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), (C14×D8)⋊3C2, (C2×D8)⋊12D7, C4.75(D4×D7), C7⋊D4.8D4, C7⋊C8.1C23, (C2×D4)⋊14D14, D8⋊D7⋊5C2, D8⋊3D7⋊6C2, (C2×C56)⋊3C22, D4⋊D7⋊1C22, C28.79(C2×D4), (D4×D7)⋊1C22, (C8×D7)⋊7C22, D4⋊6D14⋊5C2, D56⋊7C2⋊3C2, C4.4(C23×D7), D14.26(C2×D4), C4○D28⋊3C22, (C7×D8)⋊11C22, D4.D7⋊1C22, (C7×D4).2C23, (C4×D7).2C23, D4.2(C22×D7), C8.10(C22×D7), C22.20(D4×D7), D4.D14⋊7C2, D28.2C4⋊2C2, (D4×C14)⋊20C22, D4⋊2D7⋊1C22, C56⋊C2⋊14C22, C8⋊D7⋊13C22, 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
Subgroups: 1476 in 268 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×4], C22, C22 [×14], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×8], D4 [×4], D4 [×17], Q8 [×3], C23 [×6], D7 [×4], C14, C14 [×5], C2×C8, C2×C8 [×2], M4(2) [×3], D8 [×4], D8 [×5], SD16 [×6], Q16, C2×D4 [×2], C2×D4 [×10], C4○D4 [×9], Dic7 [×2], Dic7 [×2], C28 [×2], D14 [×2], D14 [×6], C2×C14, C2×C14 [×6], C8○D4, C2×D8, C2×D8 [×2], C4○D8 [×3], C8⋊C22 [×6], 2+ (1+4) [×2], C7⋊C8 [×2], C56 [×2], Dic14, Dic14 [×2], C4×D7 [×2], C4×D7 [×2], D28, D28 [×2], C2×Dic7 [×4], C7⋊D4 [×2], C7⋊D4 [×10], C2×C28, C7×D4 [×4], C7×D4 [×2], C22×D7 [×4], C22×C14 [×2], D4○D8, C8×D7 [×2], C8⋊D7 [×2], C56⋊C2 [×2], D56, Dic28, C4.Dic7, D4⋊D7 [×4], D4.D7 [×4], C2×C56, C7×D8 [×4], C4○D28, C4○D28 [×2], D4×D7 [×4], D4×D7 [×2], D4⋊2D7 [×4], D4⋊2D7 [×2], C2×C7⋊D4 [×4], D4×C14 [×2], D28.2C4, D56⋊7C2, D7×D8 [×2], D8⋊D7 [×4], D8⋊3D7 [×2], D4.D14 [×2], C14×D8, D4⋊6D14 [×2], D8⋊13D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C22×D7 [×7], D4○D8, D4×D7 [×2], C23×D7, C2×D4×D7, D8⋊13D14
Generators and relations
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 >
►On 112 points
(1 87 31 71 27 78 38 94)(2 95 39 79 28 72 32 88)(3 89 33 73 22 80 40 96)(4 97 41 81 23 74 34 90)(5 91 35 75 24 82 42 98)(6 85 29 83 25 76 36 92)(7 93 37 77 26 84 30 86)(8 109 47 57 20 64 54 102)(9 103 55 65 21 58 48 110)(10 111 49 59 15 66 56 104)(11 105 43 67 16 60 50 112)(12 99 51 61 17 68 44 106)(13 107 45 69 18 62 52 100)(14 101 53 63 19 70 46 108)
(1 100)(2 108)(3 102)(4 110)(5 104)(6 112)(7 106)(8 96)(9 90)(10 98)(11 92)(12 86)(13 94)(14 88)(15 75)(16 83)(17 77)(18 71)(19 79)(20 73)(21 81)(22 57)(23 65)(24 59)(25 67)(26 61)(27 69)(28 63)(29 60)(30 99)(31 62)(32 101)(33 64)(34 103)(35 66)(36 105)(37 68)(38 107)(39 70)(40 109)(41 58)(42 111)(43 76)(44 93)(45 78)(46 95)(47 80)(48 97)(49 82)(50 85)(51 84)(52 87)(53 72)(54 89)(55 74)(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 12)(9 11)(13 14)(16 21)(17 20)(18 19)(29 34)(30 33)(31 32)(35 42)(36 41)(37 40)(38 39)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(57 61)(58 60)(62 70)(63 69)(64 68)(65 67)(71 88)(72 87)(73 86)(74 85)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(99 109)(100 108)(101 107)(102 106)(103 105)(110 112)
G:=sub<Sym(112)| (1,87,31,71,27,78,38,94)(2,95,39,79,28,72,32,88)(3,89,33,73,22,80,40,96)(4,97,41,81,23,74,34,90)(5,91,35,75,24,82,42,98)(6,85,29,83,25,76,36,92)(7,93,37,77,26,84,30,86)(8,109,47,57,20,64,54,102)(9,103,55,65,21,58,48,110)(10,111,49,59,15,66,56,104)(11,105,43,67,16,60,50,112)(12,99,51,61,17,68,44,106)(13,107,45,69,18,62,52,100)(14,101,53,63,19,70,46,108), (1,100)(2,108)(3,102)(4,110)(5,104)(6,112)(7,106)(8,96)(9,90)(10,98)(11,92)(12,86)(13,94)(14,88)(15,75)(16,83)(17,77)(18,71)(19,79)(