metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.11D28, Q8.11D28, D56⋊11C22, C28.61C24, C56.10C23, M4(2)⋊20D14, D28.24C23, Dic28⋊10C22, Dic14.24C23, C8○D4⋊3D7, (C2×C8)⋊6D14, (C2×C56)⋊9C22, (C7×D4).23D4, C28.73(C2×D4), C7⋊1(D4○SD16), C4.27(C2×D28), (C7×Q8).23D4, D4⋊8D14⋊3C2, C8⋊D14⋊11C2, C4○D4.38D14, C4○D28⋊1C22, D56⋊7C2⋊11C2, C8.55(C22×D7), C4.58(C23×D7), C22.3(C2×D28), C8.D14⋊11C2, C56⋊C2⋊11C22, C2.30(C22×D28), C14.28(C22×D4), D4.10D14⋊3C2, (C2×C28).515C23, (C2×Dic14)⋊35C22, (C2×D28).175C22, (C7×M4(2))⋊22C22, (C7×C8○D4)⋊3C2, (C2×C56⋊C2)⋊6C2, (C2×C14).8(C2×D4), (C7×C4○D4).45C22, (C2×C4).226(C22×D7), SmallGroup(448,1204)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.11D28
G = < a,b,c,d | a4=b2=1, c28=d2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c27 >
Subgroups: 1436 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C56, C56, Dic14, Dic14, Dic14, C4×D7, D28, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×Q8, C22×D7, D4○SD16, C56⋊C2, C56⋊C2, D56, Dic28, C2×C56, C7×M4(2), C2×Dic14, C2×D28, C4○D28, D4×D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C7×C4○D4, C2×C56⋊C2, D56⋊7C2, C8⋊D14, C8.D14, C7×C8○D4, D4⋊8D14, D4.10D14, D4.11D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, D28, C22×D7, D4○SD16, C2×D28, C23×D7, C22×D28, D4.11D28
►On 112 points
(1 92 29 64)(2 93 30 65)(3 94 31 66)(4 95 32 67)(5 96 33 68)(6 97 34 69)(7 98 35 70)(8 99 36 71)(9 100 37 72)(10 101 38 73)(11 102 39 74)(12 103 40 75)(13 104 41 76)(14 105 42 77)(15 106 43 78)(16 107 44 79)(17 108 45 80)(18 109 46 81)(19 110 47 82)(20 111 48 83)(21 112 49 84)(22 57 50 85)(23 58 51 86)(24 59 52 87)(25 60 53 88)(26 61 54 89)(27 62 55 90)(28 63 56 91)
(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(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 28 29 56)(2 55 30 27)(3 26 31 54)(4 53 32 25)(5 24 33 52)(6 51 34 23)(7 22 35 50)(8 49 36 21)(9 20 37 48)(10 47 38 19)(11 18 39 46)(12 45 40 17)(13 16 41 44)(14 43 42 15)(57 70 85 98)(58 97 86 69)(59 68 87 96)(60 95 88 67)(61 66 89 94)(62 93 90 65)(63 64 91 92)(71 112 99 84)(72 83 100 111)(73 110 101 82)(74 81 102 109)(75 108 103 80)(76 79 104 107)(77 106 105 78)
G:=sub<Sym(112)| (1,92,29,64)(2,93,30,65)(3,94,31,66)(4,95,32,67)(5,96,33,68)(6,97,34,69)(7,98,35,70)(8,99,36,71)(9,100,37,72)(10,101,38,73)(11,102,39,74)(12,103,40,75)(13,104,41,76)(14,105,42,77)(15,106,43,78)(16,107,44,79)(17,108,45,80)(18,109,46,81)(19,110,47,82)(20,111,48,83)(21,112,49,84)(22,57,50,85)(23,58,51,86)(24,59,52,87)(25,60,53,88)(26,61,54,89)(27,62,55,90)(28,63,56,91), (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,28,29,56)(2,55,30,27)(3,26,31,54)(4,53,32,25)(5,24,33,52)(6,51,34,23)(7,22,35,50)(8,49,36,21)(9,20,37,48)(10,47,38,19)(11,18,39,46)(12,45,40,17)(13,16,41,44)(14,43,42,15)(57,70,85,98)(58,97,86,69)(59,68,87,96)(60,95,88,67)(61,66,89,94)(62,93,90,65)(63,64,91,92)(71,112,99,84)(72,83,100,111)(73,110,101,82)(74,81,102,109)(75,108,103,80)(76,79,104,107)(77,106,105,78)>;
