metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.40D4, Dic14.40D4, M4(2).17D14, C4○D4.8D14, (C7×D4).17D4, C4.107(D4×D7), (C7×Q8).17D4, C8.C22⋊3D7, C28.201(C2×D4), (C2×Q8).71D14, (C2×Dic7).6D4, D28⋊4C4⋊12C2, C22.38(D4×D7), C14.66C22≀C2, Dic7⋊Q8⋊7C2, D4⋊2Dic7⋊9C2, C28.C23⋊6C2, D4.12(C7⋊D4), C7⋊4(D4.10D4), (C2×C28).20C23, Q8.12(C7⋊D4), C4.12D28⋊12C2, C4○D28.26C22, (Q8×C14).98C22, C2.34(C23⋊D14), D4.10D14.2C2, (C4×Dic7).59C22, C4.Dic7.29C22, (C7×M4(2)).27C22, (C2×Dic14).137C22, C4.57(C2×C7⋊D4), (C2×C14).37(C2×D4), (C7×C8.C22)⋊7C2, (C2×C4).20(C22×D7), (C7×C4○D4).18C22, SmallGroup(448,739)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.40D4
G = < a,b,c,d | a28=b2=1, c4=d2=a14, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, bd=db, dcd-1=a14c3 >
Subgroups: 716 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, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 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, C7×Q8, D4.10D4, C4.Dic7, C4×Dic7, Dic7⋊C4, Q8⋊D7, C7⋊Q16, C7×M4(2), C7×SD16, C7×Q16, C2×Dic14, C2×Dic14, C4○D28, C4○D28, D4⋊2D7, Q8×D7, Q8×C14, C7×C4○D4, C4.12D28, D28⋊4C4, D4⋊2Dic7, C28.C23, Dic7⋊Q8, C7×C8.C22, D4.10D14, D28.40D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, D4.10D4, D4×D7, C2×C7⋊D4, C23⋊D14, D28.40D4
►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)
(1 105)(2 104)(3 103)(4 102)(5 101)(6 100)(7 99)(8 98)(9 97)(10 96)(11 95)(12 94)(13 93)(14 92)(15 91)(16 90)(17 89)(18 88)(19 87)(20 86)(21 85)(22 112)(23 111)(24 110)(25 109)(26 108)(27 107)(28 106)(29 83)(30 82)(31 81)(32 80)(33 79)(34 78)(35 77)(36 76)(37 75)(38 74)(39 73)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(56 84)
(1 85 22 92 15 99 8 106)(2 100 23 107 16 86 9 93)(3 87 24 94 17 101 10 108)(4 102 25 109 18 88 11 95)(5 89 26 96 19 103 12 110)(6 104 27 111 20 90 13 97)(7 91 28 98 21 105 14 112)(29 59 50 66 43 73 36 80)(30 74 51 81 44 60 37 67)(31 61 52 68 45 75 38 82)(32 76 53 83 46 62 39 69)(33 63 54 70 47 77 40 84)(34 78 55 57 48 64 41 71)(35 65 56 72 49 79 42 58)
(1 68 15 82)(2 69 16 83)(3 70 17 84)(4 71 18 57)(5 72 19 58)(6 73 20 59)(7 74 21 60)(8 75 22 61)(9 76 23 62)(10 77 24 63)(11 78 25 64)(12 79 26 65)(13 80 27 66)(14 81 28 67)(29 104 43 90)(30 105 44 91)(31 106 45 92)(32 107 46 93)(33 108 47 94)(34 109 48 95)(35 110 49 96)(36 111 50 97)(37 112 51 98)(38 85 52 99)(39 86 53 100)(40 87 54 101)(41 88 55 102)(42 89 56 103)
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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,84), (1,85,22,92,15,99,8,106)(2,100,23,107,16,86,9,93)(3,87,24,94,17,101,10,108)(4,102,25,109,18,88,11,95)(5,89,26,96,19,103,12,110)(6,104,27,111,20,90,13,97)(7,91,28,98,21,105,14,112)(29,59,50,66,43,73,36,80)(30,74,51,81,44,60,37,67)(31,61,52,68,45,75,38,82)(32,76,53,83,46,62,39,69)(33,63,54,70,47,77,40,84)(34,78,55,57,48,64,41,71)(35,65,56,72,49,79,42,58), (1,68,15,82)(2,69,16,83)(3,70,17,84)(4,71,18,57)(5,72,19,58)(6,73,20,59)(7,74,21,60)(8,75,22,61)(9,76,23,62)(10,77,24,63)(11,78,25,64)(12,79,26,65)(13,80,27,66)(14,81,28,67)(29,104,43,90)(30,105,44,91)(31,106,45,92)(32,107,46,93)(33,108,47,94)(34,109,48,95)(35,110,49,96)(36,111,50,97)(37,112,51,98)(38,85,52,99)(39,86,53,100)(40,87,54,101)(41,88,55,102)(42,89,56,103)>;
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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,100)(7,99)(8,98)(9,97)(10,96)(11,95)(12,94)(13,93)(14,92)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,84), (1,85,22,92,15,99,8,106)(2,100,23,107,16,86,9,93)(3,87,24,94,17,101,10,108)(4,102,25,109,18,88,11,95)(5,89,26,96,19,103,12,110)(6,104,27,111,20,90,13,97)(7,91,28,98,21,105,14,112)(29,59,50,66,43,73,36,80)(30,74,51,81,44,60,37,67)(31,61,52,68,45,75,38,82)(32,76,53,83,46,62,39,69)(33,63,54,70,47,77,40,84)(34,78,55,57,48,64,41,71)(35,65,56,72,49,79,42,58), (1,68,15,82)(2,69,16,83)(3,70,17,84)(4,71,18,57)(5,72,19,58)(6,73,20,59)(7,74,21,60)(8,75,22,61)(9,76,23,62)(10,77,24,63)(11,78,25,64)(12,79,26,65)(13,80,27,66)(14,81,28,67)(29,104,43,90)(30,105,44,91)(31,106,45,92)(32,107,46,93)(33,108,47,94)(34,109,48,95)(35,110,49,96)(36,111,50,97)(37,112,51,98)(38,85,52,99)(39,86,53,100)(40,87,54,101)(41,88,55,102)(42,89,56,103) );
