metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.2D4, Dic14.2D4, M4(2).1D14, C7⋊C8.24D4, D28.C4⋊5C2, C4.148(D4×D7), C4.D4⋊3D7, C28.93(C2×D4), (C2×D4).15D14, C8.D14⋊6C2, C7⋊1(D4.3D4), (C2×C28).5C23, C28.53D4⋊1C2, C4.12D28⋊6C2, C14.9(C4⋊D4), C4○D28.3C22, D4.D14.1C2, (D4×C14).15C22, C2.12(D14⋊D4), C4.Dic7.2C22, C22.13(C4○D28), (C2×Dic14).45C22, (C7×M4(2)).10C22, (C2×D4.D7)⋊1C2, (C2×C7⋊C8).1C22, (C7×C4.D4)⋊1C2, (C2×C4).5(C22×D7), (C2×C14).30(C4○D4), SmallGroup(448,282)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.2D4
G = < a,b,c,d | a28=b2=1, c4=a14, d2=a21, bab=a-1, cac-1=a15, ad=da, cbc-1=a14b, dbd-1=a7b, dcd-1=a7c3 >
Subgroups: 524 in 104 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, Q16, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C7⋊C8, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×C14, D4.3D4, C8×D7, C8⋊D7, C56⋊C2, Dic28, C2×C7⋊C8, C4.Dic7, D4⋊D7, D4.D7, C7×M4(2), C2×Dic14, C4○D28, D4×C14, C28.53D4, C4.12D28, C7×C4.D4, D28.C4, C8.D14, D4.D14, C2×D4.D7, D28.2D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C22×D7, D4.3D4, C4○D28, D4×D7, D14⋊D4, D28.2D4
►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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 60)(19 59)(20 58)(21 57)(22 84)(23 83)(24 82)(25 81)(26 80)(27 79)(28 78)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)(49 85)(50 112)(51 111)(52 110)(53 109)(54 108)(55 107)(56 106)
(1 35 22 42 15 49 8 56)(2 50 23 29 16 36 9 43)(3 37 24 44 17 51 10 30)(4 52 25 31 18 38 11 45)(5 39 26 46 19 53 12 32)(6 54 27 33 20 40 13 47)(7 41 28 48 21 55 14 34)(57 93 64 86 71 107 78 100)(58 108 65 101 72 94 79 87)(59 95 66 88 73 109 80 102)(60 110 67 103 74 96 81 89)(61 97 68 90 75 111 82 104)(62 112 69 105 76 98 83 91)(63 99 70 92 77 85 84 106)
(1 49 22 42 15 35 8 56)(2 50 23 43 16 36 9 29)(3 51 24 44 17 37 10 30)(4 52 25 45 18 38 11 31)(5 53 26 46 19 39 12 32)(6 54 27 47 20 40 13 33)(7 55 28 48 21 41 14 34)(57 100 78 93 71 86 64 107)(58 101 79 94 72 87 65 108)(59 102 80 95 73 88 66 109)(60 103 81 96 74 89 67 110)(61 104 82 97 75 90 68 111)(62 105 83 98 76 91 69 112)(63 106 84 99 77 92 70 85)
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,84)(23,83)(24,82)(25,81)(26,80)(27,79)(28,78)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106), (1,35,22,42,15,49,8,56)(2,50,23,29,16,36,9,43)(3,37,24,44,17,51,10,30)(4,52,25,31,18,38,11,45)(5,39,26,46,19,53,12,32)(6,54,27,33,20,40,13,47)(7,41,28,48,21,55,14,34)(57,93,64,86,71,107,78,100)(58,108,65,101,72,94,79,87)(59,95,66,88,73,109,80,102)(60,110,67,103,74,96,81,89)(61,97,68,90,75,111,82,104)(62,112,69,105,76,98,83,91)(63,99,70,92,77,85,84,106), (1,49,22,42,15,35,8,56)(2,50,23,43,16,36,9,29)(3,51,24,44,17,37,10,30)(4,52,25,45,18,38,11,31)(5,53,26,46,19,39,12,32)(6,54,27,47,20,40,13,33)(7,55,28,48,21,41,14,34)(57,100,78,93,71,86,64,107)(58,101,79,94,72,87,65,108)(59,102,80,95,73,88,66,109)(60,103,81,96,74,89,67,110)(61,104,82,97,75,90,68,111)(62,105,83,98,76,91,69,112)(63,106,84,99,77,92,70,85)>;
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,84)(23,83)(24,82)(25,81)(26,80)(27,79)(28,78)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106), (1,35,22,42,15,49,8,56)(2,50,23,29,16,36,9,43)(3,37,24,44,17,51,10,30)(4,52,25,31,18,38,11,45)(5,39,26,46,19,53,12,32)(6,54,27,33,20,40,13,47)(7,41,28,48,21,55,14,34)(57,93,64,86,71,107,78,100)(58,108,65,101,72,94,79,87)(59,95,66,88,73,109,80,102)(60,110,67,103,74,96,81,89)(61,97,68,90,75,111,82,104)(62,112,69,105,76,98,83,91)(63,99,70,92,77,85,84,106), (1,49,22,42,15,35,8,56)(2,50,23,43,16,36,9,29)(3,51,24,44,17,37,10,30)(4,52,25,45,18,38,11,31)(5,53,26,46,19,39,12,32)(6,54,27,47,20,40,13,33)(7,55,28,48,21,41,14,34)(57,100,78,93,71,86,64,107)(58,101,79,94,72,87,65,108)(59,102,80,95,73,88,66,109)(60,103,81,96,74,89,67,110)(61,104,82,97,75,90,68,111)(62,105,83,98,76,91,69,112)(63,106,84,99,77,92,70,85) );
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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,60),(19,59),(20,58),(21,57),(22,84),(23,83),(24,82),(25,81),(26,80),(27,79),(28,78),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,99),(36,98),(37,97),(38,96),(39,95),(40,94),(41,93),(42,92),(43,91),(44,90),(45,89),(46,88),(47,87),(48,86),(49,85),(50,112),(51,111),(52,110),(53,109),(54,108),(55,107),(56,106)], [(1,35,22,42,15,49,8,56),(2,50,23,29,16,36,9,43),(3,37,24,44,17,51,10,30),(4,52,25,31,18,38,11,45),(5,39,26,46,19,53,12,32),(6,54,27,33,20,40,13,47),(7,41,28,48,21,55,14,34),(57,93,64,86,71,107,78,100),(58,108,65,101,72,94,79,87),(59,95,66,88,73,109,80,102),(60,110,67,103,74,96,81,89),(61,97,68,90,75,111,82,104),(62,112,69,105,76,98,83,91),(63,99,70,92,77,85,84,106)], [(1,49,22,42,15,35,8,56),(2,50,23,43,16,36,9,29),(3,51,24,44,17,37,10,30),(4,52,25,45,18,38,11,31),(5,53,26,46,19,39,12,32),(6,54,27,47,20,40,13,33),(7,55,28,48,21,41,14,34),(57,100,78,93,71,86,64,107),(58,101,79,94,72,87,65,108),(59,102,80,95,73,88,66,109),(60,103,81,96,74,89,67,110),(61,104,82,97,75,90,68,111),(62,105,83,98,76,91,69,112),(63,106,84,99,77,92,70,85)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 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 | 4 | 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 | 2 | 2 | 28 | 56 | 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 | D4.3D4 | D4×D7 | D28.2D4 |
| kernel | D28.2D4 | C28.53D4 | C4.12D28 | C7×C4.D4 | D28.C4 | C8.D14 | D4.D14 | C2×D4.D7 | 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 D28.2D4 ►in GL8(𝔽113)
| 80 | 0 | 104 | 0 | 0 | 0 | 0 | 0 |
| 0 | 80 | 0 | 104 | 0 | 0 | 0 | 0 |
| 9 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 112 | 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 | 112 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 85 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 85 | 0 | 5 | 0 | 0 | 0 | 0 |
| 92 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 92 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 100 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 13 |
| 0 | 0 | 0 | 0 | 100 | 100 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 100 | 0 | 0 |
| 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 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(8,GF(113))| [80,0,9,0,0,0,0,0,0,80,0,9,0,0,0,0,104,0,112,0,0,0,0,0,0,104,0,112,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0],[85,0,92,0,0,0,0,0,0,85,0,92,0,0,0,0,5,0,28,0,0,0,0,0,0,5,0,28,0,0,0,0,0,0,0,0,0,0,100,13,0,0,0,0,0,0,100,100,0,0,0,0,13,13,0,0,0,0,0,0,100,13,0,0],[0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,112,0,0] >;
D28.2D4 in GAP, Magma, Sage, TeX
D_{28}._2D_4 % in TeX
G:=Group("D28.2D4"); // GroupNames label
G:=SmallGroup(448,282);
// by ID
G=gap.SmallGroup(448,282);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,64,590,219,297,136,1684,851,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^4=a^14,d^2=a^21,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^7*c^3>;
// generators/relations