metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.35D4, D4.10D28, Q8.10D28, C42.26D14, Dic14.35D4, M4(2).8D14, C4≀C2⋊4D7, (C7×D4).5D4, C28.6(C2×D4), (C7×Q8).5D4, C8⋊D14⋊9C2, C4○D4.4D14, C4.12(C2×D28), C4.128(D4×D7), D4⋊D14⋊2C2, C7⋊2(D4.8D4), (C2×Dic7).3D4, Dic14⋊C4⋊10C2, C22.32(D4×D7), C14.30C22≀C2, C28.47D4⋊2C2, C4.D28⋊11C2, (C4×C28).53C22, D4.10D14⋊1C2, (C2×C28).267C23, C4○D28.16C22, (C2×D28).71C22, C2.33(C22⋊D28), (C7×M4(2)).5C22, C4.Dic7.11C22, (C2×Dic14).77C22, (C7×C4≀C2)⋊4C2, (C2×C14).29(C2×D4), (C7×C4○D4).8C22, (C2×C4).112(C22×D7), SmallGroup(448,361)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.35D4
G = < a,b,c,d | a28=b2=c4=1, d2=a14, bab=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd-1=a21c-1 >
Subgroups: 892 in 146 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×5], C22, C22 [×5], C7, C8 [×2], C2×C4, C2×C4 [×9], D4, D4 [×7], Q8, Q8 [×5], C23, D7 [×2], C14, C14 [×2], C42, C22⋊C4 [×2], M4(2), M4(2), D8 [×2], SD16 [×2], C2×D4, C2×Q8 [×3], C4○D4, C4○D4 [×5], Dic7 [×3], C28 [×2], C28 [×2], D14 [×4], C2×C14, C2×C14, C4.10D4, C4≀C2, C4≀C2, C4.4D4, C8⋊C22 [×2], 2- 1+4, C7⋊C8, C56, Dic14, Dic14 [×4], C4×D7 [×3], D28, D28 [×2], C2×Dic7 [×2], C2×Dic7 [×2], C7⋊D4 [×3], C2×C28, C2×C28 [×2], C7×D4, C7×D4, C7×Q8, C22×D7, D4.8D4, C56⋊C2, D56, C4.Dic7, D14⋊C4 [×2], D4⋊D7, Q8⋊D7, C4×C28, C7×M4(2), C2×Dic14, C2×Dic14, C2×D28, C4○D28, C4○D28, D4⋊2D7 [×3], Q8×D7, C7×C4○D4, Dic14⋊C4, C28.47D4, C7×C4≀C2, C4.D28, C8⋊D14, D4⋊D14, D4.10D14, D28.35D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D7, C2×D4 [×3], D14 [×3], C22≀C2, D28 [×2], C22×D7, D4.8D4, C2×D28, D4×D7 [×2], C22⋊D28, D28.35D4
►On 112 points
(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 84)(2 83)(3 82)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 60)(26 59)(27 58)(28 57)(29 111)(30 110)(31 109)(32 108)(33 107)(34 106)(35 105)(36 104)(37 103)(38 102)(39 101)(40 100)(41 99)(42 98)(43 97)(44 96)(45 95)(46 94)(47 93)(48 92)(49 91)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 112)
(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)(57 64 71 78)(58 65 72 79)(59 66 73 80)(60 67 74 81)(61 68 75 82)(62 69 76 83)(63 70 77 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(1 50 15 36)(2 51 16 37)(3 52 17 38)(4 53 18 39)(5 54 19 40)(6 55 20 41)(7 56 21 42)(8 29 22 43)(9 30 23 44)(10 31 24 45)(11 32 25 46)(12 33 26 47)(13 34 27 48)(14 35 28 49)(57 91 71 105)(58 92 72 106)(59 93 73 107)(60 94 74 108)(61 95 75 109)(62 96 76 110)(63 97 77 111)(64 98 78 112)(65 99 79 85)(66 100 80 86)(67 101 81 87)(68 102 82 88)(69 103 83 89)(70 104 84 90)
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), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,111)(30,110)(31,109)(32,108)(33,107)(34,106)(35,105)(36,104)(37,103)(38,102)(39,101)(40,100)(41,99)(42,98)(43,97)(44,96)(45,95)(46,94)(47,93)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,112), (29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(57,64,71,78)(58,65,72,79)(59,66,73,80)(60,67,74,81)(61,68,75,82)(62,69,76,83)(63,70,77,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (1,50,15,36)(2,51,16,37)(3,52,17,38)(4,53,18,39)(5,54,19,40)(6,55,20,41)(7,56,21,42)(8,29,22,43)(9,30,23,44)(10,31,24,45)(11,32,25,46)(12,33,26,47)(13,34,27,48)(14,35,28,49)(57,91,71,105)(58,92,72,106)(59,93,73,107)(60,94,74,108)(61,95,75,109)(62,96,76,110)(63,97,77,111)(64,98,78,112)(65,99,79,85)(66,100,80,86)(67,101,81,87)(68,102,82,88)(69,103,83,89)(70,104,84,90)>;
