metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.3D4, Dic14.3D4, M4(2).2D14, C7⋊C8.25D4, C8⋊D14⋊6C2, D28.C4⋊6C2, C28.94(C2×D4), C4.D4⋊4D7, C4.149(D4×D7), (C2×D4).16D14, C7⋊1(D4.4D4), (C2×C28).6C23, D4.D14⋊1C2, C28.53D4⋊2C2, C28.46D4⋊6C2, C4○D28.4C22, C14.10(C4⋊D4), (C2×D28).39C22, (D4×C14).16C22, C2.13(D14⋊D4), C4.Dic7.3C22, C22.14(C4○D28), (C7×M4(2)).11C22, (C2×D4⋊D7)⋊1C2, (C2×C7⋊C8).2C22, (C7×C4.D4)⋊2C2, (C2×C4).6(C22×D7), (C2×C14).31(C4○D4), SmallGroup(448,283)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.3D4
G = < a,b,c,d | a28=b2=1, c4=a14, d2=a7, bab=a-1, cac-1=a15, ad=da, cbc-1=a14b, dbd-1=a7b, dcd-1=a21c3 >
Subgroups: 652 in 108 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4.D4, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C7⋊C8, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, D4.4D4, C8×D7, C8⋊D7, C56⋊C2, D56, C2×C7⋊C8, C4.Dic7, D4⋊D7, D4.D7, C7×M4(2), C2×D28, C4○D28, D4×C14, C28.53D4, C28.46D4, C7×C4.D4, D28.C4, C8⋊D14, C2×D4⋊D7, D4.D14, D28.3D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C22×D7, D4.4D4, C4○D28, D4×D7, D14⋊D4, D28.3D4
►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 97)(2 96)(3 95)(4 94)(5 93)(6 92)(7 91)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 112)(15 111)(16 110)(17 109)(18 108)(19 107)(20 106)(21 105)(22 104)(23 103)(24 102)(25 101)(26 100)(27 99)(28 98)(29 84)(30 83)(31 82)(32 81)(33 80)(34 79)(35 78)(36 77)(37 76)(38 75)(39 74)(40 73)(41 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)
(1 75 22 82 15 61 8 68)(2 62 23 69 16 76 9 83)(3 77 24 84 17 63 10 70)(4 64 25 71 18 78 11 57)(5 79 26 58 19 65 12 72)(6 66 27 73 20 80 13 59)(7 81 28 60 21 67 14 74)(29 95 50 102 43 109 36 88)(30 110 51 89 44 96 37 103)(31 97 52 104 45 111 38 90)(32 112 53 91 46 98 39 105)(33 99 54 106 47 85 40 92)(34 86 55 93 48 100 41 107)(35 101 56 108 49 87 42 94)
(1 105 8 112 15 91 22 98)(2 106 9 85 16 92 23 99)(3 107 10 86 17 93 24 100)(4 108 11 87 18 94 25 101)(5 109 12 88 19 95 26 102)(6 110 13 89 20 96 27 103)(7 111 14 90 21 97 28 104)(29 65 36 72 43 79 50 58)(30 66 37 73 44 80 51 59)(31 67 38 74 45 81 52 60)(32 68 39 75 46 82 53 61)(33 69 40 76 47 83 54 62)(34 70 41 77 48 84 55 63)(35 71 42 78 49 57 56 64)
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), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,112)(15,111)(16,110)(17,109)(18,108)(19,107)(20,106)(21,105)(22,104)(23,103)(24,102)(25,101)(26,100)(27,99)(28,98)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57), (1,75,22,82,15,61,8,68)(2,62,23,69,16,76,9,83)(3,77,24,84,17,63,10,70)(4,64,25,71,18,78,11,57)(5,79,26,58,19,65,12,72)(6,66,27,73,20,80,13,59)(7,81,28,60,21,67,14,74)(29,95,50,102,43,109,36,88)(30,110,51,89,44,96,37,103)(31,97,52,104,45,111,38,90)(32,112,53,91,46,98,39,105)(33,99,54,106,47,85,40,92)(34,86,55,93,48,100,41,107)(35,101,56,108,49,87,42,94), (1,105,8,112,15,91,22,98)(2,106,9,85,16,92,23,99)(3,107,10,86,17,93,24,100)(4,108,11,87,18,94,25,101)(5,109,12,88,19,95,26,102)(6,110,13,89,20,96,27,103)(7,111,14,90,21,97,28,104)(29,65,36,72,43,79,50,58)(30,66,37,73,44,80,51,59)(31,67,38,74,45,81,52,60)(32,68,39,75,46,82,53,61)(33,69,40,76,47,83,54,62)(34,70,41,77,48,84,55,63)(35,71,42,78,49,57,56,64)>;
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), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,112)(15,111)(16,110)(17,109)(18,108)(19,107)(20,106)(21,105)(22,104)(23,103)(24,102)(25,101)(26,100)(27,99)(28,98)(29,84)(30,83)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57), (1,75,22,82,15,61,8,68)(2,62,23,69,16,76,9,83)(3,77,24,84,17,63,10,70)(4,64,25,71,18,78,11,57)(5,79,26,58,19,65,12,72)(6,66,27,73,20,80,13,59)(7,81,28,60,21,67,14,74)(29,95,50,102,43,109,36,88)(30,110,51,89,44,96,37,103)(31,97,52,104,45,111,38,90)(32,112,53,91,46,98,39,105)(33,99,54,106,47,85,40,92)(34,86,55,93,48,100,41,107)(35,101,56,108,49,87,42,94), (1,105,8,112,15,91,22,98)(2,106,9,85,16,92,23,99)(3,107,10,86,17,93,24,100)(4,108,11,87,18,94,25,101)(5,109,12,88,19,95,26,102)(6,110,13,89,20,96,27,103)(7,111,14,90,21,97,28,104)(29,65,36,72,43,79,50,58)(30,66,37,73,44,80,51,59)(31,67,38,74,45,81,52,60)(32,68,39,75,46,82,53,61)(33,69,40,76,47,83,54,62)(34,70,41,77,48,84,55,63)(35,71,42,78,49,57,56,64) );
