0,14,57,81,66,86,27,80,8],[80,9,0,0,80,33,0,0,0,0,104,34,0,0,104,9] >;
D8⋊13D14 in GAP, Magma, Sage, TeX
D_8\rtimes_{13}D_{14} % in TeX
G:=Group("D8:13D14"); // GroupNames label
G:=SmallGroup(448,1210);
// by ID
G=gap.SmallGroup(448,1210);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^14=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^6*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations