metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊16D4, C4⋊C4⋊3D14, (C2×C14)⋊2D8, (C2×D4)⋊1D14, C4⋊D4⋊1D7, C4.98(D4×D7), C7⋊3(C22⋊D8), C14.54(C2×D8), (C2×C28).71D4, C22⋊2(D4⋊D7), C28.145(C2×D4), (D4×C14)⋊1C22, C14.D8⋊33C2, C14.44C22≀C2, (C22×D28)⋊13C2, (C22×C14).82D4, C28.55D4⋊10C2, (C2×C28).355C23, (C22×C4).119D14, C23.58(C7⋊D4), C2.12(D4⋊D14), C2.12(C23⋊D14), C14.114(C8⋊C22), (C2×D28).240C22, (C22×C28).159C22, (C2×D4⋊D7)⋊8C2, (C2×C7⋊C8)⋊5C22, C2.9(C2×D4⋊D7), (C7×C4⋊D4)⋊1C2, (C7×C4⋊C4)⋊5C22, (C2×C14).486(C2×D4), (C2×C4).49(C7⋊D4), (C2×C4).455(C22×D7), C22.161(C2×C7⋊D4), SmallGroup(448,570)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊16D4
G = < a,b,c,d | a28=b2=c4=d2=1, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, bd=db, dcd=c-1 >
Subgroups: 1356 in 198 conjugacy classes, 47 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, C7⋊C8, D28, D28, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22×C14, C22⋊D8, C2×C7⋊C8, D4⋊D7, C7×C22⋊C4, C7×C4⋊C4, C2×D28, C2×D28, C22×C28, D4×C14, D4×C14, C23×D7, C14.D8, C28.55D4, C2×D4⋊D7, C7×C4⋊D4, C22×D28, D28⋊16D4
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, D4×D7, C2×C7⋊D4, C2×D4⋊D7, C23⋊D14, D4⋊D14, D28⋊16D4
►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)
(1 103)(2 102)(3 101)(4 100)(5 99)(6 98)(7 97)(8 96)(9 95)(10 94)(11 93)(12 92)(13 91)(14 90)(15 89)(16 88)(17 87)(18 86)(19 85)(20 112)(21 111)(22 110)(23 109)(24 108)(25 107)(26 106)(27 105)(28 104)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)
(1 30 104 71)(2 45 105 58)(3 32 106 73)(4 47 107 60)(5 34 108 75)(6 49 109 62)(7 36 110 77)(8 51 111 64)(9 38 112 79)(10 53 85 66)(11 40 86 81)(12 55 87 68)(13 42 88 83)(14 29 89 70)(15 44 90 57)(16 31 91 72)(17 46 92 59)(18 33 93 74)(19 48 94 61)(20 35 95 76)(21 50 96 63)(22 37 97 78)(23 52 98 65)(24 39 99 80)(25 54 100 67)(26 41 101 82)(27 56 102 69)(28 43 103 84)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 84)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
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), (1,103)(2,102)(3,101)(4,100)(5,99)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,89)(16,88)(17,87)(18,86)(19,85)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65), (1,30,104,71)(2,45,105,58)(3,32,106,73)(4,47,107,60)(5,34,108,75)(6,49,109,62)(7,36,110,77)(8,51,111,64)(9,38,112,79)(10,53,85,66)(11,40,86,81)(12,55,87,68)(13,42,88,83)(14,29,89,70)(15,44,90,57)(16,31,91,72)(17,46,92,59)(18,33,93,74)(19,48,94,61)(20,35,95,76)(21,50,96,63)(22,37,97,78)(23,52,98,65)(24,39,99,80)(25,54,100,67)(26,41,101,82)(27,56,102,69)(28,43,103,84), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,84)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)>;
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), (1,103)(2,102)(3,101)(4,100)(5,99)(6,98)(7,97)(8,96)(9,95)(10,94)(11,93)(12,92)(13,91)(14,90)(15,89)(16,88)(17,87)(18,86)(19,85)(20,112)(21,111)(22,110)(23,109)(24,108)(25,107)(26,106)(27,105)(28,104)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65), (1,30,104,71)(2,45,105,58)(3,32,106,73)(4,47,107,60)(5,34,108,75)(6,49,109,62)(7,36,110,77)(8,51,111,64)(9,38,112,79)(10,53,85,66)(11,40,86,81)(12,55,87,68)(13,42,88,83)(14,29,89,70)(15,44,90,57)(16,31,91,72)(17,46,92,59)(18,33,93,74)(19,48,94,61)(20,35,95,76)(21,50,96,63)(22,37,97,78)(23,52,98,65)(24,39,99,80)(25,54,100,67)(26,41,101,82)(27,56,102,69)(28,43,103,84), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,84)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81)(55,82)(56,83)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112) );
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)], [(1,103),(2,102),(3,101),(4,100),(5,99),(6,98),(7,97),(8,96),(9,95),(10,94),(11,93),(12,92),(13,91),(14,90),(15,89),(16,88),(17,87),(18,86),(19,85),(20,112),(21,111),(22,110),(23,109),(24,108),(25,107),(26,106),(27,105),(28,104),(29,64),(30,63),(31,62),(32,61),(33,60),(34,59),(35,58),(36,57),(37,84),(38,83),(39,82),(40,81),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65)], [(1,30,104,71),(2,45,105,58),(3,32,106,73),(4,47,107,60),(5,34,108,75),(6,49,109,62),(7,36,110,77),(8,51,111,64),(9,38,112,79),(10,53,85,66),(11,40,86,81),(12,55,87,68),(13,42,88,83),(14,29,89,70),(15,44,90,57),(16,31,91,72),(17,46,92,59),(18,33,93,74),(19,48,94,61),(20,35,95,76),(21,50,96,63),(22,37,97,78),(23,52,98,65),(24,39,99,80),(25,54,100,67),(26,41,101,82),(27,56,102,69),(28,43,103,84)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,84),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77),(51,78),(52,79),(53,80),(54,81),(55,82),(56,83),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)]])
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 | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| 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 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 28 | 28 | 28 | 28 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D8 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8⋊C22 | D4×D7 | D4⋊D7 | D4⋊D14 |
| kernel | D28⋊16D4 | C14.D8 | C28.55D4 | C2×D4⋊D7 | C7×C4⋊D4 | C22×D28 | D28 | C2×C28 | C22×C14 | C4⋊D4 | C2×C14 | C4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C23 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 3 | 4 | 3 | 3 | 3 | 6 | 6 | 1 | 6 | 6 | 6 |
Matrix representation of D28⋊16D4 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 34 | 0 | 0 |
| 0 | 0 | 103 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 69 |
| 0 | 0 | 0 | 0 | 48 | 90 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 34 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 37 | 112 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 105 |
| 0 | 0 | 0 | 0 | 16 | 109 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,24,103,0,0,0,0,34,0,0,0,0,0,0,0,23,48,0,0,0,0,69,90],[1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,10,0,0,0,0,34,0,0,0,0,0,0,0,1,37,0,0,0,0,0,112],[0,1,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,16,0,0,0,0,105,109],[1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;
D28⋊16D4 in GAP, Magma, Sage, TeX
D_{28}\rtimes_{16}D_4 % in TeX
G:=Group("D28:16D4"); // GroupNames label
G:=SmallGroup(448,570);
// by ID
G=gap.SmallGroup(448,570);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations