metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.10D28, D28.35D4, 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, C4.128(D4×D7), C4.12(C2×D28), D4⋊D14⋊2C2, C7⋊2(D4.8D4), (C2×Dic7).3D4, Dic14⋊C4⋊10C2, C22.32(D4×D7), C14.30C22≀C2, C4.12D28⋊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 D4.10D28
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, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C42, C22⋊C4, M4(2), M4(2), D8, SD16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C4.10D4, C4≀C2, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, D4.8D4, C56⋊C2, D56, C4.Dic7, D14⋊C4, D4⋊D7, Q8⋊D7, C4×C28, C7×M4(2), C2×Dic14, C2×Dic14, C2×D28, C4○D28, C4○D28, D4⋊2D7, Q8×D7, C7×C4○D4, Dic14⋊C4, C4.12D28, C7×C4≀C2, C4.D28, C8⋊D14, D4⋊D14, D4.10D14, D4.10D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4.8D4, C2×D28, D4×D7, C22⋊D28, D4.10D28
►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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 84)(15 83)(16 82)(17 81)(18 80)(19 79)(20 78)(21 77)(22 76)(23 75)(24 74)(25 73)(26 72)(27 71)(28 70)(29 101)(30 100)(31 99)(32 98)(33 97)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 112)(47 111)(48 110)(49 109)(50 108)(51 107)(52 106)(53 105)(54 104)(55 103)(56 102)
(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 30 15 44)(2 31 16 45)(3 32 17 46)(4 33 18 47)(5 34 19 48)(6 35 20 49)(7 36 21 50)(8 37 22 51)(9 38 23 52)(10 39 24 53)(11 40 25 54)(12 41 26 55)(13 42 27 56)(14 43 28 29)(57 88 71 102)(58 89 72 103)(59 90 73 104)(60 91 74 105)(61 92 75 106)(62 93 76 107)(63 94 77 108)(64 95 78 109)(65 96 79 110)(66 97 80 111)(67 98 81 112)(68 99 82 85)(69 100 83 86)(70 101 84 87)
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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,78)(21,77)(22,76)(23,75)(24,74)(25,73)(26,72)(27,71)(28,70)(29,101)(30,100)(31,99)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,112)(47,111)(48,110)(49,109)(50,108)(51,107)(52,106)(53,105)(54,104)(55,103)(56,102), (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), (1,30,15,44)(2,31,16,45)(3,32,17,46)(4,33,18,47)(5,34,19,48)(6,35,20,49)(7,36,21,50)(8,37,22,51)(9,38,23,52)(10,39,24,53)(11,40,25,54)(12,41,26,55)(13,42,27,56)(14,43,28,29)(57,88,71,102)(58,89,72,103)(59,90,73,104)(60,91,74,105)(61,92,75,106)(62,93,76,107)(63,94,77,108)(64,95,78,109)(65,96,79,110)(66,97,80,111)(67,98,81,112)(68,99,82,85)(69,100,83,86)(70,101,84,87)>;
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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,78)(21,77)(22,76)(23,75)(24,74)(25,73)(26,72)(27,71)(28,70)(29,101)(30,100)(31,99)(32,98)(33,97)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,112)(47,111)(48,110)(49,109)(50,108)(51,107)(52,106)(53,105)(54,104)(55,103)(56,102), (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), (1,30,15,44)(2,31,16,45)(3,32,17,46)(4,33,18,47)(5,34,19,48)(6,35,20,49)(7,36,21,50)(8,37,22,51)(9,38,23,52)(10,39,24,53)(11,40,25,54)(12,41,26,55)(13,42,27,56)(14,43,28,29)(57,88,71,102)(58,89,72,103)(59,90,73,104)(60,91,74,105)(61,92,75,106)(62,93,76,107)(63,94,77,108)(64,95,78,109)(65,96,79,110)(66,97,80,111)(67,98,81,112)(68,99,82,85)(69,100,83,86)(70,101,84,87) );
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)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,60),(11,59),(12,58),(13,57),(14,84),(15,83),(16,82),(17,81),(18,80),(19,79),(20,78),(21,77),(22,76),(23,75),(24,74),(25,73),(26,72),(27,71),(28,70),(29,101),(30,100),(31,99),(32,98),(33,97),(34,96),(35,95),(36,94),(37,93),(38,92),(39,91),(40,90),(41,89),(42,88),(43,87),(44,86),(45,85),(46,112),(47,111),(48,110),(49,109),(50,108),(51,107),(52,106),(53,105),(54,104),(55,103),(56,102)], [(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)], [(1,30,15,44),(2,31,16,45),(3,32,17,46),(4,33,18,47),(5,34,19,48),(6,35,20,49),(7,36,21,50),(8,37,22,51),(9,38,23,52),(10,39,24,53),(11,40,25,54),(12,41,26,55),(13,42,27,56),(14,43,28,29),(57,88,71,102),(58,89,72,103),(59,90,73,104),(60,91,74,105),(61,92,75,106),(62,93,76,107),(63,94,77,108),(64,95,78,109),(65,96,79,110),(66,97,80,111),(67,98,81,112),(68,99,82,85),(69,100,83,86),(70,101,84,87)]])
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 | D4.10D28 |
| kernel | D4.10D28 | Dic14⋊C4 | C4.12D28 | 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 D4.10D28 ►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] >;
D4.10D28 in GAP, Magma, Sage, TeX
D_4._{10}D_{28} % in TeX
G:=Group("D4.10D28"); // 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