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)], [(1,105),(2,104),(3,103),(4,102),(5,101),(6,100),(7,99),(8,98),(9,97),(10,96),(11,95),(12,94),(13,93),(14,92),(15,91),(16,90),(17,89),(18,88),(19,87),(20,86),(21,85),(22,112),(23,111),(24,110),(25,109),(26,108),(27,107),(28,106),(29,83),(30,82),(31,81),(32,80),(33,79),(34,78),(35,77),(36,76),(37,75),(38,74),(39,73),(40,72),(41,71),(42,70),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(56,84)], [(1,85,22,92,15,99,8,106),(2,100,23,107,16,86,9,93),(3,87,24,94,17,101,10,108),(4,102,25,109,18,88,11,95),(5,89,26,96,19,103,12,110),(6,104,27,111,20,90,13,97),(7,91,28,98,21,105,14,112),(29,59,50,66,43,73,36,80),(30,74,51,81,44,60,37,67),(31,61,52,68,45,75,38,82),(32,76,53,83,46,62,39,69),(33,63,54,70,47,77,40,84),(34,78,55,57,48,64,41,71),(35,65,56,72,49,79,42,58)], [(1,68,15,82),(2,69,16,83),(3,70,17,84),(4,71,18,57),(5,72,19,58),(6,73,20,59),(7,74,21,60),(8,75,22,61),(9,76,23,62),(10,77,24,63),(11,78,25,64),(12,79,26,65),(13,80,27,66),(14,81,28,67),(29,104,43,90),(30,105,44,91),(31,106,45,92),(32,107,46,93),(33,108,47,94),(34,109,48,95),(35,110,49,96),(36,111,50,97),(37,112,51,98),(38,85,52,99),(39,86,53,100),(40,87,54,101),(41,88,55,102),(42,89,56,103)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 28 | 2 | 2 | 4 | 8 | 28 | ··· | 28 | 2 | 2 | 2 | 8 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
49 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 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | D4.10D4 | D4×D7 | D4×D7 | D28.40D4 |
| kernel | D28.40D4 | C4.12D28 | D28⋊4C4 | D4⋊2Dic7 | C28.C23 | Dic7⋊Q8 | C7×C8.C22 | D4.10D14 | Dic14 | D28 | C2×Dic7 | C7×D4 | C7×Q8 | C8.C22 | M4(2) | C2×Q8 | 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 | 3 |
Matrix representation of D28.40D4 ►in GL6(𝔽113)
| 85 | 97 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 41 | 0 | 0 |
| 0 | 0 | 22 | 1 | 0 | 0 |
| 0 | 0 | 29 | 48 | 1 | 22 |
| 0 | 0 | 65 | 70 | 41 | 112 |
| 85 | 97 | 0 | 0 | 0 | 0 |
| 56 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 78 | 52 | 73 |
| 0 | 0 | 90 | 36 | 40 | 17 |
| 0 | 0 | 84 | 65 | 112 | 91 |
| 0 | 0 | 64 | 16 | 101 | 68 |
| 112 | 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 42 | 6 | 59 |
| 0 | 0 | 84 | 57 | 54 | 41 |
| 0 | 0 | 12 | 7 | 112 | 0 |
| 0 | 0 | 40 | 46 | 49 | 106 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 106 | 90 | 72 | 0 |
| 0 | 0 | 0 | 112 | 111 | 91 |
| 0 | 0 | 15 | 10 | 66 | 84 |
| 0 | 0 | 50 | 30 | 66 | 55 |
G:=sub<GL(6,GF(113))| [85,0,0,0,0,0,97,4,0,0,0,0,0,0,112,22,29,65,0,0,41,1,48,70,0,0,0,0,1,41,0,0,0,0,22,112],[85,56,0,0,0,0,97,28,0,0,0,0,0,0,10,90,84,64,0,0,78,36,65,16,0,0,52,40,112,101,0,0,73,17,91,68],[112,0,0,0,0,0,112,1,0,0,0,0,0,0,64,84,12,40,0,0,42,57,7,46,0,0,6,54,112,49,0,0,59,41,0,106],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,106,0,15,50,0,0,90,112,10,30,0,0,72,111,66,66,0,0,0,91,84,55] >;
D28.40D4 in GAP, Magma, Sage, TeX
D_{28}._{40}D_4 % in TeX
G:=Group("D28.40D4"); // GroupNames label
G:=SmallGroup(448,739);
// by ID
G=gap.SmallGroup(448,739);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,184,1123,297,136,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^4=d^2=a^14,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^-1=a^14*c^3>;
// generators/relations