metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊6D4, D14⋊5D8, (C7×D4)⋊5D4, (C2×D8)⋊4D7, (C2×C8)⋊3D14, (C2×D4)⋊3D14, C4.59(D4×D7), C2.28(D7×D8), D14⋊C8⋊27C2, (C14×D8)⋊13C2, C28⋊2D4⋊3C2, C7⋊4(C22⋊D8), D4⋊2(C7⋊D4), C28.46(C2×D4), C14.45(C2×D8), (C2×C56)⋊27C22, (D4×C14)⋊3C22, C2.D56⋊29C2, C14.55C22≀C2, D4⋊Dic7⋊28C2, C4⋊Dic7⋊19C22, (C2×Dic7).65D4, (C22×D7).90D4, C22.256(D4×D7), C2.29(D8⋊D7), C14.50(C8⋊C22), (C2×C28).433C23, C2.23(C23⋊D14), (C2×D28).116C22, (C2×D4×D7)⋊2C2, (C2×C7⋊C8)⋊7C22, (C2×D4⋊D7)⋊19C2, C4.36(C2×C7⋊D4), (C2×C4×D7).44C22, (C2×C14).346(C2×D4), (C2×C4).523(C22×D7), SmallGroup(448,690)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=cac-1=a-1, dad=a13, cbc-1=a19b, dbd=a12b, dcd=c-1 >
Subgroups: 1380 in 198 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C2×D8, C22×D4, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C7×D4, C22×D7, C22×D7, C22×C14, C22⋊D8, C2×C7⋊C8, C4⋊Dic7, D4⋊D7, C23.D7, C2×C56, C7×D8, C2×C4×D7, C2×D28, D4×D7, C2×C7⋊D4, D4×C14, C23×D7, D14⋊C8, C2.D56, D4⋊Dic7, C2×D4⋊D7, C28⋊2D4, C14×D8, C2×D4×D7, D28⋊D4
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C22≀C2, C2×D8, C8⋊C22, C7⋊D4, C22×D7, C22⋊D8, D4×D7, C2×C7⋊D4, D7×D8, D8⋊D7, C23⋊D14, D28⋊D4
►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)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(52 56)(53 55)(57 62)(58 61)(59 60)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)(107 112)(108 111)(109 110)
(1 96 44 74)(2 95 45 73)(3 94 46 72)(4 93 47 71)(5 92 48 70)(6 91 49 69)(7 90 50 68)(8 89 51 67)(9 88 52 66)(10 87 53 65)(11 86 54 64)(12 85 55 63)(13 112 56 62)(14 111 29 61)(15 110 30 60)(16 109 31 59)(17 108 32 58)(18 107 33 57)(19 106 34 84)(20 105 35 83)(21 104 36 82)(22 103 37 81)(23 102 38 80)(24 101 39 79)(25 100 40 78)(26 99 41 77)(27 98 42 76)(28 97 43 75)
(1 15)(2 28)(3 13)(4 26)(5 11)(6 24)(7 9)(8 22)(10 20)(12 18)(14 16)(17 27)(19 25)(21 23)(29 31)(30 44)(32 42)(33 55)(34 40)(35 53)(36 38)(37 51)(39 49)(41 47)(43 45)(46 56)(48 54)(50 52)(57 85)(58 98)(59 111)(60 96)(61 109)(62 94)(63 107)(64 92)(65 105)(66 90)(67 103)(68 88)(69 101)(70 86)(71 99)(72 112)(73 97)(74 110)(75 95)(76 108)(77 93)(78 106)(79 91)(80 104)(81 89)(82 102)(83 87)(84 100)
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)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(52,56)(53,55)(57,62)(58,61)(59,60)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(107,112)(108,111)(109,110), (1,96,44,74)(2,95,45,73)(3,94,46,72)(4,93,47,71)(5,92,48,70)(6,91,49,69)(7,90,50,68)(8,89,51,67)(9,88,52,66)(10,87,53,65)(11,86,54,64)(12,85,55,63)(13,112,56,62)(14,111,29,61)(15,110,30,60)(16,109,31,59)(17,108,32,58)(18,107,33,57)(19,106,34,84)(20,105,35,83)(21,104,36,82)(22,103,37,81)(23,102,38,80)(24,101,39,79)(25,100,40,78)(26,99,41,77)(27,98,42,76)(28,97,43,75), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,31)(30,44)(32,42)(33,55)(34,40)(35,53)(36,38)(37,51)(39,49)(41,47)(43,45)(46,56)(48,54)(50,52)(57,85)(58,98)(59,111)(60,96)(61,109)(62,94)(63,107)(64,92)(65,105)(66,90)(67,103)(68,88)(69,101)(70,86)(71,99)(72,112)(73,97)(74,110)(75,95)(76,108)(77,93)(78,106)(79,91)(80,104)(81,89)(82,102)(83,87)(84,100)>;