G:=Group( (1,92,29,64)(2,93,30,65)(3,94,31,66)(4,95,32,67)(5,96,33,68)(6,97,34,69)(7,98,35,70)(8,99,36,71)(9,100,37,72)(10,101,38,73)(11,102,39,74)(12,103,40,75)(13,104,41,76)(14,105,42,77)(15,106,43,78)(16,107,44,79)(17,108,45,80)(18,109,46,81)(19,110,47,82)(20,111,48,83)(21,112,49,84)(22,57,50,85)(23,58,51,86)(24,59,52,87)(25,60,53,88)(26,61,54,89)(27,62,55,90)(28,63,56,91), (57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (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), (1,28,29,56)(2,55,30,27)(3,26,31,54)(4,53,32,25)(5,24,33,52)(6,51,34,23)(7,22,35,50)(8,49,36,21)(9,20,37,48)(10,47,38,19)(11,18,39,46)(12,45,40,17)(13,16,41,44)(14,43,42,15)(57,70,85,98)(58,97,86,69)(59,68,87,96)(60,95,88,67)(61,66,89,94)(62,93,90,65)(63,64,91,92)(71,112,99,84)(72,83,100,111)(73,110,101,82)(74,81,102,109)(75,108,103,80)(76,79,104,107)(77,106,105,78) );
G=PermutationGroup([[(1,92,29,64),(2,93,30,65),(3,94,31,66),(4,95,32,67),(5,96,33,68),(6,97,34,69),(7,98,35,70),(8,99,36,71),(9,100,37,72),(10,101,38,73),(11,102,39,74),(12,103,40,75),(13,104,41,76),(14,105,42,77),(15,106,43,78),(16,107,44,79),(17,108,45,80),(18,109,46,81),(19,110,47,82),(20,111,48,83),(21,112,49,84),(22,57,50,85),(23,58,51,86),(24,59,52,87),(25,60,53,88),(26,61,54,89),(27,62,55,90),(28,63,56,91)], [(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(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)], [(1,28,29,56),(2,55,30,27),(3,26,31,54),(4,53,32,25),(5,24,33,52),(6,51,34,23),(7,22,35,50),(8,49,36,21),(9,20,37,48),(10,47,38,19),(11,18,39,46),(12,45,40,17),(13,16,41,44),(14,43,42,15),(57,70,85,98),(58,97,86,69),(59,68,87,96),(60,95,88,67),(61,66,89,94),(62,93,90,65),(63,64,91,92),(71,112,99,84),(72,83,100,111),(73,110,101,82),(74,81,102,109),(75,108,103,80),(76,79,104,107),(77,106,105,78)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | ··· | 14L | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56L | 56M | ··· | 56AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D28 | D28 | D4○SD16 | D4.11D28 |
| kernel | D4.11D28 | C2×C56⋊C2 | D56⋊7C2 | C8⋊D14 | C8.D14 | C7×C8○D4 | D4⋊8D14 | D4.10D14 | C7×D4 | C7×Q8 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 3 | 1 | 3 | 9 | 9 | 3 | 18 | 6 | 2 | 12 |
Matrix representation of D4.11D28 ►in GL4(𝔽113) generated by
| 112 | 0 | 2 | 0 |
| 0 | 112 | 0 | 2 |
| 112 | 0 | 1 | 0 |
| 0 | 112 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 112 | 0 |
| 0 | 1 | 0 | 112 |
| 7 | 47 | 0 | 0 |
| 66 | 104 | 0 | 0 |
| 0 | 0 | 7 | 47 |
| 0 | 0 | 66 | 104 |
| 7 | 47 | 0 | 0 |
| 35 | 106 | 0 | 0 |
| 0 | 0 | 7 | 47 |
| 0 | 0 | 35 | 106 |
G:=sub<GL(4,GF(113))| [112,0,112,0,0,112,0,112,2,0,1,0,0,2,0,1],[1,0,1,0,0,1,0,1,0,0,112,0,0,0,0,112],[7,66,0,0,47,104,0,0,0,0,7,66,0,0,47,104],[7,35,0,0,47,106,0,0,0,0,7,35,0,0,47,106] >;
D4.11D28 in GAP, Magma, Sage, TeX
D_4._{11}D_{28} % in TeX
G:=Group("D4.11D28"); // GroupNames label
G:=SmallGroup(448,1204);
// by ID
G=gap.SmallGroup(448,1204);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,387,675,80,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^28=d^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^27>;
// generators/relations