metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.29D4, C28.9C24, SD16⋊13D14, D56⋊21C22, C56.36C23, D28.5C23, Dic14.29D4, Dic28⋊18C22, Dic14.5C23, (C2×C8)⋊11D14, C4.76(D4×D7), C7⋊D4.9D4, C7⋊C8.3C23, D56⋊C2⋊5C2, (C2×C56)⋊6C22, D4⋊D7⋊2C22, (C2×Q8)⋊11D14, (D7×SD16)⋊5C2, (C2×SD16)⋊6D7, C7⋊2(D4○SD16), C28.84(C2×D4), (C8×D7)⋊9C22, Q8⋊D7⋊1C22, D56⋊7C2⋊8C2, D4⋊6D14⋊6C2, (Q8×D7)⋊1C22, C4.9(C23×D7), (C14×SD16)⋊2C2, D14.27(C2×D4), SD16⋊D7⋊5C2, C4○D28⋊4C22, D4.D7⋊2C22, (D4×D7).1C22, C7⋊Q16⋊1C22, D4.7(C22×D7), (C7×D4).7C23, (C4×D7).5C23, C22.21(D4×D7), C8.12(C22×D7), SD16⋊3D7⋊5C2, (C2×D4).116D14, D28.2C4⋊5C2, D4.D14⋊8C2, (C7×Q8).3C23, Q8.3(C22×D7), C28.C23⋊7C2, C56⋊C2⋊19C22, C8⋊D7⋊10C22, Dic7.32(C2×D4), Q8⋊2D7⋊1C22, (Q8×C14)⋊19C22, (C2×C28).526C23, Q8.10D14⋊3C2, (C7×SD16)⋊14C22, D4⋊2D7.1C22, C14.110(C22×D4), C4.Dic7⋊29C22, (D4×C14).167C22, C2.83(C2×D4×D7), (C2×C14).399(C2×D4), (C2×C4).230(C22×D7), SmallGroup(448,1215)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.29D4
G = < a,b,c,d | a28=b2=d2=1, c4=a14, bab=a-1, ac=ca, dad=a15, bc=cb, dbd=a14b, dcd=c3 >
Subgroups: 1316 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8○D4, C2×SD16, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, C22×C14, D4○SD16, C8×D7, C8⋊D7, C56⋊C2, D56, Dic28, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×C56, C7×SD16, C4○D28, C4○D28, D4×D7, D4×D7, D4⋊2D7, D4⋊2D7, Q8×D7, Q8×D7, Q8⋊2D7, Q8⋊2D7, C2×C7⋊D4, D4×C14, Q8×C14, D28.2C4, D56⋊7C2, D7×SD16, D56⋊C2, SD16⋊D7, SD16⋊3D7, D4.D14, C28.C23, C14×SD16, D4⋊6D14, Q8.10D14, D28.29D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, D4○SD16, D4×D7, C23×D7, C2×D4×D7, D28.29D4
►On 112 points
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 33)(30 32)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(57 61)(58 60)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(86 112)(87 111)(88 110)(89 109)(90 108)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)
(1 42 110 70 15 56 96 84)(2 43 111 71 16 29 97 57)(3 44 112 72 17 30 98 58)(4 45 85 73 18 31 99 59)(5 46 86 74 19 32 100 60)(6 47 87 75 20 33 101 61)(7 48 88 76 21 34 102 62)(8 49 89 77 22 35 103 63)(9 50 90 78 23 36 104 64)(10 51 91 79 24 37 105 65)(11 52 92 80 25 38 106 66)(12 53 93 81 26 39 107 67)(13 54 94 82 27 40 108 68)(14 55 95 83 28 41 109 69)
(1 8)(2 23)(3 10)(4 25)(5 12)(6 27)(7 14)(9 16)(11 18)(13 20)(15 22)(17 24)(19 26)(21 28)(29 78)(30 65)(31 80)(32 67)(33 82)(34 69)(35 84)(36 71)(37 58)(38 73)(39 60)(40 75)(41 62)(42 77)(43 64)(44 79)(45 66)(46 81)(47 68)(48 83)(49 70)(50 57)(51 72)(52 59)(53 74)(54 61)(55 76)(56 63)(85 92)(86 107)(87 94)(88 109)(89 96)(90 111)(91 98)(93 100)(95 102)(97 104)(99 106)(101 108)(103 110)(105 112)
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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,33)(30,32)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(57,61)(58,60)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100), (1,42,110,70,15,56,96,84)(2,43,111,71,16,29,97,57)(3,44,112,72,17,30,98,58)(4,45,85,73,18,31,99,59)(5,46,86,74,19,32,100,60)(6,47,87,75,20,33,101,61)(7,48,88,76,21,34,102,62)(8,49,89,77,22,35,103,63)(9,50,90,78,23,36,104,64)(10,51,91,79,24,37,105,65)(11,52,92,80,25,38,106,66)(12,53,93,81,26,39,107,67)(13,54,94,82,27,40,108,68)(14,55,95,83,28,41,109,69), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(29,78)(30,65)(31,80)(32,67)(33,82)(34,69)(35,84)(36,71)(37,58)(38,73)(39,60)(40,75)(41,62)(42,77)(43,64)(44,79)(45,66)(46,81)(47,68)(48,83)(49,70)(50,57)(51,72)(52,59)(53,74)(54,61)(55,76)(56,63)(85,92)(86,107)(87,94)(88,109)(89,96)(90,111)(91,98)(93,100)(95,102)(97,104)(99,106)(101,108)(103,110)(105,112)>;
