metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.9D28, Q8.9D28, D28.34D4, C42.25D14, Dic14.34D4, M4(2).7D14, C4≀C2⋊3D7, (C7×D4).4D4, C28.5(C2×D4), (C7×Q8).4D4, C4○D4.3D14, C4.127(D4×D7), C4.11(C2×D28), Dic14⋊C4⋊9C2, C28⋊2Q8⋊10C2, C8.D14⋊9C2, (C2×Dic7).2D4, C22.31(D4×D7), C14.29C22≀C2, D4.9D14⋊2C2, C4.12D28⋊1C2, C7⋊2(D4.10D4), (C4×C28).52C22, (C2×C28).266C23, C4○D28.15C22, C2.32(C22⋊D28), D4.10D14.1C2, (C7×M4(2)).4C22, C4.Dic7.10C22, (C2×Dic14).76C22, (C7×C4≀C2)⋊3C2, (C2×C14).28(C2×D4), (C7×C4○D4).7C22, (C2×C4).111(C22×D7), SmallGroup(448,360)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.9D28
G = < a,b,c,d | a4=b2=1, c28=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=c27 >
Subgroups: 764 in 142 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D7, C14, C14, C42, C4⋊C4, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.10D4, C4≀C2, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, D4.10D4, C56⋊C2, Dic28, C4.Dic7, C4⋊Dic7, D4.D7, C7⋊Q16, C4×C28, C7×M4(2), C2×Dic14, C2×Dic14, C4○D28, C4○D28, D4⋊2D7, Q8×D7, C7×C4○D4, Dic14⋊C4, C4.12D28, C7×C4≀C2, C28⋊2Q8, C8.D14, D4.9D14, D4.10D14, D4.9D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4.10D4, C2×D28, D4×D7, C22⋊D28, D4.9D28
►On 112 points
Generators in S112
(1 15 29 43)(2 44 30 16)(3 17 31 45)(4 46 32 18)(5 19 33 47)(6 48 34 20)(7 21 35 49)(8 50 36 22)(9 23 37 51)(10 52 38 24)(11 25 39 53)(12 54 40 26)(13 27 41 55)(14 56 42 28)(57 99 85 71)(58 72 86 100)(59 101 87 73)(60 74 88 102)(61 103 89 75)(62 76 90 104)(63 105 91 77)(64 78 92 106)(65 107 93 79)(66 80 94 108)(67 109 95 81)(68 82 96 110)(69 111 97 83)(70 84 98 112)
(1 36)(2 23)(3 38)(4 25)(5 40)(6 27)(7 42)(8 29)(9 44)(10 31)(11 46)(12 33)(13 48)(14 35)(15 50)(16 37)(17 52)(18 39)(19 54)(20 41)(21 56)(22 43)(24 45)(26 47)(28 49)(30 51)(32 53)(34 55)(57 78)(58 93)(59 80)(60 95)(61 82)(62 97)(63 84)(64 99)(65 86)(66 101)(67 88)(68 103)(69 90)(70 105)(71 92)(72 107)(73 94)(74 109)(75 96)(76 111)(77 98)(79 100)(81 102)(83 104)(85 106)(87 108)(89 110)(91 112)
(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 60 29 88)(2 87 30 59)(3 58 31 86)(4 85 32 57)(5 112 33 84)(6 83 34 111)(7 110 35 82)(8 81 36 109)(9 108 37 80)(10 79 38 107)(11 106 39 78)(12 77 40 105)(13 104 41 76)(14 75 42 103)(15 102 43 74)(16 73 44 101)(17 100 45 72)(18 71 46 99)(19 98 47 70)(20 69 48 97)(21 96 49 68)(22 67 50 95)(23 94 51 66)(24 65 52 93)(25 92 53 64)(26 63 54 91)(27 90 55 62)(28 61 56 89)
