metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.6D28, D28.8D4, D14⋊5SD16, C4⋊C4⋊2D14, D14⋊C8⋊9C2, (C2×C8)⋊16D14, (C7×D4).1D4, C4.2(C2×D28), C28.1(C2×D4), C4.85(D4×D7), C14.D8⋊7C2, D4⋊C4⋊10D7, (C2×C56)⋊15C22, D14⋊2Q8⋊1C2, C7⋊2(C22⋊SD16), C2.11(D7×SD16), C14.20C22≀C2, (C2×D4).135D14, (C2×Dic7).20D4, C14.23(C2×SD16), (C22×D7).72D4, C22.174(D4×D7), C2.13(D8⋊D7), C14.31(C8⋊C22), (C2×C28).216C23, (D4×C14).37C22, (C2×D28).50C22, C2.23(C22⋊D28), (C2×Dic14)⋊13C22, (C2×D4×D7).5C2, (C2×C7⋊C8)⋊3C22, (C7×C4⋊C4)⋊4C22, (C2×D4.D7)⋊3C2, (C2×C56⋊C2)⋊14C2, (C2×C4×D7).9C22, (C7×D4⋊C4)⋊10C2, (C2×C14).229(C2×D4), (C2×C4).323(C22×D7), SmallGroup(448,310)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.6D28
G = < a,b,c,d | a28=b2=c4=1, d2=a7, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, dbd-1=a21b, dcd-1=a7c-1 >
Subgroups: 1332 in 188 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C22×D7, C22×D7, C22×C14, C22⋊SD16, C56⋊C2, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, D4.D7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, D4×D7, C2×C7⋊D4, D4×C14, C23×D7, C14.D8, D14⋊C8, C7×D4⋊C4, D14⋊2Q8, C2×C56⋊C2, C2×D4.D7, C2×D4×D7, D4.6D28
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C22≀C2, C2×SD16, C8⋊C22, D28, C22×D7, C22⋊SD16, C2×D28, D4×D7, C22⋊D28, D8⋊D7, D7×SD16, D4.6D28
►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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 28)(23 27)(24 26)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(57 76)(58 75)(59 74)(60 73)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(77 84)(78 83)(79 82)(80 81)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)
(1 67 47 91)(2 82 48 106)(3 69 49 93)(4 84 50 108)(5 71 51 95)(6 58 52 110)(7 73 53 97)(8 60 54 112)(9 75 55 99)(10 62 56 86)(11 77 29 101)(12 64 30 88)(13 79 31 103)(14 66 32 90)(15 81 33 105)(16 68 34 92)(17 83 35 107)(18 70 36 94)(19 57 37 109)(20 72 38 96)(21 59 39 111)(22 74 40 98)(23 61 41 85)(24 76 42 100)(25 63 43 87)(26 78 44 102)(27 65 45 89)(28 80 46 104)
(1 91 8 98 15 105 22 112)(2 92 9 99 16 106 23 85)(3 93 10 100 17 107 24 86)(4 94 11 101 18 108 25 87)(5 95 12 102 19 109 26 88)(6 96 13 103 20 110 27 89)(7 97 14 104 21 111 28 90)(29 77 36 84 43 63 50 70)(30 78 37 57 44 64 51 71)(31 79 38 58 45 65 52 72)(32 80 39 59 46 66 53 73)(33 81 40 60 47 67 54 74)(34 82 41 61 48 68 55 75)(35 83 42 62 49 69 56 76)
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(77,84)(78,83)(79,82)(80,81)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105), (1,67,47,91)(2,82,48,106)(3,69,49,93)(4,84,50,108)(5,71,51,95)(6,58,52,110)(7,73,53,97)(8,60,54,112)(9,75,55,99)(10,62,56,86)(11,77,29,101)(12,64,30,88)(13,79,31,103)(14,66,32,90)(15,81,33,105)(16,68,34,92)(17,83,35,107)(18,70,36,94)(19,57,37,109)(20,72,38,96)(21,59,39,111)(22,74,40,98)(23,61,41,85)(24,76,42,100)(25,63,43,87)(26,78,44,102)(27,65,45,89)(28,80,46,104), (1,91,8,98,15,105,22,112)(2,92,9,99,16,106,23,85)(3,93,10,100,17,107,24,86)(4,94,11,101,18,108,25,87)(5,95,12,102,19,109,26,88)(6,96,13,103,20,110,27,89)(7,97,14,104,21,111,28,90)(29,77,36,84,43,63,50,70)(30,78,37,57,44,64,51,71)(31,79,38,58,45,65,52,72)(32,80,39,59,46,66,53,73)(33,81,40,60,47,67,54,74)(34,82,41,61,48,68,55,75)(35,83,42,62,49,69,56,76)>;