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), (1,84)(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,111)(30,110)(31,109)(32,108)(33,107)(34,106)(35,105)(36,104)(37,103)(38,102)(39,101)(40,100)(41,99)(42,98)(43,97)(44,96)(45,95)(46,94)(47,93)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,112), (29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(57,64,71,78)(58,65,72,79)(59,66,73,80)(60,67,74,81)(61,68,75,82)(62,69,76,83)(63,70,77,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (1,50,15,36)(2,51,16,37)(3,52,17,38)(4,53,18,39)(5,54,19,40)(6,55,20,41)(7,56,21,42)(8,29,22,43)(9,30,23,44)(10,31,24,45)(11,32,25,46)(12,33,26,47)(13,34,27,48)(14,35,28,49)(57,91,71,105)(58,92,72,106)(59,93,73,107)(60,94,74,108)(61,95,75,109)(62,96,76,110)(63,97,77,111)(64,98,78,112)(65,99,79,85)(66,100,80,86)(67,101,81,87)(68,102,82,88)(69,103,83,89)(70,104,84,90) );
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)], [(1,84),(2,83),(3,82),(4,81),(5,80),(6,79),(7,78),(8,77),(9,76),(10,75),(11,74),(12,73),(13,72),(14,71),(15,70),(16,69),(17,68),(18,67),(19,66),(20,65),(21,64),(22,63),(23,62),(24,61),(25,60),(26,59),(27,58),(28,57),(29,111),(30,110),(31,109),(32,108),(33,107),(34,106),(35,105),(36,104),(37,103),(38,102),(39,101),(40,100),(41,99),(42,98),(43,97),(44,96),(45,95),(46,94),(47,93),(48,92),(49,91),(50,90),(51,89),(52,88),(53,87),(54,86),(55,85),(56,112)], [(29,50,43,36),(30,51,44,37),(31,52,45,38),(32,53,46,39),(33,54,47,40),(34,55,48,41),(35,56,49,42),(57,64,71,78),(58,65,72,79),(59,66,73,80),(60,67,74,81),(61,68,75,82),(62,69,76,83),(63,70,77,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(1,50,15,36),(2,51,16,37),(3,52,17,38),(4,53,18,39),(5,54,19,40),(6,55,20,41),(7,56,21,42),(8,29,22,43),(9,30,23,44),(10,31,24,45),(11,32,25,46),(12,33,26,47),(13,34,27,48),(14,35,28,49),(57,91,71,105),(58,92,72,106),(59,93,73,107),(60,94,74,108),(61,95,75,109),(62,96,76,110),(63,97,77,111),(64,98,78,112),(65,99,79,85),(66,100,80,86),(67,101,81,87),(68,102,82,88),(69,103,83,89),(70,104,84,90)])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28U | 28V | 28W | 28X | 56A | ··· | 56F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 28 | 56 | 2 | 2 | 4 | 4 | 4 | 28 | 28 | 28 | 2 | 2 | 2 | 8 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | ··· | 8 |
58 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D28 | D28 | D4.8D4 | D4×D7 | D4×D7 | D28.35D4 |
| kernel | D28.35D4 | Dic14⋊C4 | C28.47D4 | C7×C4≀C2 | C4.D28 | C8⋊D14 | D4⋊D14 | D4.10D14 | Dic14 | D28 | C2×Dic7 | C7×D4 | C7×Q8 | C4≀C2 | C42 | M4(2) | 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 | 12 |
Matrix representation of D28.35D4 ►in GL4(𝔽113) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 81 | 0 |
| 0 | 0 | 0 | 81 |
| 0 | 0 | 81 | 0 |
| 0 | 0 | 0 | 81 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(113))| [60,0,0,0,0,60,0,0,0,0,81,0,0,0,0,81],[0,0,60,0,0,0,0,60,81,0,0,0,0,81,0,0],[1,0,0,0,0,98,0,0,0,0,98,0,0,0,0,112],[0,1,0,0,112,0,0,0,0,0,0,1,0,0,112,0] >;
D28.35D4 in GAP, Magma, Sage, TeX
D_{28}._{35}D_4 % in TeX
G:=Group("D28.35D4"); // GroupNames label
G:=SmallGroup(448,361);
// by ID
G=gap.SmallGroup(448,361);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,226,1123,136,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=a^14,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=a^21*c^-1>;
// generators/relations