metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊13D4, C22⋊2D56, C23.37D28, (C2×C8)⋊1D14, (C2×C14)⋊1D8, (C2×D56)⋊2C2, C22⋊C8⋊3D7, C2.7(C2×D56), C14.5(C2×D8), C28⋊7D4⋊1C2, C7⋊1(C22⋊D8), (C2×C56)⋊1C22, (C2×C4).32D28, (C2×C28).43D4, C4.120(D4×D7), C2.D56⋊4C2, C14.8C22≀C2, C28.332(C2×D4), (C2×D28)⋊1C22, (C22×D28)⋊2C2, C4⋊Dic7⋊2C22, C14.9(C8⋊C22), (C22×C4).82D14, (C22×C14).52D4, C2.12(C8⋊D14), (C2×C28).742C23, C22.105(C2×D28), C2.11(C22⋊D28), (C22×C28).51C22, (C7×C22⋊C8)⋊5C2, (C2×C14).125(C2×D4), (C2×C4).687(C22×D7), SmallGroup(448,266)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊13D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a19b, bd=db, dcd=c-1 >
Subgroups: 1596 in 198 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C56, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C22⋊D8, D56, C4⋊Dic7, D14⋊C4, C2×C56, C2×D28, C2×D28, C2×D28, C2×C7⋊D4, C22×C28, C23×D7, C2.D56, C7×C22⋊C8, C2×D56, C28⋊7D4, C22×D28, D28⋊13D4
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C22≀C2, C2×D8, C8⋊C22, D28, C22×D7, C22⋊D8, D56, C2×D28, D4×D7, C22⋊D28, C2×D56, C8⋊D14, D28⋊13D4
►On 112 points
Generators in S112
(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 63)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 84)(9 83)(10 82)(11 81)(12 80)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 72)(21 71)(22 70)(23 69)(24 68)(25 67)(26 66)(27 65)(28 64)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)(49 85)(50 112)(51 111)(52 110)(53 109)(54 108)(55 107)(56 106)
(1 36 64 106)(2 35 65 105)(3 34 66 104)(4 33 67 103)(5 32 68 102)(6 31 69 101)(7 30 70 100)(8 29 71 99)(9 56 72 98)(10 55 73 97)(11 54 74 96)(12 53 75 95)(13 52 76 94)(14 51 77 93)(15 50 78 92)(16 49 79 91)(17 48 80 90)(18 47 81 89)(19 46 82 88)(20 45 83 87)(21 44 84 86)(22 43 57 85)(23 42 58 112)(24 41 59 111)(25 40 60 110)(26 39 61 109)(27 38 62 108)(28 37 63 107)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)
G:=sub<Sym(112)| (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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,84)(9,83)(10,82)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106), (1,36,64,106)(2,35,65,105)(3,34,66,104)(4,33,67,103)(5,32,68,102)(6,31,69,101)(7,30,70,100)(8,29,71,99)(9,56,72,98)(10,55,73,97)(11,54,74,96)(12,53,75,95)(13,52,76,94)(14,51,77,93)(15,50,78,92)(16,49,79,91)(17,48,80,90)(18,47,81,89)(19,46,82,88)(20,45,83,87)(21,44,84,86)(22,43,57,85)(23,42,58,112)(24,41,59,111)(25,40,60,110)(26,39,61,109)(27,38,62,108)(28,37,63,107), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)>;
G:=Group( (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), (1,63)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,84)(9,83)(10,82)(11,81)(12,80)(13,79)(14,78)(15,77)(16,76)(17,75)(18,74)(19,73)(20,72)(21,71)(22,70)(23,69)(24,68)(25,67)(26,66)(27,65)(28,64)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106), (1,36,64,106)(2,35,65,105)(3,34,66,104)(4,33,67,103)(5,32,68,102)(6,31,69,101)(7,30,70,100)(8,29,71,99)(9,56,72,98)(10,55,73,97)(11,54,74,96)(12,53,75,95)(13,52,76,94)(14,51,77,93)(15,50,78,92)(16,49,79,91)(17,48,80,90)(18,47,81,89)(19,46,82,88)(20,45,83,87)(21,44,84,86)(22,43,57,85)(23,42,58,112)(24,41,59,111)(25,40,60,110)(26,39,61,109)(27,38,62,108)(28,37,63,107), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84) );
G=PermutationGroup([[(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)], [(1,63),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(8,84),(9,83),(10,82),(11,81),(12,80),(13,79),(14,78),(15,77),(16,76),(17,75),(18,74),(19,73),(20,72),(21,71),(22,70),(23,69),(24,68),(25,67),(26,66),(27,65),(28,64),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,99),(36,98),(37,97),(38,96),(39,95),(40,94),(41,93),(42,92),(43,91),(44,90),(45,89),(46,88),(47,87),(48,86),(49,85),(50,112),(51,111),(52,110),(53,109),(54,108),(55,107),(56,106)], [(1,36,64,106),(2,35,65,105),(3,34,66,104),(4,33,67,103),(5,32,68,102),(6,31,69,101),(7,30,70,100),(8,29,71,99),(9,56,72,98),(10,55,73,97),(11,54,74,96),(12,53,75,95),(13,52,76,94),(14,51,77,93),(15,50,78,92),(16,49,79,91),(17,48,80,90),(18,47,81,89),(19,46,82,88),(20,45,83,87),(21,44,84,86),(22,43,57,85),(23,42,58,112),(24,41,59,111),(25,40,60,110),(26,39,61,109),(27,38,62,108),(28,37,63,107)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 28 | 56 | 2 | 2 | 4 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D8 | D14 | D14 | D28 | D28 | D56 | C8⋊C22 | D4×D7 | C8⋊D14 |
| kernel | D28⋊13D4 | C2.D56 | C7×C22⋊C8 | C2×D56 | C28⋊7D4 | C22×D28 | D28 | C2×C28 | C22×C14 | C22⋊C8 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C14 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 3 | 4 | 6 | 3 | 6 | 6 | 24 | 1 | 6 | 6 |
Matrix representation of D28⋊13D4 ►in GL4(𝔽113) generated by
| 77 | 32 | 0 | 0 |
| 100 | 90 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 8 | 109 | 0 | 0 |
| 44 | 105 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 106 | 112 |
| 6 | 72 | 0 | 0 |
| 67 | 107 | 0 | 0 |
| 0 | 0 | 1 | 81 |
| 0 | 0 | 106 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 106 | 112 |
G:=sub<GL(4,GF(113))| [77,100,0,0,32,90,0,0,0,0,112,0,0,0,0,112],[8,44,0,0,109,105,0,0,0,0,1,106,0,0,0,112],[6,67,0,0,72,107,0,0,0,0,1,106,0,0,81,112],[112,0,0,0,0,112,0,0,0,0,1,106,0,0,0,112] >;
D28⋊13D4 in GAP, Magma, Sage, TeX
D_{28}\rtimes_{13}D_4 % in TeX
G:=Group("D28:13D4"); // GroupNames label
G:=SmallGroup(448,266);
// by ID
G=gap.SmallGroup(448,266);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,226,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^19*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations