metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.6D4, Dic14.6D4, M4(2).5D14, C7⋊C8.26D4, D28.C4⋊7C2, C28.99(C2×D4), C4.152(D4×D7), (C2×Q8).7D14, C8⋊D14.1C2, C4.10D4⋊3D7, C7⋊2(D4.3D4), C28.46D4⋊8C2, C28.C23⋊1C2, C28.53D4⋊3C2, (C2×C28).11C23, C4○D28.7C22, (Q8×C14).9C22, C14.11(C4⋊D4), (C2×D28).42C22, C2.14(D14⋊D4), C4.Dic7.6C22, C22.15(C4○D28), (C7×M4(2)).14C22, (C2×Q8⋊D7)⋊1C2, (C2×C7⋊C8).3C22, (C7×C4.10D4)⋊1C2, (C2×C4).11(C22×D7), (C2×C14).32(C4○D4), SmallGroup(448,288)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.6D4
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=a7c3 >
Subgroups: 620 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, C28, D14, C2×C14, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C7⋊C8, C7⋊C8, C56, Dic14, C4×D7, D28, D28, C7⋊D4, C2×C28, C2×C28, C7×Q8, C22×D7, D4.3D4, C8×D7, C8⋊D7, C56⋊C2, D56, C2×C7⋊C8, C4.Dic7, Q8⋊D7, C7⋊Q16, C7×M4(2), C2×D28, C4○D28, Q8×C14, C28.53D4, C28.46D4, C7×C4.10D4, D28.C4, C8⋊D14, C2×Q8⋊D7, C28.C23, D28.6D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C22×D7, D4.3D4, C4○D28, D4×D7, D14⋊D4, D28.6D4
►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 96)(2 95)(3 94)(4 93)(5 92)(6 91)(7 90)(8 89)(9 88)(10 87)(11 86)(12 85)(13 112)(14 111)(15 110)(16 109)(17 108)(18 107)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 99)(27 98)(28 97)(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 63 22 70 15 77 8 84)(2 78 23 57 16 64 9 71)(3 65 24 72 17 79 10 58)(4 80 25 59 18 66 11 73)(5 67 26 74 19 81 12 60)(6 82 27 61 20 68 13 75)(7 69 28 76 21 83 14 62)(29 97 50 104 43 111 36 90)(30 112 51 91 44 98 37 105)(31 99 52 106 45 85 38 92)(32 86 53 93 46 100 39 107)(33 101 54 108 47 87 40 94)(34 88 55 95 48 102 41 109)(35 103 56 110 49 89 42 96)
(1 104 8 111 15 90 22 97)(2 105 9 112 16 91 23 98)(3 106 10 85 17 92 24 99)(4 107 11 86 18 93 25 100)(5 108 12 87 19 94 26 101)(6 109 13 88 20 95 27 102)(7 110 14 89 21 96 28 103)(29 84 36 63 43 70 50 77)(30 57 37 64 44 71 51 78)(31 58 38 65 45 72 52 79)(32 59 39 66 46 73 53 80)(33 60 40 67 47 74 54 81)(34 61 41 68 48 75 55 82)(35 62 42 69 49 76 56 83)
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), (1,96)(2,95)(3,94)(4,93)(5,92)(6,91)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(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), (1,63,22,70,15,77,8,84)(2,78,23,57,16,64,9,71)(3,65,24,72,17,79,10,58)(4,80,25,59,18,66,11,73)(5,67,26,74,19,81,12,60)(6,82,27,61,20,68,13,75)(7,69,28,76,21,83,14,62)(29,97,50,104,43,111,36,90)(30,112,51,91,44,98,37,105)(31,99,52,106,45,85,38,92)(32,86,53,93,46,100,39,107)(33,101,54,108,47,87,40,94)(34,88,55,95,48,102,41,109)(35,103,56,110,49,89,42,96), (1,104,8,111,15,90,22,97)(2,105,9,112,16,91,23,98)(3,106,10,85,17,92,24,99)(4,107,11,86,18,93,25,100)(5,108,12,87,19,94,26,101)(6,109,13,88,20,95,27,102)(7,110,14,89,21,96,28,103)(29,84,36,63,43,70,50,77)(30,57,37,64,44,71,51,78)(31,58,38,65,45,72,52,79)(32,59,39,66,46,73,53,80)(33,60,40,67,47,74,54,81)(34,61,41,68,48,75,55,82)(35,62,42,69,49,76,56,83)>;
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), (1,96)(2,95)(3,94)(4,93)(5,92)(6,91)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,112)(14,111)(15,110)(16,109)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(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), (1,63,22,70,15,77,8,84)(2,78,23,57,16,64,9,71)(3,65,24,72,17,79,10,58)(4,80,25,59,18,66,11,73)(5,67,26,74,19,81,12,60)(6,82,27,61,20,68,13,75)(7,69,28,76,21,83,14,62)(29,97,50,104,43,111,36,90)(30,112,51,91,44,98,37,105)(31,99,52,106,45,85,38,92)(32,86,53,93,46,100,39,107)(33,101,54,108,47,87,40,94)(34,88,55,95,48,102,41,109)(35,103,56,110,49,89,42,96), (1,104,8,111,15,90,22,97)(2,105,9,112,16,91,23,98)(3,106,10,85,17,92,24,99)(4,107,11,86,18,93,25,100)(5,108,12,87,19,94,26,101)(6,109,13,88,20,95,27,102)(7,110,14,89,21,96,28,103)(29,84,36,63,43,70,50,77)(30,57,37,64,44,71,51,78)(31,58,38,65,45,72,52,79)(32,59,39,66,46,73,53,80)(33,60,40,67,47,74,54,81)(34,61,41,68,48,75,55,82)(35,62,42,69,49,76,56,83) );