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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,28)(23,27)(24,26)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(77,84)(78,83)(79,82)(80,81)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105), (1,67,47,91)(2,82,48,106)(3,69,49,93)(4,84,50,108)(5,71,51,95)(6,58,52,110)(7,73,53,97)(8,60,54,112)(9,75,55,99)(10,62,56,86)(11,77,29,101)(12,64,30,88)(13,79,31,103)(14,66,32,90)(15,81,33,105)(16,68,34,92)(17,83,35,107)(18,70,36,94)(19,57,37,109)(20,72,38,96)(21,59,39,111)(22,74,40,98)(23,61,41,85)(24,76,42,100)(25,63,43,87)(26,78,44,102)(27,65,45,89)(28,80,46,104), (1,91,8,98,15,105,22,112)(2,92,9,99,16,106,23,85)(3,93,10,100,17,107,24,86)(4,94,11,101,18,108,25,87)(5,95,12,102,19,109,26,88)(6,96,13,103,20,110,27,89)(7,97,14,104,21,111,28,90)(29,77,36,84,43,63,50,70)(30,78,37,57,44,64,51,71)(31,79,38,58,45,65,52,72)(32,80,39,59,46,66,53,73)(33,81,40,60,47,67,54,74)(34,82,41,61,48,68,55,75)(35,83,42,62,49,69,56,76) );
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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(22,28),(23,27),(24,26),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(57,76),(58,75),(59,74),(60,73),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(77,84),(78,83),(79,82),(80,81),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(97,112),(98,111),(99,110),(100,109),(101,108),(102,107),(103,106),(104,105)], [(1,67,47,91),(2,82,48,106),(3,69,49,93),(4,84,50,108),(5,71,51,95),(6,58,52,110),(7,73,53,97),(8,60,54,112),(9,75,55,99),(10,62,56,86),(11,77,29,101),(12,64,30,88),(13,79,31,103),(14,66,32,90),(15,81,33,105),(16,68,34,92),(17,83,35,107),(18,70,36,94),(19,57,37,109),(20,72,38,96),(21,59,39,111),(22,74,40,98),(23,61,41,85),(24,76,42,100),(25,63,43,87),(26,78,44,102),(27,65,45,89),(28,80,46,104)], [(1,91,8,98,15,105,22,112),(2,92,9,99,16,106,23,85),(3,93,10,100,17,107,24,86),(4,94,11,101,18,108,25,87),(5,95,12,102,19,109,26,88),(6,96,13,103,20,110,27,89),(7,97,14,104,21,111,28,90),(29,77,36,84,43,63,50,70),(30,78,37,57,44,64,51,71),(31,79,38,58,45,65,52,72),(32,80,39,59,46,66,53,73),(33,81,40,60,47,67,54,74),(34,82,41,61,48,68,55,75),(35,83,42,62,49,69,56,76)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 8 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | SD16 | D14 | D14 | D14 | D28 | C8⋊C22 | D4×D7 | D4×D7 | D8⋊D7 | D7×SD16 |
| kernel | D4.6D28 | C14.D8 | D14⋊C8 | C7×D4⋊C4 | D14⋊2Q8 | C2×C56⋊C2 | C2×D4.D7 | C2×D4×D7 | D28 | C2×Dic7 | C7×D4 | C22×D7 | D4⋊C4 | D14 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 4 | 3 | 3 | 3 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of D4.6D28 ►in GL4(𝔽113) generated by
| 112 | 72 | 0 | 0 |
| 91 | 1 | 0 | 0 |
| 0 | 0 | 89 | 79 |
| 0 | 0 | 10 | 0 |
| 1 | 0 | 0 | 0 |
| 22 | 112 | 0 | 0 |
| 0 | 0 | 0 | 79 |
| 0 | 0 | 103 | 0 |
| 26 | 81 | 0 | 0 |
| 60 | 87 | 0 | 0 |
| 0 | 0 | 17 | 67 |
| 0 | 0 | 80 | 96 |
| 0 | 81 | 0 | 0 |
| 60 | 87 | 0 | 0 |
| 0 | 0 | 96 | 46 |
| 0 | 0 | 33 | 17 |
G:=sub<GL(4,GF(113))| [112,91,0,0,72,1,0,0,0,0,89,10,0,0,79,0],[1,22,0,0,0,112,0,0,0,0,0,103,0,0,79,0],[26,60,0,0,81,87,0,0,0,0,17,80,0,0,67,96],[0,60,0,0,81,87,0,0,0,0,96,33,0,0,46,17] >;
D4.6D28 in GAP, Magma, Sage, TeX
D_4._6D_{28} % in TeX
G:=Group("D4.6D28"); // GroupNames label
G:=SmallGroup(448,310);
// by ID
G=gap.SmallGroup(448,310);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,422,135,268,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=a^7,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^7*b,d*b*d^-1=a^21*b,d*c*d^-1=a^7*c^-1>;
// generators/relations