G:=sub<Sym(112)| (1,15,29,43)(2,44,30,16)(3,17,31,45)(4,46,32,18)(5,19,33,47)(6,48,34,20)(7,21,35,49)(8,50,36,22)(9,23,37,51)(10,52,38,24)(11,25,39,53)(12,54,40,26)(13,27,41,55)(14,56,42,28)(57,99,85,71)(58,72,86,100)(59,101,87,73)(60,74,88,102)(61,103,89,75)(62,76,90,104)(63,105,91,77)(64,78,92,106)(65,107,93,79)(66,80,94,108)(67,109,95,81)(68,82,96,110)(69,111,97,83)(70,84,98,112), (1,36)(2,23)(3,38)(4,25)(5,40)(6,27)(7,42)(8,29)(9,44)(10,31)(11,46)(12,33)(13,48)(14,35)(15,50)(16,37)(17,52)(18,39)(19,54)(20,41)(21,56)(22,43)(24,45)(26,47)(28,49)(30,51)(32,53)(34,55)(57,78)(58,93)(59,80)(60,95)(61,82)(62,97)(63,84)(64,99)(65,86)(66,101)(67,88)(68,103)(69,90)(70,105)(71,92)(72,107)(73,94)(74,109)(75,96)(76,111)(77,98)(79,100)(81,102)(83,104)(85,106)(87,108)(89,110)(91,112), (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), (1,60,29,88)(2,87,30,59)(3,58,31,86)(4,85,32,57)(5,112,33,84)(6,83,34,111)(7,110,35,82)(8,81,36,109)(9,108,37,80)(10,79,38,107)(11,106,39,78)(12,77,40,105)(13,104,41,76)(14,75,42,103)(15,102,43,74)(16,73,44,101)(17,100,45,72)(18,71,46,99)(19,98,47,70)(20,69,48,97)(21,96,49,68)(22,67,50,95)(23,94,51,66)(24,65,52,93)(25,92,53,64)(26,63,54,91)(27,90,55,62)(28,61,56,89)>;
G:=Group( (1,15,29,43)(2,44,30,16)(3,17,31,45)(4,46,32,18)(5,19,33,47)(6,48,34,20)(7,21,35,49)(8,50,36,22)(9,23,37,51)(10,52,38,24)(11,25,39,53)(12,54,40,26)(13,27,41,55)(14,56,42,28)(57,99,85,71)(58,72,86,100)(59,101,87,73)(60,74,88,102)(61,103,89,75)(62,76,90,104)(63,105,91,77)(64,78,92,106)(65,107,93,79)(66,80,94,108)(67,109,95,81)(68,82,96,110)(69,111,97,83)(70,84,98,112), (1,36)(2,23)(3,38)(4,25)(5,40)(6,27)(7,42)(8,29)(9,44)(10,31)(11,46)(12,33)(13,48)(14,35)(15,50)(16,37)(17,52)(18,39)(19,54)(20,41)(21,56)(22,43)(24,45)(26,47)(28,49)(30,51)(32,53)(34,55)(57,78)(58,93)(59,80)(60,95)(61,82)(62,97)(63,84)(64,99)(65,86)(66,101)(67,88)(68,103)(69,90)(70,105)(71,92)(72,107)(73,94)(74,109)(75,96)(76,111)(77,98)(79,100)(81,102)(83,104)(85,106)(87,108)(89,110)(91,112), (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), (1,60,29,88)(2,87,30,59)(3,58,31,86)(4,85,32,57)(5,112,33,84)(6,83,34,111)(7,110,35,82)(8,81,36,109)(9,108,37,80)(10,79,38,107)(11,106,39,78)(12,77,40,105)(13,104,41,76)(14,75,42,103)(15,102,43,74)(16,73,44,101)(17,100,45,72)(18,71,46,99)(19,98,47,70)(20,69,48,97)(21,96,49,68)(22,67,50,95)(23,94,51,66)(24,65,52,93)(25,92,53,64)(26,63,54,91)(27,90,55,62)(28,61,56,89) );
