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, C4.58(C23×D7), C8.55(C22×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
Subgroups: 1436 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C7, C8, C8 [×3], C2×C4 [×3], C2×C4 [×9], D4 [×3], D4 [×13], Q8, Q8 [×7], C23 [×3], D7 [×4], C14, C14 [×3], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×4], C4○D4, C4○D4 [×10], Dic7 [×4], C28, C28 [×3], D14 [×7], C2×C14 [×3], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), C56, C56 [×3], Dic14, Dic14 [×3], Dic14 [×3], C4×D7 [×6], D28, D28 [×3], D28 [×3], C2×Dic7 [×3], C7⋊D4 [×6], C2×C28 [×3], C7×D4 [×3], C7×Q8, C22×D7 [×3], D4○SD16, C56⋊C2, C56⋊C2 [×9], D56 [×3], Dic28 [×3], C2×C56 [×3], C7×M4(2) [×3], C2×Dic14 [×3], C2×D28 [×3], C4○D28 [×6], D4×D7 [×3], D4⋊2D7 [×3], Q8×D7, Q8⋊2D7, C7×C4○D4, C2×C56⋊C2 [×3], D56⋊7C2 [×3], C8⋊D14 [×3], C8.D14 [×3], C7×C8○D4, D4⋊8D14, D4.10D14, D4.11D28
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, D28 [×4], C22×D7 [×7], D4○SD16, C2×D28 [×6], C23×D7, C22×D28, D4.11D28
Generators and relations
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 >
►On 112 points
(1 57 29 85)(2 58 30 86)(3 59 31 87)(4 60 32 88)(5 61 33 89)(6 62 34 90)(7 63 35 91)(8 64 36 92)(9 65 37 93)(10 66 38 94)(11 67 39 95)(12 68 40 96)(13 69 41 97)(14 70 42 98)(15 71 43 99)(16 72 44 100)(17 73 45 101)(18 74 46 102)(19 75 47 103)(20 76 48 104)(21 77 49 105)(22 78 50 106)(23 79 51 107)(24 80 52 108)(25 81 53 109)(26 82 54 110)(27 83 55 111)(28 84 56 112)
(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 84 85 112)(58 111 86 83)(59 82 87 110)(60 109 88 81)(61 80 89 108)(62 107 90 79)(63 78 91 106)(64 105 92 77)(65 76 93 104)(66 103 94 75)(67 74 95 102)(68 101 96 73)(69 72 97 100)(70 99 98 71)
G:=sub<Sym(112)| (1,57,29,85)(2,58,30,86)(3,59,31,87)(4,60,32,88)(5,61,33,89)(6,62,34,90)(7,63,35,91)(8,64,36,92)(9,65,37,93)(10,66,38,94)(11,67,39,95)(12,68,40,96)(13,69,41,97)(14,70,42,98)(15,71,43,99)(16,72,44,100)(17,73,45,101)(18,74,46,102)(19,75,47,103)(20,76,48,104)(21,77,49,105)(22,78,50,106)(23,79,51,107)(24,80,52,108)(25,81,53,109)(26,82,54,110)(27,83,55,111)(28,84,56,112), (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)(57,84,85,112)(58,111,86,83)(59,82,87,110)(60,109,88,81)(61,80,89,108)(62,107,90,79)(63,78,91,106)(64,105,92,77)(65,76,93,104)(66,103,94,75)(67,74,95,102)(68,101,96,73)(69,72,97,100)(70,99,98,71)>;
G:=Group( (1,57,29,85)(2,58,30,86)(3,59,31,87)(4,60,32,88)(5,61,33,89)(6,62,34,90)(7,63,35,91)(8,64,36,92)(9,65,37,93)(10,66,38,94)(11,67,39,95)(12,68,40,96)(13,69,41,97)(14,70,42,98)(15,71,43,99)(16,72,44,100)(17,73,45,101)(18,74,46,102)(19,75,47,103)(20,76,48,104)(21,77,49,105)(22,78,50,106)(23,79,51,107)(24,80,52,108)(25,81,53,109)(26,82,54,110)(27,83,55,111)(28,84,56,112), (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)(57,84,85,112)(58,111,86,83)(59,82,87,110)(60,109,88,81)(61,80,89,108)(62,107,90,79)(63,78,91,106)(64,105,92,77)(65,76,93,104)(66,103,94,75)(67,74,95,102)(68,101,96,73)(69,72,97,100)(70,99,98,71) );
G=PermutationGroup([(1,57,29,85),(2,58,30,86),(3,59,31,87),(4,60,32,88),(5,61,33,89),(6,62,34,90),(7,63,35,91),(8,64,36,92),(9,65,37,93),(10,66,38,94),(11,67,39,95),(12,68,40,96),(13,69,41,97),(14,70,42,98),(15,71,43,99),(16,72,44,100),(17,73,45,101),(18,74,46,102),(19,75,47,103),(20,76,48,104),(21,77,49,105),(22,78,50,106),(23,79,51,107),(24,80,52,108),(25,81,53,109),(26,82,54,110),(27,83,55,111),(28,84,56,112)], [(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),(57,84,85,112),(58,111,86,83),(59,82,87,110),(60,109,88,81),(61,80,89,108),(62,107,90,79),(63,78,91,106),(64,105,92,77),(65,76,93,104),(66,103,94,75),(67,74,95,102),(68,101,96,73),(69,72,97,100),(70,99,98,71)])
Matrix representation ►G ⊆ 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] >;
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 |
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