20,73)(21,81)(22,57)(23,65)(24,59)(25,67)(26,61)(27,69)(28,63)(29,60)(30,99)(31,62)(32,101)(33,64)(34,103)(35,66)(36,105)(37,68)(38,107)(39,70)(40,109)(41,58)(42,111)(43,76)(44,93)(45,78)(46,95)(47,80)(48,97)(49,82)(50,85)(51,84)(52,87)(53,72)(54,89)(55,74)(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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,12)(9,11)(13,14)(16,21)(17,20)(18,19)(29,34)(30,33)(31,32)(35,42)(36,41)(37,40)(38,39)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(57,61)(58,60)(62,70)(63,69)(64,68)(65,67)(71,88)(72,87)(73,86)(74,85)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112)>;
G:=Group( (1,87,31,71,27,78,38,94)(2,95,39,79,28,72,32,88)(3,89,33,73,22,80,40,96)(4,97,41,81,23,74,34,90)(5,91,35,75,24,82,42,98)(6,85,29,83,25,76,36,92)(7,93,37,77,26,84,30,86)(8,109,47,57,20,64,54,102)(9,103,55,65,21,58,48,110)(10,111,49,59,15,66,56,104)(11,105,43,67,16,60,50,112)(12,99,51,61,17,68,44,106)(13,107,45,69,18,62,52,100)(14,101,53,63,19,70,46,108), (1,100)(2,108)(3,102)(4,110)(5,104)(6,112)(7,106)(8,96)(9,90)(10,98)(11,92)(12,86)(13,94)(14,88)(15,75)(16,83)(17,77)(18,71)(19,79)(20,73)(21,81)(22,57)(23,65)(24,59)(25,67)(26,61)(27,69)(28,63)(29,60)(30,99)(31,62)(32,101)(33,64)(34,103)(35,66)(36,105)(37,68)(38,107)(39,70)(40,109)(41,58)(42,111)(43,76)(44,93)(45,78)(46,95)(47,80)(48,97)(49,82)(50,85)(51,84)(52,87)(53,72)(54,89)(55,74)(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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,12)(9,11)(13,14)(16,21)(17,20)(18,19)(29,34)(30,33)(31,32)(35,42)(36,41)(37,40)(38,39)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(57,61)(58,60)(62,70)(63,69)(64,68)(65,67)(71,88)(72,87)(73,86)(74,85)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(99,109)(100,108)(101,107)(102,106)(103,105)(110,112) );
G=PermutationGroup([(1,87,31,71,27,78,38,94),(2,95,39,79,28,72,32,88),(3,89,33,73,22,80,40,96),(4,97,41,81,23,74,34,90),(5,91,35,75,24,82,42,98),(6,85,29,83,25,76,36,92),(7,93,37,77,26,84,30,86),(8,109,47,57,20,64,54,102),(9,103,55,65,21,58,48,110),(10,111,49,59,15,66,56,104),(11,105,43,67,16,60,50,112),(12,99,51,61,17,68,44,106),(13,107,45,69,18,62,52,100),(14,101,53,63,19,70,46,108)], [(1,100),(2,108),(3,102),(4,110),(5,104),(6,112),(7,106),(8,96),(9,90),(10,98),(11,92),(12,86),(13,94),(14,88),(15,75),(16,83),(17,77),(18,71),(19,79),(20,73),(21,81),(22,57),(23,65),(24,59),(25,67),(26,61),(27,69),(28,63),(29,60),(30,99),(31,62),(32,101),(33,64),(34,103),(35,66),(36,105),(37,68),(38,107),(39,70),(40,109),(41,58),(42,111),(43,76),(44,93),(45,78),(46,95),(47,80),(48,97),(49,82),(50,85),(51,84),(52,87),(53,72),(54,89),(55,74),(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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,12),(9,11),(13,14),(16,21),(17,20),(18,19),(29,34),(30,33),(31,32),(35,42),(36,41),(37,40),(38,39),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(57,61),(58,60),(62,70),(63,69),(64,68),(65,67),(71,88),(72,87),(73,86),(74,85),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(99,109),(100,108),(101,107),(102,106),(103,105),(110,112)])
Matrix representation ►G ⊆ 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] >;
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 |
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