G=PermutationGroup([[(1,15,29,43),(2,44,30,16),(3,17,31,45),(4,46,32,18),(5,19,33,47),(6,48,34,20),(7,21,35,49),(8,50,36,22),(9,23,37,51),(10,52,38,24),(11,25,39,53),(12,54,40,26),(13,27,41,55),(14,56,42,28),(57,99,85,71),(58,72,86,100),(59,101,87,73),(60,74,88,102),(61,103,89,75),(62,76,90,104),(63,105,91,77),(64,78,92,106),(65,107,93,79),(66,80,94,108),(67,109,95,81),(68,82,96,110),(69,111,97,83),(70,84,98,112)], [(1,36),(2,23),(3,38),(4,25),(5,40),(6,27),(7,42),(8,29),(9,44),(10,31),(11,46),(12,33),(13,48),(14,35),(15,50),(16,37),(17,52),(18,39),(19,54),(20,41),(21,56),(22,43),(24,45),(26,47),(28,49),(30,51),(32,53),(34,55),(57,78),(58,93),(59,80),(60,95),(61,82),(62,97),(63,84),(64,99),(65,86),(66,101),(67,88),(68,103),(69,90),(70,105),(71,92),(72,107),(73,94),(74,109),(75,96),(76,111),(77,98),(79,100),(81,102),(83,104),(85,106),(87,108),(89,110),(91,112)], [(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)], [(1,60,29,88),(2,87,30,59),(3,58,31,86),(4,85,32,57),(5,112,33,84),(6,83,34,111),(7,110,35,82),(8,81,36,109),(9,108,37,80),(10,79,38,107),(11,106,39,78),(12,77,40,105),(13,104,41,76),(14,75,42,103),(15,102,43,74),(16,73,44,101),(17,100,45,72),(18,71,46,99),(19,98,47,70),(20,69,48,97),(21,96,49,68),(22,67,50,95),(23,94,51,66),(24,65,52,93),(25,92,53,64),(26,63,54,91),(27,90,55,62),(28,61,56,89)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28U | 28V | 28W | 28X | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 28 | 2 | 2 | 4 | 4 | 4 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 8 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D28 | D28 | D4.10D4 | D4×D7 | D4×D7 | D4.9D28 |
| kernel | D4.9D28 | Dic14⋊C4 | C4.12D28 | C7×C4≀C2 | C28⋊2Q8 | C8.D14 | D4.9D14 | D4.10D14 | Dic14 | D28 | C2×Dic7 | C7×D4 | C7×Q8 | C4≀C2 | C42 | M4(2) | C4○D4 | D4 | Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 2 | 3 | 3 | 12 |
Matrix representation of D4.9D28 ►in GL4(𝔽113) generated by
| 9 | 46 | 0 | 0 |
| 67 | 104 | 0 | 0 |
| 0 | 0 | 104 | 67 |
| 0 | 0 | 46 | 9 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 34 |
| 0 | 0 | 79 | 25 |
| 104 | 13 | 0 | 0 |
| 100 | 94 | 0 | 0 |
| 10 | 3 | 0 | 0 |
| 4 | 103 | 0 | 0 |
| 0 | 0 | 48 | 19 |
| 0 | 0 | 69 | 65 |
G:=sub<GL(4,GF(113))| [9,67,0,0,46,104,0,0,0,0,104,46,0,0,67,9],[0,0,112,0,0,0,0,112,112,0,0,0,0,112,0,0],[0,0,104,100,0,0,13,94,112,79,0,0,34,25,0,0],[10,4,0,0,3,103,0,0,0,0,48,69,0,0,19,65] >;
D4.9D28 in GAP, Magma, Sage, TeX
D_4._9D_{28} % in TeX
G:=Group("D4.9D28"); // GroupNames label
G:=SmallGroup(448,360);
// by ID
G=gap.SmallGroup(448,360);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,58,1123,136,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^28=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=c^27>;
// generators/relations