metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.4D4, Dic14.4D4, M4(2).3D14, (C2×C4).6D28, (C2×C28).8D4, C4.80(D4×D7), C28.97(C2×D4), (C2×Q8).5D14, D28⋊4C4⋊3C2, C8.D14⋊7C2, C4.10D4⋊1D7, (C2×C28).9C23, C14.17C22≀C2, Dic7⋊Q8⋊1C2, C7⋊1(D4.10D4), C4○D28.5C22, C22.12(C2×D28), (Q8×C14).7C22, C2.20(C22⋊D28), (C4×Dic7).1C22, Q8.10D14.1C2, (C7×M4(2)).2C22, (C2×Dic14).47C22, (C2×C14).22(C2×D4), (C2×C4).9(C22×D7), (C7×C4.10D4)⋊3C2, SmallGroup(448,286)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.4D4
G = < a,b,c,d | a28=b2=1, c4=a14, d2=a7, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, dbd-1=a21b, dcd-1=a7c3 >
Subgroups: 812 in 142 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, D4.10D4, C56⋊C2, Dic28, C4×Dic7, Dic7⋊C4, C7×M4(2), C2×Dic14, C4○D28, C4○D28, Q8×D7, Q8⋊2D7, Q8×C14, D28⋊4C4, C7×C4.10D4, C8.D14, Dic7⋊Q8, Q8.10D14, D28.4D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4.10D4, C2×D28, D4×D7, C22⋊D28, D28.4D4
►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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 56)(13 55)(14 54)(15 53)(16 52)(17 51)(18 50)(19 49)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 40)(57 98)(58 97)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)
(1 47 22 54 15 33 8 40)(2 34 23 41 16 48 9 55)(3 49 24 56 17 35 10 42)(4 36 25 43 18 50 11 29)(5 51 26 30 19 37 12 44)(6 38 27 45 20 52 13 31)(7 53 28 32 21 39 14 46)(57 110 78 89 71 96 64 103)(58 97 79 104 72 111 65 90)(59 112 80 91 73 98 66 105)(60 99 81 106 74 85 67 92)(61 86 82 93 75 100 68 107)(62 101 83 108 76 87 69 94)(63 88 84 95 77 102 70 109)
(1 101 8 108 15 87 22 94)(2 102 9 109 16 88 23 95)(3 103 10 110 17 89 24 96)(4 104 11 111 18 90 25 97)(5 105 12 112 19 91 26 98)(6 106 13 85 20 92 27 99)(7 107 14 86 21 93 28 100)(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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99), (1,47,22,54,15,33,8,40)(2,34,23,41,16,48,9,55)(3,49,24,56,17,35,10,42)(4,36,25,43,18,50,11,29)(5,51,26,30,19,37,12,44)(6,38,27,45,20,52,13,31)(7,53,28,32,21,39,14,46)(57,110,78,89,71,96,64,103)(58,97,79,104,72,111,65,90)(59,112,80,91,73,98,66,105)(60,99,81,106,74,85,67,92)(61,86,82,93,75,100,68,107)(62,101,83,108,76,87,69,94)(63,88,84,95,77,102,70,109), (1,101,8,108,15,87,22,94)(2,102,9,109,16,88,23,95)(3,103,10,110,17,89,24,96)(4,104,11,111,18,90,25,97)(5,105,12,112,19,91,26,98)(6,106,13,85,20,92,27,99)(7,107,14,86,21,93,28,100)(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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,56)(13,55)(14,54)(15,53)(16,52)(17,51)(18,50)(19,49)(20,48)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,40)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99), (1,47,22,54,15,33,8,40)(2,34,23,41,16,48,9,55)(3,49,24,56,17,35,10,42)(4,36,25,43,18,50,11,29)(5,51,26,30,19,37,12,44)(6,38,27,45,20,52,13,31)(7,53,28,32,21,39,14,46)(57,110,78,89,71,96,64,103)(58,97,79,104,72,111,65,90)(59,112,80,91,73,98,66,105)(60,99,81,106,74,85,67,92)(61,86,82,93,75,100,68,107)(62,101,83,108,76,87,69,94)(63,88,84,95,77,102,70,109), (1,101,8,108,15,87,22,94)(2,102,9,109,16,88,23,95)(3,103,10,110,17,89,24,96)(4,104,11,111,18,90,25,97)(5,105,12,112,19,91,26,98)(6,106,13,85,20,92,27,99)(7,107,14,86,21,93,28,100)(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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,56),(13,55),(14,54),(15,53),(16,52),(17,51),(18,50),(19,49),(20,48),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,40),(57,98),(58,97),(59,96),(60,95),(61,94),(62,93),(63,92),(64,91),(65,90),(66,89),(67,88),(68,87),(69,86),(70,85),(71,112),(72,111),(73,110),(74,109),(75,108),(76,107),(77,106),(78,105),(79,104),(80,103),(81,102),(82,101),(83,100),(84,99)], [(1,47,22,54,15,33,8,40),(2,34,23,41,16,48,9,55),(3,49,24,56,17,35,10,42),(4,36,25,43,18,50,11,29),(5,51,26,30,19,37,12,44),(6,38,27,45,20,52,13,31),(7,53,28,32,21,39,14,46),(57,110,78,89,71,96,64,103),(58,97,79,104,72,111,65,90),(59,112,80,91,73,98,66,105),(60,99,81,106,74,85,67,92),(61,86,82,93,75,100,68,107),(62,101,83,108,76,87,69,94),(63,88,84,95,77,102,70,109)], [(1,101,8,108,15,87,22,94),(2,102,9,109,16,88,23,95),(3,103,10,110,17,89,24,96),(4,104,11,111,18,90,25,97),(5,105,12,112,19,91,26,98),(6,106,13,85,20,92,27,99),(7,107,14,86,21,93,28,100),(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 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 28 | 28 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D28 | D4.10D4 | D4×D7 | D28.4D4 |
| kernel | D28.4D4 | D28⋊4C4 | C7×C4.10D4 | C8.D14 | Dic7⋊Q8 | Q8.10D14 | Dic14 | D28 | C2×C28 | C4.10D4 | M4(2) | C2×Q8 | C2×C4 | C7 | C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 6 | 3 | 12 | 2 | 6 | 3 |
Matrix representation of D28.4D4 ►in GL6(𝔽113)
| 70 | 104 | 0 | 0 | 0 | 0 |
| 18 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 88 | 25 | 112 | 22 |
| 0 | 0 | 0 | 105 | 41 | 1 |
| 105 | 80 | 0 | 0 | 0 | 0 |
| 43 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 88 | 25 | 112 | 22 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 23 | 49 | 8 | 25 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 25 | 88 | 1 | 91 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 23 | 49 | 0 | 25 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 36 | 77 | 15 | 9 |
| 0 | 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 6 | 107 | 7 | 36 |
G:=sub<GL(6,GF(113))| [70,18,0,0,0,0,104,9,0,0,0,0,0,0,0,112,88,0,0,0,1,0,25,105,0,0,0,0,112,41,0,0,0,0,22,1],[105,43,0,0,0,0,80,8,0,0,0,0,0,0,88,0,0,23,0,0,25,0,1,49,0,0,112,1,0,8,0,0,22,0,0,25],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,25,0,23,0,0,0,88,1,49,0,0,1,1,0,0,0,0,0,91,0,25],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,36,0,6,0,0,0,77,98,107,0,0,98,15,0,7,0,0,0,9,0,36] >;
D28.4D4 in GAP, Magma, Sage, TeX
D_{28}._4D_4 % in TeX
G:=Group("D28.4D4"); // GroupNames label
G:=SmallGroup(448,286);
// by ID
G=gap.SmallGroup(448,286);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,58,1123,570,136,1684,438,18822]);
// 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^7*b,d*b*d^-1=a^21*b,d*c*d^-1=a^7*c^3>;
// generators/relations