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)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(52,56)(53,55)(57,62)(58,61)(59,60)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(107,112)(108,111)(109,110), (1,96,44,74)(2,95,45,73)(3,94,46,72)(4,93,47,71)(5,92,48,70)(6,91,49,69)(7,90,50,68)(8,89,51,67)(9,88,52,66)(10,87,53,65)(11,86,54,64)(12,85,55,63)(13,112,56,62)(14,111,29,61)(15,110,30,60)(16,109,31,59)(17,108,32,58)(18,107,33,57)(19,106,34,84)(20,105,35,83)(21,104,36,82)(22,103,37,81)(23,102,38,80)(24,101,39,79)(25,100,40,78)(26,99,41,77)(27,98,42,76)(28,97,43,75), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,31)(30,44)(32,42)(33,55)(34,40)(35,53)(36,38)(37,51)(39,49)(41,47)(43,45)(46,56)(48,54)(50,52)(57,85)(58,98)(59,111)(60,96)(61,109)(62,94)(63,107)(64,92)(65,105)(66,90)(67,103)(68,88)(69,101)(70,86)(71,99)(72,112)(73,97)(74,110)(75,95)(76,108)(77,93)(78,106)(79,91)(80,104)(81,89)(82,102)(83,87)(84,100) );
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),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(52,56),(53,55),(57,62),(58,61),(59,60),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,100),(92,99),(93,98),(94,97),(95,96),(107,112),(108,111),(109,110)], [(1,96,44,74),(2,95,45,73),(3,94,46,72),(4,93,47,71),(5,92,48,70),(6,91,49,69),(7,90,50,68),(8,89,51,67),(9,88,52,66),(10,87,53,65),(11,86,54,64),(12,85,55,63),(13,112,56,62),(14,111,29,61),(15,110,30,60),(16,109,31,59),(17,108,32,58),(18,107,33,57),(19,106,34,84),(20,105,35,83),(21,104,36,82),(22,103,37,81),(23,102,38,80),(24,101,39,79),(25,100,40,78),(26,99,41,77),(27,98,42,76),(28,97,43,75)], [(1,15),(2,28),(3,13),(4,26),(5,11),(6,24),(7,9),(8,22),(10,20),(12,18),(14,16),(17,27),(19,25),(21,23),(29,31),(30,44),(32,42),(33,55),(34,40),(35,53),(36,38),(37,51),(39,49),(41,47),(43,45),(46,56),(48,54),(50,52),(57,85),(58,98),(59,111),(60,96),(61,109),(62,94),(63,107),(64,92),(65,105),(66,90),(67,103),(68,88),(69,101),(70,86),(71,99),(72,112),(73,97),(74,110),(75,95),(76,108),(77,93),(78,106),(79,91),(80,104),(81,89),(82,102),(83,87),(84,100)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 14 | 14 | 28 | 28 | 2 | 2 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D8 | D14 | D14 | C7⋊D4 | C8⋊C22 | D4×D7 | D4×D7 | D7×D8 | D8⋊D7 |
| kernel | D28⋊D4 | D14⋊C8 | C2.D56 | D4⋊Dic7 | C2×D4⋊D7 | C28⋊2D4 | C14×D8 | C2×D4×D7 | D28 | C2×Dic7 | C7×D4 | C22×D7 | C2×D8 | D14 | 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 | 6 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of D28⋊D4 ►in GL4(𝔽113) generated by
| 1 | 9 | 0 | 0 |
| 104 | 33 | 0 | 0 |
| 0 | 0 | 112 | 111 |
| 0 | 0 | 1 | 1 |
| 112 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 112 | 112 |
| 60 | 28 | 0 | 0 |
| 53 | 53 | 0 | 0 |
| 0 | 0 | 51 | 51 |
| 0 | 0 | 31 | 62 |
| 80 | 9 | 0 | 0 |
| 80 | 33 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [1,104,0,0,9,33,0,0,0,0,112,1,0,0,111,1],[112,9,0,0,0,1,0,0,0,0,1,112,0,0,0,112],[60,53,0,0,28,53,0,0,0,0,51,31,0,0,51,62],[80,80,0,0,9,33,0,0,0,0,112,0,0,0,0,112] >;
D28⋊D4 in GAP, Magma, Sage, TeX
D_{28}\rtimes D_4 % in TeX
G:=Group("D28:D4"); // GroupNames label
G:=SmallGroup(448,690);
// by ID
G=gap.SmallGroup(448,690);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^13,c*b*c^-1=a^19*b,d*b*d=a^12*b,d*c*d=c^-1>;
// generators/relations