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)], [(1,97),(2,96),(3,95),(4,94),(5,93),(6,92),(7,91),(8,90),(9,89),(10,88),(11,87),(12,86),(13,85),(14,112),(15,111),(16,110),(17,109),(18,108),(19,107),(20,106),(21,105),(22,104),(23,103),(24,102),(25,101),(26,100),(27,99),(28,98),(29,84),(30,83),(31,82),(32,81),(33,80),(34,79),(35,78),(36,77),(37,76),(38,75),(39,74),(40,73),(41,72),(42,71),(43,70),(44,69),(45,68),(46,67),(47,66),(48,65),(49,64),(50,63),(51,62),(52,61),(53,60),(54,59),(55,58),(56,57)], [(1,75,22,82,15,61,8,68),(2,62,23,69,16,76,9,83),(3,77,24,84,17,63,10,70),(4,64,25,71,18,78,11,57),(5,79,26,58,19,65,12,72),(6,66,27,73,20,80,13,59),(7,81,28,60,21,67,14,74),(29,95,50,102,43,109,36,88),(30,110,51,89,44,96,37,103),(31,97,52,104,45,111,38,90),(32,112,53,91,46,98,39,105),(33,99,54,106,47,85,40,92),(34,86,55,93,48,100,41,107),(35,101,56,108,49,87,42,94)], [(1,105,8,112,15,91,22,98),(2,106,9,85,16,92,23,99),(3,107,10,86,17,93,24,100),(4,108,11,87,18,94,25,101),(5,109,12,88,19,95,26,102),(6,110,13,89,20,96,27,103),(7,111,14,90,21,97,28,104),(29,65,36,72,43,79,50,58),(30,66,37,73,44,80,51,59),(31,67,38,74,45,81,52,60),(32,68,39,75,46,82,53,61),(33,69,40,76,47,83,54,62),(34,70,41,77,48,84,55,63),(35,71,42,78,49,57,56,64)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14L | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 8 | 28 | 56 | 2 | 2 | 28 | 2 | 2 | 2 | 4 | 4 | 8 | 14 | 14 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C4○D28 | D4.4D4 | D4×D7 | D28.3D4 |
| kernel | D28.3D4 | C28.53D4 | C28.46D4 | C7×C4.D4 | D28.C4 | C8⋊D14 | C2×D4⋊D7 | D4.D14 | C7⋊C8 | Dic14 | D28 | C4.D4 | C2×C14 | M4(2) | C2×D4 | C22 | C7 | C4 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 6 | 3 | 12 | 2 | 6 | 3 |
Matrix representation of D28.3D4 ►in GL8(𝔽113)
| 33 | 0 | 34 | 0 | 0 | 0 | 0 | 0 |
| 0 | 33 | 0 | 34 | 0 | 0 | 0 | 0 |
| 71 | 0 | 104 | 0 | 0 | 0 | 0 | 0 |
| 0 | 71 | 0 | 104 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 91 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 0 | 112 | 2 |
| 0 | 0 | 0 | 0 | 0 | 28 | 112 | 1 |
| 80 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 80 | 0 | 1 | 0 | 0 | 0 | 0 |
| 42 | 0 | 33 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 62 | 51 |
| 0 | 0 | 0 | 0 | 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 112 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 109 | 0 | 0 | 91 |
| 0 | 0 | 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 36 | 82 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 51 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 62 | 85 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 85 | 0 | 1 |
G:=sub<GL(8,GF(113))| [33,0,71,0,0,0,0,0,0,33,0,71,0,0,0,0,34,0,104,0,0,0,0,0,0,34,0,104,0,0,0,0,0,0,0,0,1,91,62,0,0,0,0,0,72,112,0,28,0,0,0,0,0,0,112,112,0,0,0,0,0,0,2,1],[80,0,42,0,0,0,0,0,0,80,0,42,0,0,0,0,1,0,33,0,0,0,0,0,0,1,0,33,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,28,28,0,0,0,0,62,109,0,112,0,0,0,0,51,0,0,1],[0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,109,1,1,0,0,0,0,0,0,72,36,0,0,0,0,1,0,0,82,0,0,0,0,0,91,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,112,0,62,62,0,0,0,0,0,0,85,85,0,0,0,0,0,4,0,0,0,0,0,0,51,0,0,1] >;
D28.3D4 in GAP, Magma, Sage, TeX
D_{28}._3D_4 % in TeX
G:=Group("D28.3D4"); // GroupNames label
G:=SmallGroup(448,283);
// by ID
G=gap.SmallGroup(448,283);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,64,590,555,297,136,1684,851,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^4=a^14,d^2=a^7,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^14*b,d*b*d^-1=a^7*b,d*c*d^-1=a^21*c^3>;
// generators/relations