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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,33)(30,32)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(57,61)(58,60)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,112)(87,111)(88,110)(89,109)(90,108)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100), (1,42,110,70,15,56,96,84)(2,43,111,71,16,29,97,57)(3,44,112,72,17,30,98,58)(4,45,85,73,18,31,99,59)(5,46,86,74,19,32,100,60)(6,47,87,75,20,33,101,61)(7,48,88,76,21,34,102,62)(8,49,89,77,22,35,103,63)(9,50,90,78,23,36,104,64)(10,51,91,79,24,37,105,65)(11,52,92,80,25,38,106,66)(12,53,93,81,26,39,107,67)(13,54,94,82,27,40,108,68)(14,55,95,83,28,41,109,69), (1,8)(2,23)(3,10)(4,25)(5,12)(6,27)(7,14)(9,16)(11,18)(13,20)(15,22)(17,24)(19,26)(21,28)(29,78)(30,65)(31,80)(32,67)(33,82)(34,69)(35,84)(36,71)(37,58)(38,73)(39,60)(40,75)(41,62)(42,77)(43,64)(44,79)(45,66)(46,81)(47,68)(48,83)(49,70)(50,57)(51,72)(52,59)(53,74)(54,61)(55,76)(56,63)(85,92)(86,107)(87,94)(88,109)(89,96)(90,111)(91,98)(93,100)(95,102)(97,104)(99,106)(101,108)(103,110)(105,112) );
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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,33),(30,32),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(57,61),(58,60),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(86,112),(87,111),(88,110),(89,109),(90,108),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100)], [(1,42,110,70,15,56,96,84),(2,43,111,71,16,29,97,57),(3,44,112,72,17,30,98,58),(4,45,85,73,18,31,99,59),(5,46,86,74,19,32,100,60),(6,47,87,75,20,33,101,61),(7,48,88,76,21,34,102,62),(8,49,89,77,22,35,103,63),(9,50,90,78,23,36,104,64),(10,51,91,79,24,37,105,65),(11,52,92,80,25,38,106,66),(12,53,93,81,26,39,107,67),(13,54,94,82,27,40,108,68),(14,55,95,83,28,41,109,69)], [(1,8),(2,23),(3,10),(4,25),(5,12),(6,27),(7,14),(9,16),(11,18),(13,20),(15,22),(17,24),(19,26),(21,28),(29,78),(30,65),(31,80),(32,67),(33,82),(34,69),(35,84),(36,71),(37,58),(38,73),(39,60),(40,75),(41,62),(42,77),(43,64),(44,79),(45,66),(46,81),(47,68),(48,83),(49,70),(50,57),(51,72),(52,59),(53,74),(54,61),(55,76),(56,63),(85,92),(86,107),(87,94),(88,109),(89,96),(90,111),(91,98),(93,100),(95,102),(97,104),(99,106),(101,108),(103,110),(105,112)]])
64 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 | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| 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 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D4○SD16 | D4×D7 | D4×D7 | D28.29D4 |
| kernel | D28.29D4 | D28.2C4 | D56⋊7C2 | D7×SD16 | D56⋊C2 | SD16⋊D7 | SD16⋊3D7 | D4.D14 | C28.C23 | C14×SD16 | D4⋊6D14 | Q8.10D14 | Dic14 | D28 | C7⋊D4 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 12 | 3 | 3 | 2 | 3 | 3 | 12 |
Matrix representation of D28.29D4 ►in GL4(𝔽113) generated by
| 0 | 0 | 54 | 82 |
| 0 | 0 | 31 | 91 |
| 59 | 31 | 0 | 0 |
| 82 | 22 | 0 | 0 |
| 88 | 79 | 0 | 0 |
| 25 | 25 | 0 | 0 |
| 0 | 0 | 88 | 79 |
| 0 | 0 | 25 | 25 |
| 100 | 0 | 13 | 0 |
| 0 | 100 | 0 | 13 |
| 100 | 0 | 100 | 0 |
| 0 | 100 | 0 | 100 |
| 0 | 0 | 22 | 101 |
| 0 | 0 | 12 | 91 |
| 22 | 101 | 0 | 0 |
| 12 | 91 | 0 | 0 |
G:=sub<GL(4,GF(113))| [0,0,59,82,0,0,31,22,54,31,0,0,82,91,0,0],[88,25,0,0,79,25,0,0,0,0,88,25,0,0,79,25],[100,0,100,0,0,100,0,100,13,0,100,0,0,13,0,100],[0,0,22,12,0,0,101,91,22,12,0,0,101,91,0,0] >;
D28.29D4 in GAP, Magma, Sage, TeX
D_{28}._{29}D_4 % in TeX
G:=Group("D28.29D4"); // GroupNames label
G:=SmallGroup(448,1215);
// by ID
G=gap.SmallGroup(448,1215);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=d^2=1,c^4=a^14,b*a*b=a^-1,a*c=c*a,d*a*d=a^15,b*c=c*b,d*b*d=a^14*b,d*c*d=c^3>;
// generators/relations