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)], [(1,96),(2,95),(3,94),(4,93),(5,92),(6,91),(7,90),(8,89),(9,88),(10,87),(11,86),(12,85),(13,112),(14,111),(15,110),(16,109),(17,108),(18,107),(19,106),(20,105),(21,104),(22,103),(23,102),(24,101),(25,100),(26,99),(27,98),(28,97),(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)], [(1,63,22,70,15,77,8,84),(2,78,23,57,16,64,9,71),(3,65,24,72,17,79,10,58),(4,80,25,59,18,66,11,73),(5,67,26,74,19,81,12,60),(6,82,27,61,20,68,13,75),(7,69,28,76,21,83,14,62),(29,97,50,104,43,111,36,90),(30,112,51,91,44,98,37,105),(31,99,52,106,45,85,38,92),(32,86,53,93,46,100,39,107),(33,101,54,108,47,87,40,94),(34,88,55,95,48,102,41,109),(35,103,56,110,49,89,42,96)], [(1,104,8,111,15,90,22,97),(2,105,9,112,16,91,23,98),(3,106,10,85,17,92,24,99),(4,107,11,86,18,93,25,100),(5,108,12,87,19,94,26,101),(6,109,13,88,20,95,27,102),(7,110,14,89,21,96,28,103),(29,84,36,63,43,70,50,77),(30,57,37,64,44,71,51,78),(31,58,38,65,45,72,52,79),(32,59,39,66,46,73,53,80),(33,60,40,67,47,74,54,81),(34,61,41,68,48,75,55,82),(35,62,42,69,49,76,56,83)]])
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 | 28A | ··· | 28F | 28G | ··· | 28L | 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 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 56 | 2 | 2 | 8 | 28 | 2 | 2 | 2 | 4 | 4 | 8 | 14 | 14 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 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 | D28.6D4 |
| kernel | D28.6D4 | C28.53D4 | C28.46D4 | C7×C4.10D4 | D28.C4 | C8⋊D14 | C2×Q8⋊D7 | C28.C23 | C7⋊C8 | Dic14 | D28 | C4.10D4 | C2×C14 | M4(2) | C2×Q8 | 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 D28.6D4 ►in GL6(𝔽113)
| 112 | 89 | 0 | 0 | 0 | 0 |
| 24 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 87 | 0 | 0 |
| 0 | 0 | 87 | 112 | 0 | 0 |
| 0 | 0 | 62 | 68 | 112 | 77 |
| 0 | 0 | 89 | 77 | 44 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 89 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 | 87 |
| 0 | 0 | 1 | 36 | 69 | 111 |
| 0 | 0 | 96 | 21 | 44 | 99 |
| 0 | 0 | 53 | 84 | 48 | 33 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 69 | 69 | 0 |
| 0 | 0 | 0 | 23 | 0 | 87 |
| 0 | 0 | 0 | 105 | 112 | 26 |
| 0 | 0 | 1 | 3 | 0 | 90 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 87 | 104 | 14 | 26 |
| 0 | 0 | 112 | 77 | 44 | 2 |
| 0 | 0 | 98 | 5 | 95 | 14 |
| 0 | 0 | 86 | 29 | 65 | 80 |
G:=sub<GL(6,GF(113))| [112,24,0,0,0,0,89,10,0,0,0,0,0,0,1,87,62,89,0,0,87,112,68,77,0,0,0,0,112,44,0,0,0,0,77,1],[1,89,0,0,0,0,0,112,0,0,0,0,0,0,0,1,96,53,0,0,23,36,21,84,0,0,0,69,44,48,0,0,87,111,99,33],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,1,0,0,69,23,105,3,0,0,69,0,112,0,0,0,0,87,26,90],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,87,112,98,86,0,0,104,77,5,29,0,0,14,44,95,65,0,0,26,2,14,80] >;
D28.6D4 in GAP, Magma, Sage, TeX
D_{28}._6D_4 % in TeX
G:=Group("D28.6D4"); // GroupNames label
G:=SmallGroup(448,288);
// by ID
G=gap.SmallGroup(448,288);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,64,590,219,184,297,136,1684,851,18822]);
// 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^7*c^3>;
// generators/relations